CHAPTER 1 Semiparametric Models: a Review of Progress since BKRW (1993)
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paper_title: The asymptotic information in censored survival data

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Various procedures are considered for fitting a regression model to censored survival data in continuous time with time-dependent covariate functions. These include maximum likelihood with the underlying hazard function known completely and known up to a multiplicative constant, and the maximization of Cox's partial likelihood. Explicit formulae for the asymptotic variances of the estimators are derived informally and compared. It is shown how sample second derivatives may be used to estimate the amount of information lost through lack of knowledge of the underlying hazard function. Corresponding results for a more general parameterization which includes the Weibull hazard function are indicated.
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paper_title: Timely Topics in Statistical Methods for Clinical Trials

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Clinical trials provide information on the absolute and relative efficacy and side effects of medical treatments. These studies range from the well-designed, randomized controlled trial to a study of a single treat­ ment on a series of patients compared to the responses of previous patients treated another way. Statistical methods and concepts play a central role in the design, conduct, analysis, and reporting of these investigations. This article provides a discussion of current developments in the application of these methods and concepts to clinical investiga­ tions. We have made a detailed review of the literature published from 1976 to 1979 and have provided coverage of the major journals for 1980 and part of 1981. Our bibliography refers to the most recent and important literature. These citations should provide sufficient entry points for a more extensive study. The current literature on statistical applications to clinical trials di­ vides into two primary components: presentation of methodologic devel­ opments and discussion of the controversies associated with choosing
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paper_title: Some further asymptotic efficiency calculations for survival data regression models

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SUMMARY The asymptotic estimating efficiency of a nonparametric proportional hazard regression model for survival data versus the corresponding exponential model is examined for one and two independent variables and the results extended to incorporate censoring. The nonparametric model estimators are shown to have high efficiency in cases of practical importance.
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paper_title: ASYMPTOTICALLY MOST POWERFUL RANK-ORDER TESTS'

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= 0 against the alternative j3 > 0. We suppose that the square root of the probability density f(x) of the residuals Yi possesses a quadratically integrable derivative and define a class of rank order tests, which are asymptotically most powerful for given f. The main result is exposed in the following succession: theorem, corollaries and examples, comments, preliminaries and proof. The proof is based on results by Hajek [6] and LeCam [8], [9]. Section 6 deals with asymptotic efficiency of rank-order tests, which is shown, on the basis of Mikulski's results [10], to be presumably never less than the asymptotic efficiency of corresponding parametric tests of Neyman's type [11]. This would extend the wellknown result obtained by Chernoff and Savage [2] for the Student t-test. Furthermore, it is shown that the efficiency may be negative, i.e., asymptotic power may be less than the asymptotic size. In Section 7 we consider parallel rank-order tests of symmetry for judging paired comparisons. Section 8 is devoted to rankorder tests for densities such that (f(x))' does not possess a quadratically integrable derivative. In Section 9, we construct a test which is asymptotically most powerful simultaneously for all densities f(x) such that (f(x)) 2 possesses a quadratically integrable derivative. 1. The main theorem. Consider a sequence of random vectors (XV v*... , XVN,), 1 0, and ,B is the parameter under test. We test the hypothesis ,B = 0 against the alternative , > 0. Assume that the density f(x) = F'(x) exists and that (f(x) )i possesses a quadratically integrable derivative. As
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paper_title: Starting age and subsequent birth intervals in cohabitational unions in current danish cohorts, 1975

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This paper presents some main results of an investigation by life table methods of birth interval data in cohabitational unions (marriages as well as consensual unions) in current Danish cohorts. Our results confirm previous findings that an early age at the start of recorded exposure to childbearing risk is indicative of a rapid pace and high level of subsequent fertility. The analysis modifies previous results and adds several new details regarding cohort trends and the effect of parity at the start of reported cohabitation. For each parity within a period of cohabitation, fertility differentials by reported starting age seem to have diminished from our older cohorts (of age up to 49 years in 1975) to our younger ones (of age less than 30 years in 1975). There are indications of a dramatic change in childbearing behaviour following the arrival of novel attitudes to non-marital cohabitation and childbearing in Denmark about 1967.
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paper_title: Information and Asymptotic Efficiency in Parametric-Nonparametric Models

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Asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems (for regular estimates) and asymptotic minimax bounds. The methods used involve: (i) the notion of a "Hellinger-differentiable (root-) density", where part of the differentiation is with respect to the nonparametric part of the model, to obtain appropriate scores; and (ii) calculation of the "effective score" for the real or vector (finite-dimensional) parameter of interest as that component of the score function orthogonal to all nuisance parameter "scores" (perhaps infinite-dimensional). The resulting asymptotic information for estimation of the parametric component of the model is just (4 times) the squared L'-norm of the "effective score". A corollary of these results is a simple necessary condition for "adaptive estimation": adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model. Examples considered include the one-sample location model with and without symmetry, mixture models, the two-sample shift model, and Cox's proportional hazards model.
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paper_title: Fitting Cox's Regression Model to Survival Data Using Glim

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The proportional hazard regression model is reviewed, and its analysis using GLIM is described. Methods of estimating the underlying survivor functions are discussed. The Poisson model which allows the use of GLIM is introduced and interpreted. Two different treatments of tied observations are mentioned, and their properties are compared in the context of a specific example.
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paper_title: Testing Generalized Linear and Semiparametric Models Against Smooth Alternatives

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We propose goodness of fit tests for testing generalized linear models and semiparametric regression models against smooth alternatives. The focus is on models having both, continuous and factorial covariates. As smooth extension of a parametric or semiparametric model we use generalized varying coefficient models as proposed by Hastie&Tibshirani (JRSS B, 1993). A likelihood ratio statistic is used for testing, and asymptotic normality of the test statistic is proven. Due to a slow asymptotic convergence rate a bootstrap approach is pursued. Asymptotic expansions allow to write the estimates as linear smoothers which in turn guarantees simple and fast bootstrapping. The test is shown to have sqrt(n) power, but in contrast to parametric tests it is powerful against smooth alternatives in general.
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paper_title: Testing a Parametric Model Against a Semiparametric Alternative

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This paper describes a method for testing a parametric model of the mean of a random variable Y conditional on a vector of explanatory variables X against a semiparametric alternative. The test is motivated by a conditional moment test against a parametric alternative and amounts to replacing the parametric alternative model with a semiparametric model. The resulting semiparametric test is consistent against a larger set of alternatives than are parametric conditional moments tests based on finitely many moment conditions. The results of Monte Carlo experiments and an application illustrate the usefulness of the new test.
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paper_title: Consistent covariate selection and post model selection inference in semiparametric regression

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This paper presents a model selection technique of estimation in semiparametric regression models of the type Y i = β'X i + f(T i ) + W i , i = 1,...,n. The parametric and nonparametric components are estimated simultaneously by this procedure. Estimation is based on a collection of finite-dimensional models, using a penalized least squares criterion for selection. We show that by tailoring the penalty terms developed for nonparametric regression to semiparametric models, we can consistently estimate the subset of nonzero coefficients of the linear part. Moreover, the selected estimator of the linear component is asymptotically normal.
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paper_title: Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES

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Let Y 1 ,...,Y n be independent identically distributed with density p 0 and let F be a space of densities. We show that the supremum of the likelihood ratios Π i=1 n p(Y i )/p 0 (Y i ), where the supremum is over p ∈ F with ∥p l/2 - p 0 1/2 ∥ 2 ≥ e, is exponentially small with probability exponentially close to 1. The exponent is proportional to ne 2 . The only condition required for this to hold is that e exceeds a value determined by the bracketing Hellinger entropy of F. A similar inequality also holds if we replace F by F n and p 0 by q n , where q n is an approximation to p 0 in a suitable sense. These results are applied to establish rates of convergence of sieve MLEs. Furthermore, weak conditions are given under which the optimal rate e n defined by H(e n ,F)=ne n 2 , where H(.,F) is the Hellinger entropy of F, is nearly achievable by sieve estimators.
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paper_title: Variable Selection for Cox's Proportional Hazards Model and Frailty Model

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A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in Fan and Li (2001a). It has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance. Such a property is called an oracle property. The proposed procedures were illustrated in the context of linear regression, robust linear regression and generalized linear models. In this paper, the nonconcave penalized likelihood approach is extended further to the Cox proportional hazards model and the Cox proportional hazards frailty model, two commonly used semi-parametric models in survival analysis. As a result, new variable selection procedures for these two commonly-used models are proposed. It is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function. Further, with a proper choice of the regularization parameter and the penalty function, the proposed estimators possess an oracle property. Standard error formulae are derived and their accuracies are empirically tested. Simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection, and they reduce model complexity as effectively as the best subset variable selection. Compared with the LASSO, which is the penalized likelihood method with the L 1 -penalty, proposed by Tibshirani, the newly proposed approaches have better theoretic properties and finite sample performance.
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paper_title: ON METHODS OF SIEVES AND PENALIZATION

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We develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates (MLE's) in parametric, semiparametric and nonparametric models. We find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices: the degree of smoothness of the functional and the local size of the underlying parameter space. We show that when the local size of the parameter space is not very large, the substitution standard (nonsieve), substitution sieve and substitution penalized MLE's are asymptotically efficient in the Fisher sense, under certain stochastic equicontinuity conditions of the loglikelihood. Moreover, when the convergence rate of the estimate is slow, the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency. When the size of the parameter space is very large, the standard and penalized maximum likelihood procedures may be inefficient, whereas the method of sieves may be able to overcome this difficulty. This phenomenon is particularly manifested when the functional of interest is very smooth, especially in the semiparametric case.
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paper_title: Rates of convergence for minimum contrast estimators

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We shall present here a general study of minimum contrast estimators in a nonparametric setting (although our results are also valid in the classical parametric case) for independent observations. These estimators include many of the most popular estimators in various situations such as maximum likelihood estimators, least squares and other estimators of the regression function, estimators for mixture models or deconvolution... The main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters. Optimal rates depending on entropy conditions are already known, at least for some of the models involved, and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large. But, under some circumstances (“large” entropies or changes in the entropy structure due to local perturbations), the resulting the rates are only suboptimal. Counterexamples are constructed which show that the phenomenon is real for non-parametric maximum likelihood or regression. This proves that, under purely metric assumptions, our theorem is optimal and that minimum contrast estimators happen to be suboptimal.
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paper_title: Efficient estimation for the proportional hazards model with interval censoring

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The maximum likelihood estimator (MLE) for the proportional hazards model with case 1 interval censored data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with √n convergence rate and achieves the information bound, even though the MLE for the baseline cumulative hazard function only converges at n 1/3 rate. Estimation of the asymptotic variance matrix for the MLE of the regression parameter is also considered. To prove our main results, we also establish a general theorem showing that the MLE of the finite-dimensional parameter in a class of semiparametric models is asymptotically efficient even though the MLE of the infinite-dimensional parameter converges at a rate slower than √n. The results are iliustrated by applying them to a data set from a tumorigenicity study.
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paper_title: Covariate Selection for Semiparametric Hazard Function Regression

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We study a flexible class of nonproportional hazard function regression models in which the influence of the covariates splits into the sum of a parametric part and a time-dependent nonparametric part. We develop a method of covariate selection for the parametric part by adjusting for the implicit fitting of the nonparametric part. Asymptotic consistency of the proposed covariate selection method is established, leading to asymptotically normal estimators of both parametric and nonparametric parts of the model in the presence of covariate selection. The approach is applied to a real data set and a simulation study is presented.
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paper_title: Empirical Likelihood Ratio Confidence Regions

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than are those based on Student's t statistic. An effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable. The method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain. Algorithms exist for finding the unique global minimum at a superlinear rate of convergence. A byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points. The multivariate empirical likelihood regions are justified for functions of several means, such as variances, correlations and regression parameters and for statistics with linear estimating equations. An algorithm is given for computing profile empirical likelihoods for these statistics.
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paper_title: Efficient Estimation of Semiparametric Multivariate Copula Models

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We propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions. A joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions. This class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable, and its circumvention of the “curse of dimensionality” associated with purely nonparametric multivariate distributions. We show that the plug-in sieve maximum likelihood estimators (MLEs) of all smooth functionals, including the finite-dimensional copula parameters and the unknown marginal distributions, are semiparametrically efficient, and that their asymptotic variances can be estimated consistently. Moreover, prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve fu...
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paper_title: Semiparametric methods in econometrics

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1. Introduction.- 2. Single-Index Models.- 2.1 Definition of a Single-Index Model.- 2.2 Why Single-Index Models Are Useful.- 2.3 Other Approaches to Dimension Reduction.- 2.4 Identification of Single-Index Models.- 2.5 EstimatingGin a Single-Index Modei.- 2.6 Optimization Estimators ofss.- 2.7 Direct Semiparametric Estimators.- 2.8 Bandwidth Selection.- 2.9 An Empirical Example.- 3. Binary Response Models.- 3.1 Random-Coefficients Models.- 3.2 Identification.- 3.3 Estimation.- 3.4 Extensions of the Maximum Score and Smoothed Maximum Score Estimators.- 3.5 An Empirical Example.- 4. Deconvolution Problems.- 4.1 A Model of Measurement Error.- 4.2 Models for Panel Data.- 4.3 Extensions.- 4.4 An Empirical Example.- 5. Transformation Models.- 5.1 Estimation with ParametricTand NonparametricF.- 5.2 Estimation with NonparametricTand ParametricF.- 5.3 Estimation when BothTandFare Nonparametric.- 5.4 Predicting Y Conditional onX.- 5.5 An Empirical Example.- Appendix: Nonparametric Estimation.- A.1 Nonparametric Density Estimation.- A.2 Nonparametric Mean Regression.- References.
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paper_title: Nonparametric and semiparametric estimation of the receiver operating characteristic curve

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curve (ODC) given by F(G-'(t)). First we consider nonparametric estimators based on empirical distribution functions and derive asymptotic properties. Next we consider the so-called semiparametric "binormal" model, in which it is assumed that the distributions F and G are normal after some unknown monotonic transformation of the measurement scale. For this model, we propose a generalized least squares procedure and compare it with the estimation algorithm of Dorfman and Alf, which is based on grouped data. Asymptotic results are obtained; small sample properties are examined via a simulation study. Finally, we describe a minimum distance estimator for the ROC curve, which does not require grouping the data.
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paper_title: Asymptotic theory for the correlated gamma-frailty model

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The frailty model is a generalization of Cox's proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. Murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. In this paper we extend this result to the correlated gamma-frailty model, and we allow for covariates. We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to Wald. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.
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paper_title: Semi‐parametric estimation of the binormal ROC curve for a continuous diagnostic test

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SUMMARY Not until recently has much attention been given to deriving maximum likelihood methods for estimating the intercept and slope parameters from a binormal ROC curve that assesses the accuracy of a continuous diagnostic test. We propose two new methods for estimating these parameters. The first method uses the profile likelihood and a simple algorithm to produce fully efficient estimates. The second method is based on a pseudo-maximum likelihood that can easily accommodate adjusting for covariates that could affect the accuracy of the continuous test.
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paper_title: Modeling spatial survival data using semiparametric frailty models.

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We propose a new class of semiparametric frailty models for spatially correlated survival data. Specifically, we extend the ordinary frailty models by allowing random effects accommodating spatial correlations to enter into the baseline hazard function multiplicatively. We prove identifiability of the models and give sufficient regularity conditions. We propose drawing inference based on a marginal rank likelihood. No parametric forms of the baseline hazard need to be assumed in this semiparametric approach. Monte Carlo simulations and the Laplace approach are used to tackle the intractable integral in the likelihood function. Different spatial covariance structures are explored in simulations and the proposed methods are applied to the East Boston Asthma Study to detect prognostic factors leading to childhood asthma.
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paper_title: Asymptotic Theory for the Frailty Model

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The frailty model is a generalization of Cox's proportional hazards model which includes a random effect. Nielsen, Gill, Andersen and Sorensen (1992) proposed an EM algorithm to estimate the cumulative baseline hazard and the variance of the random effect. Here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance.
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paper_title: Two transformation models for estimating an ROC curve derived from continuous data

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A receiver operating characteristic (ROC) curve is a plot of two survival functions, derived separately from the diseased and healthy samples. A special feature is that the ROC curve is invariant to any monotone transformation of the measurement scale. We propose and analyse semiparametric and parametric transformation models for this two-sample problem. Following an unspecified or specified monotone transformation, we assume that the healthy and diseased measurements have two normal distributions with different means and variances. Maximum likelihood algorithms for estimating ROC curve parameters are developed. The proposed methods are illustrated on the marker CA125 in the diagnosis of gastric cancer.
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paper_title: The semi-parametric Pareto model for regression analysis of survival times

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A device for moving logs from a first longitudinal conveyor onto a crossing second conveyor and for turning the logs the desired end first on said second conveyor. The device comprises two side walls positioned at each side of said second conveyor, said side walls forming each two oppositely facing sliding surfaces inclined in the longitudinal direction of said second conveyor. Each side wall is provided with a guiding means, said guiding means being pivotable in opposite directions for guiding each log selectively upon two oppositely facing sliding surfaces of said side walls to cause the log to turn in the desired direction while sliding along said sliding surfaces onto said second conveyor.
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paper_title: Semiparametric and parametric transformation models for comparing diagnostic markers with paired design

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We develop semiparametric and parametric transformation models for estimation and comparison of ROC curves derived from measurements from two diagnostic tests on the same subjects. We assume the existence of transformed measurement scales, one for each test, on which the paired measurements have bivariate normal distributions. The resulting pair of ROC curves are estimated by maximum likelihood algorithms, using joint rank data in the semiparametric model with unspecified transformations and using Box-Cox transformations in the parametric transformation case. Several hypothesis tests for comparing the two ROC curves, or characteristics of them, are developed. Two clinical examples are presented and simulation results are provided.
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paper_title: Adaptive estimation in time-series models

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In a framework particularly suited for many time-series models we obtain a LAN result under quite natural and economical conditions. This enables us to construct adaptive estimators for (part of) the Euclidean parameter in these semiparametric models. Special attention is directed to group models in time series with the important subclass of models with time varying location and scale. Our set-up is confronted with the existing literature and, as examples, we reconsider linear regression and ARMA, TAR and ARCH models.
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paper_title: Application of convolution theorems in semiparametric models with noni . i . d . data

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A useful approach to asymptotic efficiency for estimators in semiparametric models is the study of lower bounds on asymptotic variances via convolution theorems. Such theorems are often applicable in models in which the classical assumptions of independence and identical distributions fail to hold, but to date, much of the research has focused on semiparametric models with independent and identically distributed (i.i.d.) data because tools are available in the i.i.d. setting for verifying pre-conditions of the convolution theorems. We develop tools for non-i.i.d. data that are similar in spirit to those for i.i.d. data and also analogous to the approaches used in parametric models with dependent data. This involves extending the notion of the tangent vector figuring so prominently in the i.i.d. theory and providing conditions for smoothness, or differentiability, of the parameter of interest as a function of the underlying probability measures. As a corollary to the differentiability result we obtain sufficient conditions for equivalence, in terms of asymptotic variance bounds, of two models. Regularity and asymptotic linearity of estimators are also discussed.
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paper_title: Adaptive estimation in a random coefficient autoregressive model

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This paper proves the local asymptotic normality of a stationary and ergodic first order random coefficient autoregressive model in a semiparametric setting. This result is used to show that Stein's necessary condition for adaptive estimation of the mean of the random coefficient is satisfied if the distributions of the innovations and the errors in the random coefficients are symmetric around zero. Under these symmetry assumptions, a locally asymptotically minimax adaptive estimator of the mean of the random coefficient is constructed. The paper also proves the asymptotic normality of generalized M-estimators of the parameter of interest. These estimators are used as preliminary estimators in the above construction.
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paper_title: Efficient estimation of invariant distributions of some semiparametric Markov chain models

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We characterize efficient estimators for the expectation of a function under the invariant distribution of a Markov chain and outline ways of constructing such estimators. We consider two models. The first is described by a parametric family of constraints on the transition distribution; the second is the heteroscedastic nonlinear autoregressive model. The efficient estimator for the first model adds a correction term to the empirical estimator. In the second model, the suggested efficient estimator is a one-step improvement of an initial estimator which might be obtained by a version of Markov chain Monte Carlo. AMS 1991 subject classifications. Primary 62F12, 62G20; secondary 62J09, 62M05.
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paper_title: Efficient estimation in nonlinear autoregressive time-series models

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This paper discusses efficient estimation for a class of nonlinear time-series models with unknown error densities. It establishes local asymptotic normality in this semi-parametric setting. This is then used to describe efficient estimates and to discuss the question of adaptation. Stein's necessary condition for adaptive estimation is satisfied if the error densities are symmetric, but is also satisfied in some models with asymmetric error densities. The paper gives several methods of constructing efficient estimates. These results are then applied to construct efficient estimators in SETAR(2; 1, 1), EXPAR(1) and ARMA(1, 1) models. We observe that adaptation is not possible in the SETAR(2; 1, 1) model with asymmetric errors while the efficient estimators in the ARMA(1, 1) model are adaptive even for asymmetric error densities. Section 8 contains a result that is useful in verifying the continuity of the stationary density with respect to the underlying parameters.
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paper_title: Estimation of Integral Functionals of a Density

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Let φ be a smooth function of k + 2 variables. We shall investigate in this paper the rates of convergence of estimators of T(f)= fφ(f(x), f'(x),..., f (k) (x), x) dx when f belongs to some class of densities of smoothness s. We prove that, when δ ≥ 2k + 1/4, one can define an estimator T n of T(f), based on n i.i.d. observations of density f on the real line, which converges at the semiparametric rate 1/√n. On the other hand, when s < 2k + 1/4, T(f) cannot be estimated at a rate faster than n -γ with γ = 4(s-k)/[4s + 1]. We shall also provide some extensions to the multidimensional case. Those results extend previous works of Levit, of Bickel and Ritov and of Donoho and Nussbaum on estimation of quadratic functionals.
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Title: CHAPTER 1 Semiparametric Models: a Review of Progress since BKRW (1993)

Section 1: Introduction
Description 1: Introduce the scope of the review, mentioning the focus on asymptotic efficiency and the selective nature of the review.

Section 2: Testing and profile likelihood theory
Description 2: Discuss the developments in semiparametric testing and profile likelihood methods, highlighting key contributions and challenges.

Section 3: Semiparametric mixture model theory
Description 3: Review progress in the theory of semiparametric mixture models, including consistency and asymptotic normality of maximum likelihood estimators.

Section 4: Rates of convergence via empirical process methods
Description 4: Examine the advancements in understanding rates of convergence for various estimators in semiparametric models using empirical process methods.

Section 5: Empirical likelihood
Description 5: Detail the developments and applications of empirical likelihood methods in semiparametric models.

Section 6: Transformation and frailty models
Description 6: Highlight the progress in transformation and frailty models, including efficiency and asymptotic theory.

Section 7: Semiparametric regression models
Description 7: Survey the advancements in semiparametric regression models, emphasizing efficient estimation methods.

Section 8: Extensions to non-i.i.d. data
Description 8: Explore the extensions of semiparametric theory to non-i.i.d. data, covering time series models and Markov processes.

Section 9: Critiques and possible alternative theories
Description 9: Address critiques of semiparametric theory and discuss possible alternative theories and their implications.