Goodness-of-Fit Tests Based on Sample Space Partitions: a Unifying Overview
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paper_title: An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

paper_content:

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2 × k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c ∈ ℕ, which is based on similarly constructed c × k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.
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paper_title: A Contingency Table Approach to Nonparametric Testing

paper_content:

INTRODUCTION Parametric or Nonparametric? Instructors Example Quadratic Differences and Ranking Outline and Scope Applications of Nonparametric Methods to Sensory Evaluation MODELLING TIES Introduction The Sign Test and Ties Modelling Partitioned Ties in the Sign Test Modelling Unpartitioned Ties in the Sign Test McNemar's Test Partitioning into Components Ties in a Multinomial Test Ties When Testing for Independence TESTS ON ONE-WAY LAYOUT DATA: EXTENSIONS TO THE MEDIAN AND KRUSKAL-WALLIS TESTS Introduction A Model and Pearson's c2 Test Partitioning Pearson's Statistic The Kruskal-Wallis Test with No Ties The Kruskal-Wallis Test with Ties Generalised Median Tests TESTS BASED ON A PRODUCT MULTINOMIAL MODEL: YATES' TEST AND ITS EXTENSIONS Introduction One-Way Tables Partitioning c2p Using Score Statistics Other Methods for Ordered Data Small Sample Size and Power Comparisons Examples FURTHER TESTS BASED ON A PRODUCT MULTINOMIAL MODEL: ORDER IN THE SIGN TEST AND ORDINAL CATEGORICAL DATA WITH A FACTORIAL RESPONSE Introduction How Order Affects the Sign Test The Sign Test and Gart's Tests A New Model and Score Test Comparison of the Sign and Score Tests Sports Drink Example Recommendations Nonparametric Analysis of Ordinal Categorical Data with Factorial Response Olives Data Example Cross Cultural Study Example TESTS ON COMPLETE RANDOMISED BLOCKS: EXTENSIONS TO THE FRIEDMAN AND COCHRAN TESTS Peach Example Friedman's Test and Its Extensions Derivations Page's Test and Its Relationship to Friedman's, Anderson's and Pearson's Tests An Alternative Partition of the Anderson Statistic: An Umbrella Test Ties Cochran's Test Stuart's Test and Its Extensions FURTHER TESTS ON RANDOMISED BLOCKS: EXTENSIONS TO DURBIN'S TEST Introduction Durbin's Test and Its Extensions Derivations A Page-Type Test Paired Comparisons with a 2n Factorial Structure EXTENSIONS TO A NONPARMETRIC CORRELATION TEST: SPEARMAN'S TEST Introduction A Smooth Model and Tests for Independence Smooth Extensions Interpretation of the Components Discussion Multi-way Tables ONE AND S-SAMPLE SMOOTH TESTS OF GOODNESS OF FIT Introduction One-Sample Testing for Uncategorised Distributions One-Sample Testing for Categorised Distributions S-Sample Testing Derivations and Simulation Study CONCLUSION APPENDICES
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paper_title: Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes

paper_content:

The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function F(x). If F_n(x) is the empirical cumulative distribution function and \psi(t) is some nonnegative weight function (0 \leqq t \leqq 1), we consider n^{\frac{1}{2}} \sup_{-\infty and n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x). A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including \psi = 1 and \psi = 1/\lbrack t(1 - t) \rbrack we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises \omega^2 criterion is given.
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paper_title: A two-sample Anderson-Darling rank statistic

paper_content:

SUMMARY A two-sample Anderson-Darling statistic is introduced and small-sample percentage points are given. An approximation to the distribution is also given. The statistic is related to Wilcoxon's and Mood's rank statistics. Asymptotic power comparisons are made with other two-sample rank statistics for shifts in location and scale.
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paper_title: A Nonparametric Test for Independence Based on Sample Space Partitions

paper_content:

In this paper, a class of nonparametric tests for independence between two continuous random variables is proposed. The members of the class are characterized by the size of the partitions of the sample space on which the test statistics are based. The test statistic that corresponds to the smallest partition size may be seen as an extension of Hoeffding's statistic with an Anderson-Darling-type weight function. In this simplest case, we have proven its asymptotic null distribution as well as its omnibus consistency. In a simulation study, the powers of the new tests are compared to those of some other tests for independence. It is concluded that overall high powers are obtained with these new tests.
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paper_title: K-Sample Anderson–Darling Tests

paper_content:

Abstract Two k-sample versions of an Anderson–Darling rank statistic are proposed for testing the homogeneity of samples. Their asymptotic null distributions are derived for the continuous as well as the discrete case. In the continuous case the asymptotic distributions coincide with the (k – 1)-fold convolution of the asymptotic distribution for the Anderson–Darling one-sample statistic. The quality of this large sample approximation is investigated for small samples through Monte Carlo simulation. This is done for both versions of the statistic under various degrees of data rounding and sample size imbalances. Tables for carrying out these tests are provided, and their usage in combining independent one- or k-sample Anderson–Darling tests is pointed out. The test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. One weighting adjusts for the possibly different s...
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paper_title: An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

paper_content:

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2 × k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c ∈ ℕ, which is based on similarly constructed c × k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.
---
paper_title: Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes

paper_content:

The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function F(x). If F_n(x) is the empirical cumulative distribution function and \psi(t) is some nonnegative weight function (0 \leqq t \leqq 1), we consider n^{\frac{1}{2}} \sup_{-\infty and n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x). A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including \psi = 1 and \psi = 1/\lbrack t(1 - t) \rbrack we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises \omega^2 criterion is given.
---
paper_title: A two-sample Anderson-Darling rank statistic

paper_content:

SUMMARY A two-sample Anderson-Darling statistic is introduced and small-sample percentage points are given. An approximation to the distribution is also given. The statistic is related to Wilcoxon's and Mood's rank statistics. Asymptotic power comparisons are made with other two-sample rank statistics for shifts in location and scale.
---
paper_title: A Nonparametric Test for Independence Based on Sample Space Partitions

paper_content:

In this paper, a class of nonparametric tests for independence between two continuous random variables is proposed. The members of the class are characterized by the size of the partitions of the sample space on which the test statistics are based. The test statistic that corresponds to the smallest partition size may be seen as an extension of Hoeffding's statistic with an Anderson-Darling-type weight function. In this simplest case, we have proven its asymptotic null distribution as well as its omnibus consistency. In a simulation study, the powers of the new tests are compared to those of some other tests for independence. It is concluded that overall high powers are obtained with these new tests.
---
paper_title: K-Sample Anderson–Darling Tests

paper_content:

Abstract Two k-sample versions of an Anderson–Darling rank statistic are proposed for testing the homogeneity of samples. Their asymptotic null distributions are derived for the continuous as well as the discrete case. In the continuous case the asymptotic distributions coincide with the (k – 1)-fold convolution of the asymptotic distribution for the Anderson–Darling one-sample statistic. The quality of this large sample approximation is investigated for small samples through Monte Carlo simulation. This is done for both versions of the statistic under various degrees of data rounding and sample size imbalances. Tables for carrying out these tests are provided, and their usage in combining independent one- or k-sample Anderson–Darling tests is pointed out. The test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. One weighting adjusts for the possibly different s...
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paper_title: An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

paper_content:

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2 × k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c ∈ ℕ, which is based on similarly constructed c × k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.
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paper_title: An analysis of variance test for normality (complete samples)

paper_content:

The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order
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paper_title: An Approximate Analysis of Variance Test for Non-Normality Suitable for Machine Calculation

paper_content:

Replacement by a simple approximation of expected values of order statistics in the Shapiro-Francis (1972) W′ statistics, yield a statistics that is more srlitable for machine computation. We show that is equivalent to W′ and discuss its percentage points.
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paper_title: Modern Applied Statistics With S-Plus

paper_content:

S-Plus is a powerful environment for statistical and graphical analysis of data. It provides the tools to implement many statistical ideas which have been made possible by the widespread availability of workstations having good graphics and computational capabilities. This book is a guide to using S-Plus to perform statistical analyses and provides both an introduction to the use of S-Plus and a course in modern statistical methods. All data sets and S-Plus functions used are supplied with the book on a diskette.
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Title: Goodness-of-Fit Tests Based on Sample Space Partitions: a Unifying Overview

Section 1: Introduction
Description 1: Introduce the background, motivation, and aims of the paper, including an overview of sample space partition-based goodness-of-fit tests.

Section 2: The Sample Space Partition Tests
Description 2: Describe the construction of the tests for different goodness-of-fit problems, including detailed distinctions between one-sample, k-sample, and independence problems.

Section 3: The Observation-Based Sample Space Partitions
Description 3: Explain the methodology for constructing observation-based sample space partitions for different types of goodness-of-fit tests.

Section 4: The Sample Space Partition Test Statistic
Description 4: Detail the computation of the Pearson χ2 goodness-of-fit statistic, and provide specific forms and properties of the test statistics for different problems.

Section 5: Power Characteristics
Description 5: Summarize the power characteristics of the SSP-based tests through extensive simulation studies, comparing them against other tests.

Section 6: Example of the One-Sample SSP Test
Description 6: Present an example application of the one-sample SSP test using the Singer dataset, discussing results and potential reasons for observed differences.