Methods of Statistical Inference for Median Regression Models with Doubly Censored Data

Recently, least absolute deviations (LAD) estimator for median regression models with doubly censored data was proposed and the asymptotic normality of the estimator was established. However, it is invalid to make inference on the regression parameter vectors, because the asymptotic covariance matrices are difficult to estimate reliably since they involve conditional densities of error terms. In this article, three methods, which are based on bootstrap, random weighting, and empirical likelihood, respectively, and do not require density estimation, are proposed for making inference for the doubly censored median regression models. Simulations are also done to assess the performance of the proposed methods.     

Sequential estimation for semiparametric models with application to the proportional hazards model

In this paper, we show that if the Euclidean parameter of a semiparametric model can be estimated through an estimating function, we can extend straightforwardly conditions by Dmitrienko and Govindarajulu [2000. Ann. Statist. 28 (5), 1472–1501] in order to prove that the estimator indexed by any regular sequence (sequential estimator), has the same asymptotic behavior as the non-sequential estimator. These conditions also allow us to obtain the asymptotic normality of the stopping rule, for the special case of sequential confidence sets. These results are applied to the proportional hazards model, for which we show that after slight modifications, the classical assumptions given by Andersen and Gill [1982. Ann. Statist. 10(4), 1100–1120] are sufficient to obtain the asymptotic behavior of the sequential version of the well-known [Cox, 1972. J. Roy. Statist. Soc. Ser. B (34), 187–220] partial maximum likelihood estimator. To prove this result we need to establish a strong convergence result for the regression parameter estimator, involving mainly exponential inequalities for both continuous martingales and some basic empirical processes. A typical example of a fixed-width confidence interval is given and illustrated by a Monte Carlo study.     

Accelerated Recurrence Time Models

Abstract.  For the analysis with recurrent events, we propose a generalization of the accelerated failure time model to allow for evolving covariate effects. These so-called accelerated recurrence time models postulate that the time to expected recurrence frequency, upon transformation, is a linear function of covariates with frequency-dependent coefficients. This modelling strategy shares the same spirit as quantile regression. An estimation and inference procedure is developed by generalizing the celebrated Powell's ( J. Econometrics 25, 1984, 303; J. Econometrics 32, 1986, 143) estimator for censored quantile regression. Consistency and asymptotic normality of the proposed estimator are established. An algorithm is devised to attain good computational efficiency. Simulations demonstrate that this proposal performs well under practical settings. This methodology is illustrated in an application to the well-known bladder cancer study.     

An inverse-probability-weighted approach to estimation of the bivariate survival function under left-truncation and right-censoring

In this note, we consider estimating the bivariate survival function when both survival times are subject to random left truncation and one of the survival times is subject to random right censoring. Motivated by Satten and Datta [2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. 55, 207–210], we propose an inverse-probability-weighted (IPW) estimator. It involves simultaneous estimation of the bivariate survival function of the truncation variables and that of the censoring variable and the truncation variable of the uncensored components. We prove that (i) when there is no censoring, the IPW estimator reduces to NPMLE of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131] and (ii) when there is random left truncation and right censoring on only one of the components and the other component is always observed, the IPW estimator reduces to the estimator of Gijbels and Gürler [1998. Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Statist. Sin. 1219–1232]. Based on Theorem 3.1 of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627], we prove that the IPW estimator is consistent under certain conditions. Finally, we examine the finite sample performance of the IPW estimator in some simulation studies. For the special case that censoring time is independent of truncation time, a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627]. For the special case (i), a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by Huang et al. (2001. Nonnparametric estimation of marginal distributions under bivariate truncation with application to testing for age-of-onset application. Statist. Sin. 11, 1047–1068).     

Alternative estimators of the common regression matrix in two GMANOVA models under weighted quadratic losses

We consider the problem of estimating the common regression matrix of two GMANOVA models with different unknown covariance matrices under certain type of loss functions which include a weighted quadratic loss function as a special case. We consider a class of estimators, which contains the Graybill–Deal-type estimator proposed by Sugiura and Kubokawa (Ann. Inst. Statist. Math. 40 (1988) 119), and we give its risk representation via Kubokawa and Srivastava's (Ann. Statist. 27 (1999) 600; J. Multivariate Anal. 76 (2001) 138) identities when the error matrices follow the elliptically contoured distributions. Using the method similar to an approximate minimization of the unbiased risk estimate due to Stein (Studies in the Statistical Theory of Estimation, vol. 74, Nauka, Leningrad, 1977, p. 4), we obtain an alternative estimator to the Graybill–Deal-type estimator which was given under the normality assumption. However, it seems difficult to evaluate the risk of our proposed estimator analytically because of complex nature of its risk function. Instead, we conduct a Monte-Carlo simulation to evaluate the performance of our proposed estimator. The results indicate that our proposed estimator compares favorably with the Graybill–Deal-type estimator.     

Adaptive Warped Kernel Estimators

In this work, we develop a method of adaptive non‐parametric estimation, based on ‘warped’ kernels. The aim is to estimate a real‐valued function s from a sample of random couples (X,Y). We deal with transformed data (Φ(X),Y), with Φ a one‐to‐one function, to build a collection of kernel estimators. The data‐driven bandwidth selection is performed with a method inspired by Goldenshluger and Lepski (Ann. Statist., 39, 2011, 1608). The method permits to handle various problems such as additive and multiplicative regression, conditional density estimation, hazard rate estimation based on randomly right‐censored data, and cumulative distribution function estimation from current‐status data. The interest is threefold. First, the squared‐bias/variance trade‐off is automatically realized. Next, non‐asymptotic risk bounds are derived. Lastly, the estimator is easily computed, thanks to its simple expression: a short simulation study is presented.     

Estimating coefficients of single-index regression models by minimizing variation

This paper proposes a novel estimation of coefficients in single-index regression models. Unlike the traditional average derivative estimation [Powell JL, Stock JH, Stoker TM. Semiparametric estimation of index coefficients. Econometrica. 1989;57(6):1403–1430; Hardle W, Thomas M. Investigating smooth multiple regression by the method of average derivatives. J Amer Statist Assoc. 1989;84(408):986–995] and semiparametric least squares estimation [Ichimura H. Semiparametric least squares (sls) and weighted sls estimation of single-index models. J Econometrics. 1993;58(1):71–120; Hardle W, Hall P, Ichimura H. Optimal smoothing in single-index models. Ann Statist. 1993;21(1):157–178], the procedure developed in this paper is to estimate the coefficients directly by minimizing the mean variation function and does not involve estimating the link function nonparametrically. As a result, it avoids the selection of the bandwidth or the number of knots, and its implementation is more robust and easier. The resultant estimator is shown to be consistent. Numerical results and real data analysis also show that the proposed procedure is more applicable against model free assumptions.     

Quantile regression based on a weighted approach under semi-competing risks data

In this article, we investigate the quantile regression analysis for semi-competing risks data in which a non-terminal event may be dependently censored by a terminal event. Due to the dependent censoring, the estimation of quantile regression coefficients on the non-terminal event becomes difficult. In order to handle this problem, we assume Archimedean Copula to specify the dependence of the non-terminal event and the terminal event. Portnoy [Censored regression quantiles. J Amer Statist Assoc. 2003;98:1001–1012] considered the quantile regression model under right-censoring data. We extend his approach to construct a weight function, and then impose the weight function to estimate the quantile regression parameter for the non-terminal event under semi-competing risks data. We also prove the consistency and asymptotic properties for the proposed estimator. According to the simulation studies, the performance of our proposed method is good. We also apply our suggested approach to analyse a real data.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.     

A Wald-type test statistic based on robust modified median estimator in logistic regression models

In this paper a new robust estimator, modified median estimator, is introduced and studied for the logistic regression model. This estimator is based on the median estimator considered in Hobza et al. [Robust median estimator in logistic regression. J Stat Plan Inference. 2008;138:3822–3840]. Its asymptotic distribution is obtained. Using the modified median estimator, we also consider a Wald-type test statistic for testing linear hypotheses in the logistic regression model and we obtain its asymptotic distribution under the assumption of random regressors. An extensive simulation study is presented in order to analyse the efficiency as well as the robustness of the modified median estimator and Wald-type test based on it.     

Wavelet estimation of density for censored data with censoring indicator missing at random

In this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.     

Estimation of the scale parameter of the half-logistic distribution under progressively type II censored sample

Based on a progressively type II censored sample, the maximum likelihood and Bayes estimators of the scale parameter of the half-logistic distribution are derived. However, since the maximum likelihood estimator (MLE) and Bayes estimator do not exist in an explicit form for the scale parameter, we consider a simple method of deriving an explicit estimator by approximating the likelihood function and derive the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney and Kadane in J Am Stat Assoc 81:82–86, 1986) and importance sampling methods are used for obtaining the Bayes estimator. In order to compare the performance of the MLE, approximate MLE and Bayes estimates of the scale parameter, we use Monte Carlo simulation.     

Doubly robust empirical likelihood inference in covariate-missing data problems

Missing covariate data occurs often in regression analysis. We study methods for estimating the regression coefficients in an assumed conditional mean function when some covariates are completely observed but other covariates are missing for some subjects. We adopt the semiparametric perspective of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866] on regression analyses with missing covariates, in which they pioneered the use of two working models, the working propensity score model and the working conditional score model. A recent approach to missing covariate data analysis is the empirical likelihood method of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503], which effectively combines unbiased estimating equations. In this paper, we consider an alternative likelihood approach based on the full likelihood of the observed data. This full likelihood-based method enables us to generate estimators for the vector of the regression coefficients that are (a) asymptotically equivalent to those of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the working propensity score model is correctly specified, and (b) doubly robust, like the augmented inverse probability weighting (AIPW) estimators of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Am Statist Assoc. 1994;89:846–866]. Thus, the proposed full likelihood-based estimators improve on the efficiency of the AIPW estimators when the working propensity score model is correct but the working conditional score model is possibly incorrect, and also improve on the empirical likelihood estimators of Qin, Zhang and Leung [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the reverse is true, that is, the working conditional score model is correct but the working propensity score model is possibly incorrect. In addition, we consider a regression method for estimation of the regression coefficients when the working conditional score model is correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866]. Finally, we compare the finite-sample performance of various estimators in a simulation study.     

Random coefficient regressions: parametric goodness-of-fit tests

Random coefficient regression models have been applied in different fields during recent years and they are a unifying frame for many statistical models. Recently, Beran and Hall (Ann. Statist. 20 (1992) 1970) raised the question of the nonparametric study of the coefficients distribution. Nonparametric goodness-of-fit tests were considered in Delicado and Romo (Ann. Inst. Statist. Math. 51 (1999) 125). In this nonparametric framework, the study of parametric families for the coefficient distributions was started by Beran (Ann. Inst. Statist. Math. (1993) 639). Here we propose statistics for parametric goodness-of-fit tests and we obtain their asymptotic distributions. Moreover, we construct bootstrap approximations to these distributions, proving their validity. Finally, a simulation study illustrates our results.     

On local likelihood density estimation when the bandwidth is large

In this paper, we provide a large bandwidth analysis for a class of local likelihood methods. This work complements the small bandwidth analysis of Park et al. (Ann. Statist. 30 (2002) 1480). Our treatment is more general than the large bandwidth analysis of Eguchi and Copas (J. Roy. Statist. Soc. B 60 (1998) 709). We provide a higher-order asymptotic analysis for the risk of the local likelihood density estimator, from which a direct comparison between various versions of local likelihood can be made. The present work, being combined with the small bandwidth results of Park et al. (2002), gives an optimal size of the bandwidth which depends on the degree of departure of the underlying density from the proposed parametric model.     

Weak convergence for the conditional distribution function in a Koziol–Green model under dependent censoring

In this paper we consider the conditional Koziol–Green model of Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] in which they generalized the Koziol–Green model of Veraverbeke and Cadarso Suárez [2000. Estimation of the conditional distribution in a conditional Koziol–Green model. Test 9, 97–122] by assuming that the association between a censoring time and a time until an event is described by a known Archimedean copula function. They got in this way, an informative censoring model with two different types of informative censoring. Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] derived in this model a non-parametric Koziol–Green estimator for the conditional distribution function of the time until an event, for which they showed the uniform consistency and the asymptotic normality. In this paper we extend their results and prove the weak convergence of the process associated to this estimator. Furthermore we show that the conditional Koziol–Green estimator is asymptotically more efficient in this model than the general copula-graphic estimator of Braekers and Veraverbeke [2005. A copula-graphic estimator for the conditional survival function under dependent censoring. Canad. J. Statist. 33, 429–447]. As last result, we construct an asymptotic confidence band for the conditional Koziol–Green estimator. Through a simulation study, we investigate the small sample properties of this asymptotic confidence band. Afterwards we apply this estimator and its confidence band on a practical data set.     

Maximin clusters for nonreplicated multiresponse lack of fit tests

Khuri (Technometrics 27 (1985) 213) and Levy and Neill (Comm. Statist. A 19 (1990) 1987) presented regression lack of fit tests for multiresponse data with replicated observations available at points in the experimental region, thereby extending the classical univariate lack of fit test given by Fisher (J. Roy. Statist. Soc. 85 (1922) 597). In this paper, multivariate tests for lack of fit in a linear multiresponse model are derived for the common circumstance in which replicated observations are not obtained. The tests are based on the union–intersection principle, and provide multiresponse extensions of the univariate tests for between- and within-cluster lack of fit introduced by Christensen (Ann. of Statist. 17 (1989) 673; J. Amer. Statist. Assoc. 86 (1991) 752). Since the properties of these tests depend on the choice of multivariate clusters of the observations, a multiresponse generalization of the maximin power clustering criterion given by Miller, Neill and Sherfey (Ann. of Statist. 26 (1998) 1411; J. Amer. Statist. Assoc. 94 (1999) 610) is also developed.     

Case-cohort analysis with semiparametric transformation models

Semiparametric transformation models provide flexible regression models for survival analysis, including the Cox proportional hazards and the proportional odds models as special cases. We consider the application of semiparametric transformation models in case-cohort studies, where the covariate data are observed only on cases and on a subcohort randomly sampled from the full cohort. We first propose an approximate profile likelihood approach with full-cohort data, which amounts to the pseudo-partial likelihood approach of Zucker [2005. A pseudo-partial likelihood method for semiparametric survival regression with covariate errors. J. Amer. Statist. Assoc. 100, 1264–1277]. Simulation results show that our proposal is almost as efficient as the nonparametric maximum likelihood estimator. We then extend this approach to the case-cohort design, applying the Horvitz–Thompson weighting method to the estimating equations from the approximated profile likelihood. Two levels of weights can be utilized to achieve unbiasedness and to gain efficiency. The resulting estimator has a closed-form asymptotic covariance matrix, and is found in simulations to be substantially more efficient than the estimator based on martingale estimating equations. The extension to left-truncated data will be discussed. We illustrate the proposed method on data from a cardiovascular risk factor study conducted in Taiwan.