Warranty Data Analysis: A Review
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paper_title: Statistical Analysis of Product Warranty Data

paper_content:

Summary Data bases regarding warranty claims for manufactured products record claims experience and information about concomitant factors. If constructed and maintained properly, warranty data bases may be used for a variety of purposes that include the prediction of future claims, the comparison of claims experience for different groups of products, the estimation of field reliability, and the identification of opportunities for quality and reliability improvement. This paper reviews some methods of analyzing warranty data and of addressing these objectives. Some extensions of previous work and suggestions for future development are included. Examples involving warranty claims for automobiles and refrigerators are considered.
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paper_title: Lifetime Estimation Using Only Failure Information From Warranty Database

paper_content:

Knowledge about product lifetime obtained from actual usage (field data) is of great importance to manufacturers who want to get information about the true reliability of their products. Warranty data are frequently used to estimate reliability characteristics because such databases are automatically generated and updated at no additional cost from repair claims during warranty coverage. For engineering purposes, usage time (e.g., mileage) is more relevant, and lifetime parameters measured in usage time is an integral part of reliability analysis using warranty data. Usually, warranty data consist of only failure information. Censored data (e.g., mileage of non-failure automobiles) are not obtainable, of which usage time distributions are different from those of failed ones. Effective usage-based estimation thus requires supplementary information about usage accumulation for non-failure units such as follow-up studies, or a usage time distribution which includes both failure, and non-failure products. However, sometimes usage-based data for non-failed units are expensive and difficult to obtain. Thus, the unavailability of the usage time of censored units makes it difficult to estimate the usage-based lifetime distribution of products. This paper deals with that problem, and discusses how to estimate the lifetime distribution using warranty data which consist of only failure information. The practical consequence of this finding is that supplemental information is not needed to obtain correct estimates of the lifetime parameters.
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paper_title: Product warranty logistics: Issues and challenges

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Warranty is an important element of marketing new products. The servicing of warranty results in additional costs to the manufacturer. Warranty logistics deals with various issues relating to the servicing of warranty. Proper management of warranty logistics is needed not only to reduce the warranty servicing cost but also to ensure customer satisfaction as customer dissatisfaction has a negative impact on sales and revenue. Unfortunately, warranty logistics has received very little attention. The paper links the literature on warranty and on logistics and then discusses the different issues in warranty logistics. It highlights the challenges and identifies some research topics of potential interest to operational researchers.
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paper_title: Warranty cost analysis for nonrepairable services products

paper_content:

Many service products are installed in a complex system that is operated only when the entire system is completed. The time from their installation to commissioning, called a dormant state in this article, may take several years for systems such as complete buildings or aircraft. Warranties for the products may cover the time starting from their installation to a certain time. Warranty cost on replacements for such products is different from normal products without any dormant state. This article analyses the replacement cost for nonrepairable services products from a manufacturer perspective. We consider four warranty policies, which include two types of warranty terms (i.e., nonrenewing or renewing) and two types of replacements (i.e., with preventive replacement or replacement only upon failures). Relationships between the failure patterns at the dormant state and at the operating state are also discussed. Numerical examples and sensitivity analysis are presented to demonstrate the applicability of the methodology described in this article.
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paper_title: Models and estimation for systems with recurrent events and usage processes.

paper_content:

This paper considers settings where populations of units may experience recurrent events, termed failures for convenience, and where the units are subject to varying levels of usage. We provide joint models for the recurrent events and usage processes, which facilitate analysis of their relationship as well as prediction of failures. Data on usage are often incomplete and we show how to implement maximum likelihood estimation in such cases. Random effects models with linear usage processes and gamma usage processes are considered in some detail. Data on automobile warranty claims are used to illustrate the proposed models and estimation methodology.
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paper_title: Analysis of warranty claim data: a literature review

paper_content:

Purpose – To provide a brief survey of the literature directed towards the analysis of warranty claim data.Design/methodology/approach – For convenience, this survey of the analysis of warranty claims data is somewhat arbitrarily be classified by topics as follows: age‐based claims analysis, aggregated warranty claims analysis, marginal counts of claims analysis, warranty claims analysis by using covariates, estimation of lifetime distribution using supplementary data, two‐dimensional warranty, warranty costs analysis, sales lag and reporting lag analysis, and forecasts of warranty claims.Findings – Emphasis is placed on a discussion of different kinds of warranty claims data selected from reviews and on a comparison of the statistical models and methods used to analyze such data.Research limitations/implications – Since the literature on product warranty data is vast, more work on this problem is needed.Practical implications – This review points out why warranty claims data is important and gives a surv...
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paper_title: REVIEW OF LONG-TERM WARRANTY POLICIES

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A large number of products are now being sold with long-term warranty policies. This is mainly due to fierce competition in the market and customer demand. Offering long-term warranty results in additional complexities in estimating costs associated with warranty servicing due to the uncertainties associated with usage, preventive maintenance and warranty servicing strategies. This paper reviews various long-term warranty policies and mathematical models for estimation of warranty servicing costs.
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paper_title: Methods for the estimation of failure distributions and rates from automobile warranty data

paper_content:

We consider the occurrence of warranty claims for automobiles when both age and mileage accumulation may affect failure. The presence of both age and mileage limits on warranties creates interesting problems for the analysis of failures. We propose a family of models that relates failure to time and mileage accumulation. Methods for fitting the models based on warranty data and supplementary information about mileage accumulation are presented and illustrated on some real data. The general problem of modelling failures in equipment when both time and usage are factors is discussed.
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paper_title: Warranty Management and Product Manufacture

paper_content:

Introduction and Overview Products and Product Quality Product Warranty Warranty Management Systems Approach to Warranty Management The Role and Use of Data in Warranty Management Warranty Cost Analysis Warranty Considerations in Product Design and Development Implications of Warranty on Production Decisions The Role of Warranty in Marketing Warranty Logistics Reliability Improvement Warranties Financial, Societal, and Legal Aspects of Warranty Warranty Management System Epilogue
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paper_title: Warranty cost analysis for products with a dormant state

paper_content:

Most building services products are installed while a building is constructed, but they are not operated until the building is commissioned. The warranty of the products may cover the time starting from their installation to the end of the warranty period. Prior to the commissioning of the building, the products are at a dormant mode (i.e., not operated) but protected by the warranty. For such products, both the usage intensity and the failure patterns are different from those with continuous usage intensity and failure patterns. This paper develops warranty cost models for repairable products with a dormant mode from both the manufacturer's and buyer's perspectives. Relationships between the failure patterns at the dormant mode and at the operational mode are also discussed. Numerical examples and sensitivity analysis are used to demonstrate the applicability of the methodology derived in the paper.
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paper_title: Optimising age-replacement and extended non-renewing warranty policies in lifecycle costing

paper_content:

This paper analyses the life cycle cost of equipment protected by both base and extended warranty policies from a consumer's perspective. We assume that the equipment has two types of failure: minor and catastrophic. A minor failure can be corrected with minimal repair whereas a catastrophic failure can only be removed by a replacement. It is assumed that equipment is maintained at no charge to the consumer during the warranty period, whereas the consumer is fully charged for any maintenance on failures after the extended warranty expires. We formulate the expected life cycle cost of the equipment under a general failure time distribution, and then for special cases we prove that the optimal replacement and extended warranty policies exists where the expected life cycle cost per unit time is minimised. This is examined with numerical examples.
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paper_title: Some useful statistical methods for truncated data

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Truncated data arise when a variable is observable only over some portion of its range. This paper describes how truncated data arise in studies of the field performance or reliability of manufactured items. Failure to account for truncation can lead to..
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paper_title: Warranty Economic Decision Models: A Summary and Some Suggested Directions for Future Research

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Product guarantees or warranties have been around for generations, but formal approaches for establishing and examining warranties have been considered only during the past 20 years. A review of the literature on warranty models and analysis methods is provided, along with some suggestions for further research.
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paper_title: Warranties: Planning, Analysis, and Implementation

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What and Why of Warranties. Warranty Principles. Warranty Requirements. Repair Warranties. Performance Guarantees. Warranty Tradeoffs. Warranty Costing Analysis. Warranty Negotiating. Warranty Cost-Effectiveness Analysis. Warranty Implementation and Administration. Warranty Management--Planning and Execution. The Future of Warranties and Competition.
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paper_title: Warranty and preventive maintenance

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For products sold with warranty, preventive maintenance actions by manufacturers and/or buyers have an impact on the total costs for both parties. This paper develops a framework to study preventive maintenance actions when items are sold under warranty and reviews the models that have appeared in the literature. It then develops a new model and carries out its analysis.
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paper_title: Warranty Spend Forecasting for Subsystem Failures Influenced by Calendar Month Seasonality

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Although the need for collecting warranty data originated from financial reasons, it is also extensively used for modeling and analysis to support managerial decision-making in industries. Strategic, tactical, and operational level decisions involving warranty cost very often use warranty spending forecasts that are developed using statistical methods. Existing literature provides warranty forecasting approaches involving variables such as mileage accumulation rate, failure rate, repeat repair rate, and cost per repair. However, there are several key failure modes that are known to be influenced by seasonality. For example, `engine slow to start' conditions drive a higher claim rate in colder months than in warmer months. Accommodation of such failure modes influenced by seasonality has not been considered in the warranty cost modeling literature. This paper presents a flexible approach for developing a monthly warranty spend forecasting model that incorporates calendar month seasonality, business days per month for authorized service centers, and sales ramp-up in addition to the earlier mentioned variables. On one hand, the model allows development of warranty spend forecasts for entire warranty coverage to support strategic level decisions; on the other hand, forecasts for monthly warranty spend help support tactical and operational level decisions. The workability of the proposed methodology is illustrated using an application example.
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paper_title: Product warranty management -- I: A taxonomy for warranty policies

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Effective management of product warranty requires proper evaluation of alternative warranty policies. Many different types of warranty policies are in common use. Still others are discussed in the literature on product warranty. In this paper we formulate a taxonomy for warranty to assist the manager responsible for product warranty in choosing appropriate alternatives for evaluation before a final choice is made.
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paper_title: Product warranty and reliability

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Product warranty is an important part of new product marketing and sales. Offering warranty implies additional costs in the form of warranty servicing cost. Product reliability has a serious impact on the warranty servicing cost. As such, effective management of product reliability must take into account the link between warranty and reliability. This paper deals with this topic and develops a framework needed for effective management of product reliability. It reviews the relevant literature and defines topics for future research.
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paper_title: Estimation of Lifetime Parameters From Incomplete Field Data

paper_content:

When automobile failures occur within the automotive warranty period, a manufacturer can develop a record of mileages to failure from owners' requests for repair. When no failures occur during the warranty period the owner naturally will not report mileages, and it may be inferred that “no record of failures” means “no failures.” By using a follow-up survey or postal reply cards, data can be acquired to include a partial record of nonfailures. A method of estimating lifetime parameters is proposed for analyzing this kind of data. Assuming a Weibull and an exponential lifetime distribution under the random censoring model, I derive a consistent estimator that can be applied to any censoring distribution.
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paper_title: Analysis of Reliability and Warranty Claims in Products With Age and Usage Scales

paper_content:

Failures or other adverse events in systems or products may depend on the age and usage history of the unit. Motivated by motor vehicle reliability and warranty data issues, we present models that may be used to assess the dependence on age or usage in heterogeneous populations of products, and show how to estimate model parameters based on different types of field data. The setting in which the events in question are warranty claims is complicated because of the sparseness and incompleteness of the data, and we examine it in some detail. We consider some North American automobile warranty data and use these data to illustrate the methodology.
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paper_title: Warranty Cost Analysis

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For products sold with warranty, the manufacturer incurs costs resulting from the servicing of failures occurring under warranty. These costs depend on several factors, the most important being the reliability of the product. For planning purposes, the manufacturer must estimate (a) the expected cost of warranty per unit sold and (b) the expected cost per unit time over the product life cycle. This must be done in the product development stage. This article deals with models that may be used to obtain these cost estimates for products sold with free-replacement and pro rata warranties. ::: ::: ::: Keywords: ::: ::: warranty; ::: reliability; ::: cost analysis; ::: cost models; ::: free-replacement warranty; ::: pro rata warranty
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paper_title: New product warranty: A literature review

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Warranty is an important element of marketing new products as better warranty signals higher product quality and provides greater assurance to customers. Servicing warranty involves additional costs to the manufacturer and this cost depends on product reliability and warranty terms. Product reliability is influenced by the decisions made during the design and manufacturing of the product. As such warranty is very important in the context of new products. Product warranty has received the attention of researchers from many different disciplines and the literature on warranties is vast. This paper carries out a review of the literature that has appeared in the last ten years. It highlights issues of interest to manufacturers in the context of managing new products from an overall business perspective. (C) 2002 Elsevier Science B.V. All rights reserved.
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paper_title: Reliability and Warranties: Methods for Product Development and Quality Improvement

paper_content:

Product Quality and Warranty Introduction Background and Scope References Preliminary Results for Analyzing Product Failures Failure Time Distributions Fitting Distributions Counting the Number of Failures Exercises References Quality and Reliability Quality Concepts Product Reliability Load and Capacity Models Exercises References Product Warranties and Quality Definitions and Types of Warranties Warranty Cost Models Determining Optimum Warranty Periods Estimating Warranty Reserve Exercises References Product Monitoring and Feedback Quality System Monitoring and Control Multiattribute Quality Assessment Warranty Information Feedback Models MOP/MIS Charts Exercises References The Quality Improvement Process Introduction Identifying Quality Problems Developing Quality Improvement Goals Decision Analysis Framework Exercises References Toward an Integrated Product Quality System Introduction The Quality Movement A QRW (Quality, Reliability, Warranty) Paradigm Some Concluding Remarks References Appendix A: Notations and Acronyms Appendix B: Selected Tables Appendix C: Counting Distributions Binomial Counting Distribution Poisson Counting Distribution Renewal Function for Erlang (lambda,2) Distributed Times Index
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paper_title: An Early Warning System for Vehicle Related Quality Data

paper_content:

Vehicle production audit tests, warranty claims and car control unit data are stored in a central data warehouse for data mining analysis. Neural network based part failure rate estimations, adjusted for mileage and seasonality, are used for monitoring warranty claims. Association and sequence analysis connect production audit data, car control unit data and warranty claims for an early detection of quality changes both in production state and car field usage. Calculations are performed via grid computing.
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paper_title: Evaluating product and process design changes with warranty data

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Abstract This paper suggests evaluating field performance by analyzing hazard plots of the random variable time to first warranty claim. This approach corrects biases that arise when a single member contributes many repairs to the repair population and allows comparing the empirical lifetime distribution with the parametric model assumed at product design (e.g., Exponential or Weibull). Using automobile warranty data we demonstrate how using too broad a scope can mitigate changes in field performance by aggregating improving and degrading components. When comparing warranty performance across model years, the analyst should adjust the hazard plots to reflect common field service time.
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paper_title: Association rule-generation algorithm for mining automotive warranty data

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A new association rule-generation algorithm is presented for mining automotive warranty data. The algorithm uses elementary set concept and database manipulation techniques to develop useful relationships between product attributes and causes of failure. These relationships (knowledge) are represented using IF–THEN association rules, where the IF portion of the rule includes set of attributes representing product features (e.g. production date, repair date, mileage-at-repair, transmission, engine type, etc.) and the THEN portion of the rule includes set of attributes that represent decision outcome (e.g. problem-related labor code). Once association rules are developed, the algorithm applies a statistical analysis technique to evaluate the significance of each rule. The rules that pass the significance test are reported in a solution. Application of the association rule-generation algorithm is presented with a data-mining case study from the automotive industry. The knowledge (rules) extracted from the au...
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paper_title: Sequential pattern mining algorithm for automotive warranty data

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This paper presents a sequential pattern mining algorithm that allows product and quality engineers to extract hidden knowledge from a large automotive warranty database. The algorithm uses the elementary set concept and database manipulation techniques to search for patterns or relationships among occurrences of warranty claims over time. These patterns are represented as IF-THEN sequential rules, where the IF portion of the rule includes one or more occurrences of warranty problems at one time and the THEN portion includes warranty problem(s) that occur at a later time. Once sequential patterns are generated, the algorithm uses rule strength parameters to filter out insignificant patterns, so that only important (significant) rules are reported. Significant patterns provide knowledge of one or more product failures that leads to future product fault(s). The effectiveness of the algorithm is illustrated with the warranty data mining application from the automotive industry. A discussion on the sequential patterns generated by the algorithm and their interpretation for the automotive example are also provided.
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paper_title: Statistical Analysis of Product Warranty Data

paper_content:

Summary Data bases regarding warranty claims for manufactured products record claims experience and information about concomitant factors. If constructed and maintained properly, warranty data bases may be used for a variety of purposes that include the prediction of future claims, the comparison of claims experience for different groups of products, the estimation of field reliability, and the identification of opportunities for quality and reliability improvement. This paper reviews some methods of analyzing warranty data and of addressing these objectives. Some extensions of previous work and suggestions for future development are included. Examples involving warranty claims for automobiles and refrigerators are considered.
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paper_title: Warranty claim analysis considering human factors

paper_content:

Abstract Warranty claims are not always due to product failures. They can also be caused by two types of human factors. On the one hand, consumers might claim warranty due to misuse and/or failures caused by various human factors. Such claims might account for more than 10% of all reported claims. On the other hand, consumers might not be bothered to claim warranty for failed items that are still under warranty, or they may claim warranty after they have experienced several intermittent failures. These two types of human factors can affect warranty claim costs. However, research in this area has received rather little attention. In this paper, we propose three models to estimate the expected warranty cost when the two types of human factors are included. We consider two types of failures: intermittent and fatal failures, which might result in different claim patterns. Consumers might report claims after a fatal failure has occurred, and upon intermittent failures they might report claims after a number of failures have occurred. Numerical examples are given to validate the results derived.
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paper_title: Estimation from truncated lifetime data with supplementary information on covariates and censoring times

paper_content:

SUMMARY In epidemiology, reliability and other areas one encounters situations where response times and covariates for individuals in a population are observed only if the response time does not exceed an associated censoring time. Methods for obtaining supplementary information about covariates and censoring times for nonresponding individuals are considered. Estimation procedures are developed for the resulting combination of truncated data and supplementary data. Theoretical and simulation results are given, and the methodology illustrated on some automobile failure data collected under a warranty plan.
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paper_title: Some Issues in the Collection and Analysis of Field Reliability Data

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Sources of data on field reliability or performance of manufactured products include field tracking (follow-up) studies, warranty and failure reports, and surveys. We discuss these sources in relation to the estimation of failure time distributions and event rates. Important issues include missing information, reporting delays, the need to combine information from different sources, and the possibility of selection bias.
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paper_title: Field data analyses with additional after-warranty failure data

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This paper proposes methods of estimating the lifetime distribution for situations where additional field data can be gathered after the warranty expires in a parametric time to failure distribution. For satisfactory inference about parameters involved, it is desirable to incorporate these after-warranty data in the analysis. It is assumed that after-warranty data are reported with probability p1(<1), while within-warranty data are reported with probability 1. Methods of obtaining maximum likelihood estimators are outlined, their asymptotic properties are studied, and specific formulas for Weibull distribution are obtained. An estimation procedure using the expectation and maximization algorithm is also proposed when the reporting probability is unknown. Simulation studies are performed to investigate the properties of the estimates.
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paper_title: A mixture model for automobile warranty data

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Abstract This paper proposes a general mixture model framework for automobile warranty data that includes parameters for product field performance, the manufacturing and assembly process, and dealer preparation process. The model fits warranty claims as a mixture of manufacturing or assembly defects (quality problems) and usage related failures (reliability problems). The model also estimates the fraction of vehicles containing a manufacturing or assembly defect when leaving the assembly plant. This parameter measures the quality of the entire vehicle production process, i.e. component manufacturing and final assembly. The model also measures the proportion of manufacturing or assembly defects repaired by the automobile dealer prior to customer delivery. This conditional probability quantifies the ability of the vehicle preparation process to identify and repair defects prior to customer delivery. To apply the model to field failure or warranty data, the practitioner must identify parametric distributions for each of the two failure processes. To demonstrate the model, this paper develops a Weibull-Uniform mixture for manufacturer supplied warranty claim data.
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paper_title: Estimation of Reliability in Field-Performance Studies

paper_content:

Likelihood-based methods are developed for the analysis of field-performance studies with particular attention centered on the estimation of regression coefficients in parametric models. Failure-record data are those in which the time to failure and the regressor variables are observed only for those items that fail in some prespecified follow-up or warranty period (0, T]. It is noted that for satisfactory inference about baseline failure rates or regression effects it is usually necessary to supplement the failure-record data either by incorporating specific prior information about x or by taking a supplementary sample of items that survive T ○. General methods are outlined and specific formulas for various likelihood-based methods are obtained when the failure-time model is exponential or Weibull. In these models the methods are compared with respect to asymptotic efftciency of estimation. Several extensions to more complicated sampling plans are considered.
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paper_title: On analysing warranty data from repairable items

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Fitting models to failure data is an important topic in reliability. The resulting models can be useful both for manufacturers as well as for end-users. In this paper we provide details of some methods from the literature which can be used as a starting point when analysing and fitting models to failure data from repairable items. In particular we focus on obtaining analytical estimates of the intensity of a non-homogeneous Poisson process. We illustrate some of these methods on failure data from the warranty database of a major car manufacturer.
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paper_title: Estimation of Lifetime Parameters From Incomplete Field Data

paper_content:

When automobile failures occur within the automotive warranty period, a manufacturer can develop a record of mileages to failure from owners' requests for repair. When no failures occur during the warranty period the owner naturally will not report mileages, and it may be inferred that “no record of failures” means “no failures.” By using a follow-up survey or postal reply cards, data can be acquired to include a partial record of nonfailures. A method of estimating lifetime parameters is proposed for analyzing this kind of data. Assuming a Weibull and an exponential lifetime distribution under the random censoring model, I derive a consistent estimator that can be applied to any censoring distribution.
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paper_title: Reliability estimation from field return data.

paper_content:

In this article statistical inference for the failure time distribution of a product from "field return data", that records the time between the product being shipped and returned for repair or replacement, is described. The problem that is addressed is that the data are not failure times because they also include the time that it took to ship and install the product and then to return it to the manufacturer for repair or replacement. The inference attempts to infer the distribution of time to failure (that is, from installation to failure) from the data when in addition there are separate data on the times from shipping to installation, and from failure to return. The method is illustrated with data from units installed in a telecommunications network.
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paper_title: Nonparametric estimation of a lifetime distribution when censoring times are missing

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We consider datasets for which lifetimes associated with the units in a population are observed if they occur within certain time intervals but for which lengths of the time intervals, or censoring times of unfailed units, are missing. We consider nonparametric estimation of the lifetime distribution for the population from such data; a maximum likelihood estimator and a simple moment estimator are obtained. An example involving automobile warranty data is discussed at some length.
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paper_title: Statistical Analysis of Product Warranty Data

paper_content:

Summary Data bases regarding warranty claims for manufactured products record claims experience and information about concomitant factors. If constructed and maintained properly, warranty data bases may be used for a variety of purposes that include the prediction of future claims, the comparison of claims experience for different groups of products, the estimation of field reliability, and the identification of opportunities for quality and reliability improvement. This paper reviews some methods of analyzing warranty data and of addressing these objectives. Some extensions of previous work and suggestions for future development are included. Examples involving warranty claims for automobiles and refrigerators are considered.
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paper_title: Estimation from truncated lifetime data with supplementary information on covariates and censoring times

paper_content:

SUMMARY In epidemiology, reliability and other areas one encounters situations where response times and covariates for individuals in a population are observed only if the response time does not exceed an associated censoring time. Methods for obtaining supplementary information about covariates and censoring times for nonresponding individuals are considered. Estimation procedures are developed for the resulting combination of truncated data and supplementary data. Theoretical and simulation results are given, and the methodology illustrated on some automobile failure data collected under a warranty plan.
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paper_title: A mixed-Weibull regression model for the analysis of automotive warranty data

paper_content:

Abstract This paper presents a case study regarding the reliability analysis of some automotive components based on field failure warranty data. The components exhibit two different failure modes, namely early and wearout failures, and are mounted on different vehicles, which differ among themselves for car model and engine type, thus involving different operating conditions. Hence, the failure time of each component is a random variable with a bimodal pdf which also depends upon a vector of covariates that indexes the specific operating condition. Then, a mixed-Weibull distribution, where the pdf of each subpopulation (namely the ‘weak’ and ‘strong’ subpopulation) depends on the covariates through the scale parameter, is used to analyze the component lifetime. A Fortran algorithm for the maximum likelihood estimation of model parameters has been implemented and a stepwise procedure, in its backwards version, has been used to test the significance of covariates and to construct the regression model. The presence of a weak subpopulation has been verified and the fraction of weak units in the population has also been estimated. Finally, the adequacy of the proposed model to fit the observed data has been assessed.
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paper_title: Estimation of Reliability in Field-Performance Studies

paper_content:

Likelihood-based methods are developed for the analysis of field-performance studies with particular attention centered on the estimation of regression coefficients in parametric models. Failure-record data are those in which the time to failure and the regressor variables are observed only for those items that fail in some prespecified follow-up or warranty period (0, T]. It is noted that for satisfactory inference about baseline failure rates or regression effects it is usually necessary to supplement the failure-record data either by incorporating specific prior information about x or by taking a supplementary sample of items that survive T ○. General methods are outlined and specific formulas for various likelihood-based methods are obtained when the failure-time model is exponential or Weibull. In these models the methods are compared with respect to asymptotic efftciency of estimation. Several extensions to more complicated sampling plans are considered.
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paper_title: Analysis of data to predict warranty cost for various regions

paper_content:

Nowadays, automobile manufacturers are assembling vehicles at one location or multiple locations and selling them in the global markets. The Original Equipment Manufacturers (OEM) are asking the suppliers to share the warranty return cost for their returned components. This recent phenomenon is putting the supplier in precarious position, financially, by sharing the warranty cost when supplier margins are very thin. It is creating a challenge to suppliers how to predict the warranty cost based on components sold and forecasted to sell in coming months and years when vehicles are selling world wide carrying different warranty terms. Typically, luxury vehicles sold in North America carries warranty period of 4 years or 48,000 miles whichever comes first. In Asia, Australia, Europe and Africa warranty period is 2 years irrespective of mileage on the vehicle. Vehicle sale volumes are going to change based on currency fluctuations and demand for the particular product. This poses a challenge to the supplier's in predicting the warranty costs for various regions. This study shows that, how to take the complex failure information for different regions and break it down to main categories. The vehicle sale information has been divided into North America and rest of the World (ROW). In the same way, failure data related to particular system warranty separated by regions. ROW market failure data has been analyzed by using the Reliasoft Weibull ++ software with forecasted quantities using time as univariate. Two variables were identified to calculate the warranty forecast for the North American market. One is time and another one is mileage on the vehicle. From our experience, the failure rate of a sub system is more dependent on usage rather than time. The data clearly shows that, customers are putting 40 miles per day on average and 80 % of vehicles are reaching the warranty usage limit before the time limit. Based on this data, and failure rate is dependent on usage rather than time. It is easy to do calculations by employing the univariate plan rather than complex analysis by taking time and usage parameters. Warranty costs are calculated based on Maximum likelihood estimation. This approach is relatively easy to follow for small to medium suppliers in the automotive industry.
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paper_title: FAILURE MODELS INDEXED BY TWO SCALES

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Much of the literature in reliability and survival analysis considers failure models indexed by a single scale. There are situations which require that failure be described by several scales. An example from reliability is items under warranty whose failure is recorded by time and amount of use. An example from survival analysis is the death of a mine worker which is noted by age and the duration of exposure to dust. This paper proposes an approach for developing probabilistic models indexed by two scales: time, and usage, a quantity that is related to time. The relationship between the scales is described by an additive hazards model. The evolution of usage is described by stochastic processes like the Poisson, the gamma and the Markov additive. The paper concludes with an application involving the setting of warranties. Two features differentiate this work from related efforts: a use of specific processes for describing usage, and a use of Monte Carlo techniques for generating the models.
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paper_title: Methods for the estimation of failure distributions and rates from automobile warranty data

paper_content:

We consider the occurrence of warranty claims for automobiles when both age and mileage accumulation may affect failure. The presence of both age and mileage limits on warranties creates interesting problems for the analysis of failures. We propose a family of models that relates failure to time and mileage accumulation. Methods for fitting the models based on warranty data and supplementary information about mileage accumulation are presented and illustrated on some real data. The general problem of modelling failures in equipment when both time and usage are factors is discussed.
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paper_title: Reliability assessment based on two-dimensional warranty data and an accelerated failure time model

paper_content:

Many products (including automobiles) are sold with two-dimensional warranties. Warranty data is the data collected during the servicing of failures under warranty and can be used to estimate the reliability of the product and of its components. In this paper we discuss the challenges in the estimation of reliability based on warranty data and propose a method for estimating component reliability using an accelerated life test model.
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paper_title: Forecasting the cost of unreliability for products with two-dimensional warranties

paper_content:

This paper addresses the effect of two- dimensional warranty policies on the procedure for forecasting the cost of unreliability. Automotive warranties are characterized by age or time in service and vehicle usage mileage. This paper presents a model where the usage time is a primary variable and the mileage accumulation is estimated from field return data. At each time interval the probability of exceeding the upper mileage limit on the warranty is calculated based on the mileage distribution and an analytical cumulative distribution function (CDF) of exceeding the upper mileage limit at any point of time is constructed. This modeling procedure accounts for an observed reduction in the number of warranty claims in the second half of the warranty period thus making it a more realistic evaluation of products with two-dimensional warranty policies. A case study using an automotive parts example in order to illustrate the methodology is provided.
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paper_title: The warranty problem: its statistical and game theoretic aspects

paper_content:

Manufacturers of consumer products, such as automobiles, usually offer warranties guaranteeing the product or its parts, for example, for five years or 50,000 miles, whichever comes first. There are at least two issues of interest to applied mathematicians and statisticians that arise from warranty considerations. The first is the specification of an optimum price-warranty combination, and the second is the forecast of a reserve fund to meet warranty claims against the product. The former involves a consideration of the item’s reliability, its rate of usage, the consumer’s attitude toward a specific warranty, and the competitor’s actions, all of which lead towards a game theoretic formulation of the problem. The latter involves the analysis of a special kind of time series in two-dimensions, for which the use of dynamic linear models appears to be natural. This is an expository paper. Due to the timeliness of the topic, the focus here is on problem definition, its scope, and its formulation in a manner th...
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paper_title: Models and estimation for systems with recurrent events and usage processes.

paper_content:

This paper considers settings where populations of units may experience recurrent events, termed failures for convenience, and where the units are subject to varying levels of usage. We provide joint models for the recurrent events and usage processes, which facilitate analysis of their relationship as well as prediction of failures. Data on usage are often incomplete and we show how to implement maximum likelihood estimation in such cases. Random effects models with linear usage processes and gamma usage processes are considered in some detail. Data on automobile warranty claims are used to illustrate the proposed models and estimation methodology.
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paper_title: Bivariate reliability and availability modeling

paper_content:

Equipment longevity is a resource that is consumed in device operation. For many types of equipment, the resource can be reasonably represented in terms of two (or more) measures. We construct a general framework for specifying bivariate longevity models and incorporating them in availability models. Following an introduction to the domain of study a short taxonomy of model types is provided and indicates which types of models have not been studied. For two classes of bivariate models, their construction is shown, along with how they should be interpreted in relation to commonly used univariate concepts such as hazard functions. Several example models illustrate these points. The construction of bivariate renewal models is shown and their inclusion in basic and preventive maintenance-based availability models is explained. All of these analyses show the development of the bivariate Laplace transforms of the probability measures being studied. Unfortunately, in nearly all cases, the inverse transforms are not yet available. The chief contributions of the paper are the definition of a structure for classifying bivariate problems, description of methods for building and analyzing bivariate models, general presentation of a new domain of research in equipment reliability. Potential extensions and improvements to this work are numerous and the resulting realism in reliability and maintenance models will be important.
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paper_title: Analysis of Reliability and Warranty Claims in Products With Age and Usage Scales

paper_content:

Failures or other adverse events in systems or products may depend on the age and usage history of the unit. Motivated by motor vehicle reliability and warranty data issues, we present models that may be used to assess the dependence on age or usage in heterogeneous populations of products, and show how to estimate model parameters based on different types of field data. The setting in which the events in question are warranty claims is complicated because of the sparseness and incompleteness of the data, and we examine it in some detail. We consider some North American automobile warranty data and use these data to illustrate the methodology.
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paper_title: Alternative Time Scales and Failure Time Models

paper_content:

In many reliability applications, there may not be a unique plausible scale in which to measure time to failure or assess performance. This is especially the case when several measures of usage are available on each unit. For example, the age, the total number of flight hours, and the number of landings are usage measures that are often considered important in aircraft reliability. Similarly, in medical or biological applications of survival analysis there are often alternative scales (e.g., Oakes, 1995). This paper considers the definition of a "good" time scale, along with methods of determining a time scale.
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paper_title: Parallel time scales and two-dimensional manufacturer and individual customer warranties

paper_content:

We deal with a system whose failures depend on several parallel effects, such as the time in use L and the mileage H. Manufacturer warranties are typically described by a two-dimensional region in the (L,H)-plane. A proper determination of the warranty limits must be based on a two-dimensional distribution of time to failure on this plane. The aim of this paper is to demonstrate the possibility of designing individual warranties for a “;nontypical” customer who has a very low or very high usage rate b=H/L, and to show a simple way to calculate warranty limits by minimizing the lifetime coefficient of variation. The latter is carried out by introducing the “best” combined time scale in the form K=(1−∈)L+∈H which provides the minimal lifetime coefficient of variation.
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paper_title: Estimating Parameters of the Power Law Process With Two Measures of Failure Time

paper_content:

We introduce a method of combining two time indices, such as mileage and age, into a single index that resembles the Cobb-Douglas production function. Then we present the power law process with this synthesized time index as a model for the reliability ..
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paper_title: Warranty Spend Forecasting for Subsystem Failures Influenced by Calendar Month Seasonality

paper_content:

Although the need for collecting warranty data originated from financial reasons, it is also extensively used for modeling and analysis to support managerial decision-making in industries. Strategic, tactical, and operational level decisions involving warranty cost very often use warranty spending forecasts that are developed using statistical methods. Existing literature provides warranty forecasting approaches involving variables such as mileage accumulation rate, failure rate, repeat repair rate, and cost per repair. However, there are several key failure modes that are known to be influenced by seasonality. For example, `engine slow to start' conditions drive a higher claim rate in colder months than in warmer months. Accommodation of such failure modes influenced by seasonality has not been considered in the warranty cost modeling literature. This paper presents a flexible approach for developing a monthly warranty spend forecasting model that incorporates calendar month seasonality, business days per month for authorized service centers, and sales ramp-up in addition to the earlier mentioned variables. On one hand, the model allows development of warranty spend forecasts for entire warranty coverage to support strategic level decisions; on the other hand, forecasts for monthly warranty spend help support tactical and operational level decisions. The workability of the proposed methodology is illustrated using an application example.
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paper_title: A warranty forecasting model based on piecewise statistical distributions and stochastic simulation

paper_content:

Abstract This paper presents a warranty forecasting method based on stochastic simulation of expected product warranty returns. This methodology is presented in the context of a high-volume product industry and has a specific application to automotive electronics. The warranty prediction model is based on a piecewise application of Weibull and exponential distributions, having three parameters, which are the characteristic life and shape parameter of the Weibull distribution and the time coordinate of the junction point of the two distributions. This time coordinate is the point at which the reliability ‘bathtub’ curve exhibits a transition between early life and constant hazard rate behavior. The values of the parameters are obtained from the optimum fitting of data on past warranty claims for similar products. Based on the analysis of past warranty returns it is established that even though the warranty distribution parameters vary visibly between product lines they stay approximately consistent within the same product family, which increases the overall accuracy of the simulation-based warranty forecasting technique. The method is demonstrated using a case study of automotive electronics warranty returns. The approach developed and demonstrated in this paper represents a balance between correctly modeling the failure rate trend changes and practicality for use by reliability analysis organizations.
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paper_title: The warranty problem: its statistical and game theoretic aspects

paper_content:

Manufacturers of consumer products, such as automobiles, usually offer warranties guaranteeing the product or its parts, for example, for five years or 50,000 miles, whichever comes first. There are at least two issues of interest to applied mathematicians and statisticians that arise from warranty considerations. The first is the specification of an optimum price-warranty combination, and the second is the forecast of a reserve fund to meet warranty claims against the product. The former involves a consideration of the item’s reliability, its rate of usage, the consumer’s attitude toward a specific warranty, and the competitor’s actions, all of which lead towards a game theoretic formulation of the problem. The latter involves the analysis of a special kind of time series in two-dimensions, for which the use of dynamic linear models appears to be natural. This is an expository paper. Due to the timeliness of the topic, the focus here is on problem definition, its scope, and its formulation in a manner th...
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paper_title: Finite-Horizon Prediction of Recurrent Events, With Application to Forecasts of Warranty Claims

paper_content:

In this article we present prediction methods for recurrent events that occur for individuals or units in some population. The events are modeled using flexible nonhomogeneous Poisson processes, and possible heterogeneity among the individuals is modeled using random effects. We also present effective calibration techniques that provide prediction intervals with coverage probabilities close to a desired nominal level. We apply these methods to a particular warranty data setting from the automobile industry. The number of processes in this context being very large, we emphasize methods providing rapid computation of prediction intervals.
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paper_title: A comparison of three strategies for forecasting warranty claims

paper_content:

The economic necessity for developing accurate forecasts of extended warranty costs on durable goods is established. Three strategies for forecasting warranty failures are outlined. The first strategy encompasses the use of linear, predictive models, wherein forecasts are based upon an extrapolation of a least squares fit of the data. These include regression and time series approaches. The risks associated with the use of these models when localized or highly nonlinear phenomena exists are illustrated. The second strategy is based upon the use of dynamic linear models wherein the model parameter estimates are updated sequentially by using the equations for the Kalman filter. The adaptive nature of these models leads to improved forecasts in the presence of localized phenomena. The third strategy encompasses the use of nonparametric modeling approaches, including the use of neural network models to provide a generalized, form-free modeling framework. With a constructed example, the comparative advantage i...
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paper_title: A USEFUL GENERALIZATION OF RENEWAL THEORY: COUNTING PROCESSES GOVERNED BY NON-NEGATIVE MARKOVIAN INCREMENTS

paper_content:

Let N(t) be a counting process associated with a sequence of non-negative random variables (X,)' where the distribution of X+i, depends only on the value of the partial sum S = X.., X,. In this paper, we study the structure of the function H(t)= E[N(t)], extending the ordinary renewal theory. It is shown under certain conditions that h(t) = (d/dt)H(t) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h(t) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h(t) and H(t) as t-oo is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.
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paper_title: Statistical Analysis of Product Warranty Data

paper_content:

Summary Data bases regarding warranty claims for manufactured products record claims experience and information about concomitant factors. If constructed and maintained properly, warranty data bases may be used for a variety of purposes that include the prediction of future claims, the comparison of claims experience for different groups of products, the estimation of field reliability, and the identification of opportunities for quality and reliability improvement. This paper reviews some methods of analyzing warranty data and of addressing these objectives. Some extensions of previous work and suggestions for future development are included. Examples involving warranty claims for automobiles and refrigerators are considered.
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paper_title: Estimation of Rate and Mean Functions from Truncated Recurrent Event Data

paper_content:

Abstract This article investigates data on recurrent events that arise from sources such as warranty claims, where the observation period for a unit is unknown until it experiences at least one event. This creates a type of truncation in the data. We consider nonparametric estimation of means and rates of the event occurrences with such “zero-truncated” data and examine the case where the population size and the distribution of observation times across units are at least approximately known. We study the behaviors of the proposed estimators by simulation. We examine some car warranty data by applying the methodology developed here and considering the underlying assumptions.
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paper_title: Some Issues in the Collection and Analysis of Field Reliability Data

paper_content:

Sources of data on field reliability or performance of manufactured products include field tracking (follow-up) studies, warranty and failure reports, and surveys. We discuss these sources in relation to the estimation of failure time distributions and event rates. Important issues include missing information, reporting delays, the need to combine information from different sources, and the possibility of selection bias.
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paper_title: Age-based warranty data analysis without date-specific sales information

paper_content:

Age-based analysis of warranty claim data forms the basis for the estimation, prediction, and comparison of claims across different product groups or across time. The age of a product means the elapsed calendar time since the product was sold or entered service. In this article, we develop a method to deal with situations when the information on date-specific sales amounts is not available but the total sales quantity can be determined from other sources. Date-specific sales amounts, i.e. the number of products sold on different dates, are difficult to determine for many products from many wholesalers and retailers around the world. We consider the problem of maximum likelihood estimation of the expected number of claims for repairable items based on a non-homogeneous Poisson process model, and of the probability of a claim for non-repairable items based on a multinomial model. These estimators derived for the above two cases are in a closed-form expression and are the same. We carry out numerical simulation to evaluate the effectiveness of these proposed estimators by comparing them with those obtained under the assumption that date-specific sales amounts are available.
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paper_title: Statistical analysis of marginal count failure data.

paper_content:

Manufacturers want to assess the quality andreliability of their products. Specifically, they want to knowthe exact number of failures from the sales transacted duringa particular month. Information available today is sometimesincomplete as many companies analyze their failure data simplycomparing sales for a total month from a particular departmentwith the total number of claims registered for that given month.This information—called marginal count data—is, thus,incomplete as it does not give the exact number of failures ofthe specific products that were sold in a particular month. Inthis paper we discuss nonparametric estimation of the mean numbersof failures for repairable products and the failure probabilitiesfor nonrepairable products. We present a nonhomogeneous Poissonprocess model for repairable products and a multinomial modeland its Poisson approximation for nonrepairable products. A numericalexample is given and a simulation is carried out to evaluatethe proposed methods of estimating failure probabilities undera number of possible situations.
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paper_title: Statistical Analysis of Product Warranty Data

paper_content:

Summary Data bases regarding warranty claims for manufactured products record claims experience and information about concomitant factors. If constructed and maintained properly, warranty data bases may be used for a variety of purposes that include the prediction of future claims, the comparison of claims experience for different groups of products, the estimation of field reliability, and the identification of opportunities for quality and reliability improvement. This paper reviews some methods of analyzing warranty data and of addressing these objectives. Some extensions of previous work and suggestions for future development are included. Examples involving warranty claims for automobiles and refrigerators are considered.
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paper_title: New product warranty: A literature review

paper_content:

Warranty is an important element of marketing new products as better warranty signals higher product quality and provides greater assurance to customers. Servicing warranty involves additional costs to the manufacturer and this cost depends on product reliability and warranty terms. Product reliability is influenced by the decisions made during the design and manufacturing of the product. As such warranty is very important in the context of new products. Product warranty has received the attention of researchers from many different disciplines and the literature on warranties is vast. This paper carries out a review of the literature that has appeared in the last ten years. It highlights issues of interest to manufacturers in the context of managing new products from an overall business perspective. (C) 2002 Elsevier Science B.V. All rights reserved.
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paper_title: A practical method for failure analysis using incomplete warranty data

paper_content:

The use of warranty claims data to determine the failure characteristics of a product is well documented. Typically, the failure distribution and its parameters are determined using product manufacturing data for each month of production and the corresponding monthly failure counts derived from the warranty claims. If the data is collected systematically, the product ages at the times of failure can be derived. Classical methods are then used to determine the failure time distribution and parameters. However, our experience shows that, in many cases, it may not be possible to know the failure ages of components. The information available each month might be limited to the volume of shipments and total claims or product returns. In such cases, the data hides the component age at the time of failure. In this paper, we show that when the failure history information is incomplete, the failure distribution of the product can be determined using Bayesian analysis techniques applicable for handling incomplete data. We apply the popular Expectation-Maximization (EM) algorithm to find the Maximum Likelihood Estimates (MLE) of the failure distribution parameters using incomplete data. The effectiveness of the EM algorithm is compared using several sets of incomplete warranty data generated using simulation. We observed that the EM algorithm is powerful in capturing the hidden failure patterns from the incomplete warranty data.
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paper_title: Field Failure Data: an Alternative Proposal for Reliability Estimation

paper_content:

Product reliability is a very important issue for the competitive strategy of industries. In order to estimate a product's reliability, parametric inferential methods are required to evaluate survival test data, which happens to be a fairly expensive data source. Such costly information usually imposes additional compromises in the product development and new challenges to be overcome throughout the product's life cycle. However, manufacturers also keep field failure data for warranty and maintenance purposes, which can be a low-cost data source for reliability estimation. Field-failure data are very difficult to evaluate using parametric inferential methods due to their small and highly censored samples, quite often representing mixed modes of failure. In this paper a method for reliability estimation using field failure data is proposed. The proposal is based on the use of non-parametric inferential methods, associated with resampling techniques to derive confidence intervals for the reliability estimates. Test results show the adequacy of the proposed method to calculate reliability estimates and their confidence interval for different populations, including cases with highly right-censored failure data. The method is shown to be particularly useful when the sampling distribution is not known, which happens to be the case in a large number of practical reliability evaluations. Copyright © 2006 John Wiley & Sons, Ltd.
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paper_title: Estimation for censored exponential data when the censoring times are subject to error

paper_content:

The problem of estimating the parameter of the negative exponential distribution with censored data may be complicated by a lack of complete knowledge of the censoring times. The effect of ignoring these errors in the censoring times rather than using a complete maximum likelihood estimation approach is investigated and it is shown that the loss in efficiency is generally small for the inverted gamma model considered in this paper. The theory is supplemented by some numerical results based on an example of failure data.
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paper_title: Estimation from Censored Data with Incomplete Information

paper_content:

Phillips and Sweeting [J. R. Statist. Soc. B 58 (1996) 775–783.] considered estimation of the parameter of the exponential distribution with censored failure time data when there is incomplete knowledge of the censoring times. It was shown that, under particular models for the censoring mechanism and censoring errors, it will usually be safe to ignore such errors provided they are not expected to be too large. A flexible model is introduced which includes the extreme cases of no censoring errors and no information on the censoring values. The effect of alternative assumptions about knowledge of the censoring values on the estimation of failure rate is investigated.
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paper_title: An examination of correlation effects among warranty claims

paper_content:

This paper models & examines the impact of correlation in a failure arrival process which generates a warranty claim with each occurring failure. We show that even slight positive correlation in the warranty claims arrival process can significantly impact the expected number of claims observed, and the length of the transient period. We are not suggesting that renewal processes are necessarily inappropriate or incorrect as models for the arriving warranty claims process. Rather, we emphasize the importance of considering correlation in the claims arrival process, and suggest that more research is needed to explore mathematical models which explicitly consider correlation in the warranty analysis framework.
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paper_title: Bayesian Estimation from Censored Data with Incomplete Information

paper_content:

Phillips and Sweeting (Journal of the Royal Statistical Society B 1996; 58:775–783) considered estimation of the parameter of the exponential distribution with censored failure time data when there is incomplete knowledge of the censoring times. It was shown that, under particular models for the censoring mechanism and censoring errors, it will usually be safe to ignore such errors provided they are not expected to be too large. The results depended on using an inverted Gamma model for the conditional distribution of the concomitant variable which provided the supplementary information used when the failure times were censored. The effect of alternative assumptions about knowledge of the censoring values on the estimation of failure rate is investigated through the use of Bayesian analysis. By putting prior distributions on the critical parameters of the censoring mechanism it is possible to incorporate and investigate the influence of the use of prior knowledge. Copyright © 2004 John Wiley & Sons, Ltd.
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paper_title: Using Accelerated Life Tests Results to Predict Product Field Reliability

paper_content:

Accelerated life tests (ALTs) provide timely assessments of the reliability of materials, components, and subsystems. ALTs can be run at any of these levels or at the full-system level. Sometimes ALTs generate multiple failure modes. A frequently asked question near the end of an ALT program is “what do these test results say about field performance?” ALTs are carefully controlled, whereas the field environment is highly variable. For example, products in the field have different average use rates across the product population. With good characterization of field use conditions, it may be possible to use ALT results to predict the failure time distribution in the field. When such information is not available but both life test data and field data (from, e.g., warranty returns) are available, it may be possible to find a model to relate the two data sets. Under a reasonable set of practical assumptions, this model then can be used to predict the failure time distribution for a future component or product o...
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paper_title: Nonparametric estimation of marginal failure distributions from dually censored automotive data

paper_content:

It is not uncommon that a component's reliability characteristics depend on two usage variables. As an example, for automotive components such variables are time in service and accumulated mileage. For certain airplane chassis components, these variables are accumulated flight hours and the number of landings. The problem of failure time distribution estimation from dually censored (at warranty limits) automotive data has been addressed by many authors. However, most of published work is focused on estimating marginal in time failure distributions or (less frequently) joint failure distributions. This paper focuses on nonparametric estimation of the marginal in mileage failure distributions, since for the majority of the automotive components mileage is considered a more relevant survival variable than time in service.
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paper_title: Analysis of Window-Observation Recurrence Data

paper_content:

Many systems experience recurrent events. Recurrence data are collected to analyze quantities of interest, such as the mean cumulative number of events. Methods of analysis are available for recurrence data with left and/or right censoring. Due to practical constraints, however, recurrence data are sometimes recorded only in windows. Between the windows are gaps over which the process cannot be observed. This article extends existing statistical methods, both nonparametric and parametric, to window-observation recurrence data. The nonparametric estimator requires minimum assumptions but will be inconsistent if the size of the risk set is not positive over the entire period of interest. No such difficulty arises when using a parametric model for the recurrence data. For cases in which the size of the risk set is zero for some period of time, we propose and compare two alternative hybrid estimators. We illustrate the methods with two example applications.
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Title: Warranty Data Analysis: A Review

Section 1: Introduction
Description 1: Provide an overview of warranty data, its importance, composition, and objectives of warranty data analysis (WDA). Mention the need for a comprehensive review on recent developments in WDA.

Section 2: Warranty claims
Description 2: Discuss the process and types of warranty claims, the quality of warranty data, and potential causes of warranty claims.

Section 3: Warranty data analysis
Description 3: Explain various models and methods proposed in the literature for analyzing warranty data. This section can be subdivided into early detection of reliability problems, design modification, field reliability estimation, claim prediction, and claim estimation.

Section 4: Early detection of reliability problems
Description 4: Describe techniques and methodologies for early detection of reliability problems using warranty data, including statistical and AI-based methods.

Section 5: Suggestion on design modification
Description 5: Explore how warranty data can be used to assist engineers in improving product design by analyzing failure causes and providing feedback.

Section 6: Field reliability estimation
Description 6: Review methods for estimating product reliability from warranty claims data, including one-dimensional and two-dimensional approaches, and covariate analysis.

Section 7: Warranty claim prediction
Description 7: Discuss techniques for predicting the expected number of warranty claims and respective warranty costs using lifetime distributions, stochastic processes, and time series models.

Section 8: Warranty claim estimation
Description 8: Cover the methodologies used for estimating the number of warranty claims, drawing distinctions between prediction and estimation processes.

Section 9: Conclusions and further research
Description 9: Summarize the state-of-the-art in WDA, identify gaps in current research, and suggest directions for future research, focusing on early detection, design modification, field reliability estimation, and long-term warranty claims.

Section 10: Remarks
Description 10: Mention any additional areas not covered in the review, and apologize for any pertinent articles that might have been omitted.

Section 11: For further readings
Description 11: List additional studies and articles of interest to readers seeking more information on WDA.