Weighted empirical likelihood for quantile regression with non ignorable missing covariates

In this paper, we propose an empirical likelihood-based weighted estimator of regression parameter in quantile regression model with non ignorable missing covariates. The proposed estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness on the fully observed variables is correctly specified. The efficiency gain of the proposed estimator over the complete-case-analysis estimator is quantified theoretically and illustrated via simulation and a real data application.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Statistical Engineering

This article compares the accuracy of the median unbiased estimator with that of the maximum likelihood estimator for a logistic regression model with two binary covariates. The former estimator is shown to be uniformly more accurate than the latter for small to moderately large sample sizes and a broad range of parameter values. In view of the recently developed efficient algorithms for generating exact distributions of sufficient statistics in binary-data problems, these results call for a serious consideration of median unbiased estimation as an alternative to maximum likelihood estimation, especially when the sample size is not large, or when the data structure is sparse.     

Variable selection in heterogeneous panel data models with cross-sectional dependence

This paper studies the Bridge estimator for a high-dimensional panel data model with heterogeneous varying coefficients, where the random errors are assumed to be serially correlated and cross-sectionally dependent. We establish oracle efficiency and the asymptotic distribution of the Bridge estimator, when the number of covariates increases to infinity with the sample size in both dimensions. A BIC-type criterion is also provided for tuning parameter selection. We further generalise the marginal Bridge estimator for our model to asymptotically correctly identify the covariates with zero coefficients even when the number of covariates is greater than the sample size under a partial orthogonality condition. The finite sample performance of the proposed estimator is demonstrated by simulated data examples, and an empirical application with the US stock dataset is also provided.     

Non-parametric Estimation of a Survival Function with Two-stage Design Studies

Abstract.  The two-stage design is popular in epidemiology studies and clinical trials due to its cost effectiveness. Typically, the first stage sample contains cheaper and possibly biased information, while the second stage validation sample consists of a subset of subjects with accurate and complete information. In this paper, we study estimation of a survival function with right-censored survival data from a two-stage design. A non-parametric estimator is derived by combining data from both stages. We also study its large sample properties and derive pointwise and simultaneous confidence intervals for the survival function. The proposed estimator effectively reduces the variance and finite-sample bias of the Kaplan–Meier estimator solely based on the second stage validation sample. Finally, we apply our method to a real data set from a medical device postmarketing surveillance study.     

An Alternative to Efron's Redistribution-of-Mass Construction of the Kaplan—Meier Estimator

Kaplan and Meier (1958) derived the nonparametric maximum likelihood estimator of the survival function for the case in which some survival times are right-censored. Efron (1967) proposed a redistribution-of-mass construction of the Kaplan—Meier estimator that emphasized and illustrated the contribution of the censored observations. This article presents an alternative construction that, unlike Efron's method, redistributes the mass initially associated with each censored observation directly to the uncensored observations. The proposed construction avoids distributing a given mass more than once and provides additional insight into the nature of the Kaplan—Meier estimator.     

A Bezier curve method in interval-censored data

In this paper we propose a Bezier curve method to estimate the survival function and the median survival time in interval-censored data. We compare the proposed estimator with other existing methods such as the parametric method, the single point imputation method, and the nonparametric maximum likelihood estimator through extensive numerical studies, and it is shown that the proposed estimator performs better than others in the sense of mean squared error and mean integrated squared error. An illustrative example based on a real data set is given.     

Nonlinear regression models with general distortion measurement errors

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.     

Proportion estimation in mixtures with covariates

Linear maps of a single unclassified observation are used to estimate the mixing proportion in a mixture of two populations with homogeneous variances in the presence of covariates. with complete knowledge of the parameters of the individual populations, the linear map for which the estimator is unbiased and has minimum variance amongst all similar estimators can be determined. Plug-in estimator based on independent training samples from the component populations can be constructed and is asymptotically equivalent to Cochran's classification statistic V* for covariate classification; see Memon and Okamoto (1970). Under normality assumptions, asymptotic expansion of the distribution of the plug-in estimator is available. In the absence of covariates, our estimator reduces to that suggested by Walker (1980) who has investigated the problem based on information on large unclassified samples from a mixture of two populations with heterogeneous variances. In contrast, distribution of Walker's estimator seems intractable in moderate sample sizes even with normality assumption.     

Nonparametric estimation with left-truncated and right-censored data when the sample size before truncation is known

In this article, we consider nonparametric estimation of the survival function when the data are subject to left-truncation and right-censoring and the sample size before truncation is known. We propose two estimators. The first estimator is derived based on a self-consistent estimating equation. The second estimator is obtained by using the constrained expectation-maximization algorithm. Simulation results indicate that both estimators are more efficient than the product-limit estimator. When there is no censoring, the performance of the proposed estimators is compared with that of the estimator proposed by Li and Qin [Semiparametric likelihood-based inference for biased and truncated data when total sample size is known, J. R. Stat. Soc. B 60 (1998), pp. 243–254] via simulation study.     

Estimating the conditional tail index by integrating a kernel conditional quantile estimator

This paper deals with the estimation of the tail index of a heavy-tailed distribution in the presence of covariates. A class of estimators is proposed in this context and its asymptotic normality established under mild regularity conditions. These estimators are functions of a kernel conditional quantile estimator depending on some tuning parameters. The finite sample properties of our estimators are illustrated on a small simulation study.     

Large sample properties of nonparametric copula estimators under bivariate censoring

A new nonparametric estimator is proposed for the copula function of a bivariate survival function for data subject to random right-censoring. We consider two censoring models: univariate and copula censoring. We show strong consistency and we obtain an i.i.d. representation for the copula estimator. In a simulation study we compare the new estimator to the one of Gribkova and Lopez [Nonparametric copula estimation under bivariate censoring; doi:10.1111/sjos.12144].     

Restricted Two Parameter Ridge Estimator

In 2005 Lipovetsky and Conklin proposed an estimator, the two parameter ridge estimator (TRE), as an alternative to the ordinary least squares estimator (OLSE) and the ordinary ridge estimator (RE) in the presence of multicollinearity, and in 2006 Lipovetsky improved the two parameter model. In this paper, we introduce two new estimators, one of which is the modified two parameter ridge estimator (MTRE) defined by following Swindel's paper of 1976. The other one is the restricted two parameter ridge estimator (RTRE) which is derived by setting additional linear restrictions on the parameter vectors. This estimator is a generalization of the restricted least squares estimator (RLSE) and includes the restricted ridge estimator (RRE) proposed by Groß in 2003. A numerical example is provided and a simulation study is conducted for the comparisons of the RTRE with the OLSE, RLSE, RE, RRE and TRE.     

A two-parameter estimator in the negative binomial regression model

In this article, a two-parameter estimator is proposed to combat multicollinearity in the negative binomial regression model. The proposed two-parameter estimator is a general estimator which includes the maximum likelihood (ML) estimator, the ridge estimator (RE) and the Liu estimator as special cases. Some properties on the asymptotic mean-squared error (MSE) are derived and necessary and sufficient conditions for the superiority of the two-parameter estimator over the ML estimator and sufficient conditions for the superiority of the two-parameter estimator over the RE and the Liu estimator in the asymptotic mean-squared error (MSE) matrix sense are obtained. Furthermore, several methods and three rules for choosing appropriate shrinkage parameters are proposed. Finally, a Monte Carlo simulation study is given to illustrate some of the theoretical results.     

Generalized Concept of Relative Risk and Wider Applications of the Proportional Hazards Model and the Kaplan–Meier Estimator

The concepts of relative risk and hazard ratio are generalized for ordinary ordinal and continuous response variables, respectively. Under the generalized concepts, the Cox proportional hazards model with the Breslow's and Efron's methods can be regarded as generalizations of the Mantel–Haenszel estimator for dealing with broader types of covariates and responses. When ordinal responses can be regarded as discretized observations of a hypothetical continuous variable, the estimated relative risks from the Cox model reflect the associations between the responses and covariates. Examples are given to illustrate the generalized concepts and wider applications of the Cox model and the Kaplan–Meier estimator.     

Buckley–James Type Estimator for Censored Data with Covariates Missing by Design

Abstract. The Buckley–James estimator (BJE) is a well‐known estimator for linear regression models with censored data. Ritov has generalized the BJE to a semiparametric setting and demonstrated that his class of Buckley–James type estimators is asymptotically equivalent to the class of rank‐based estimators proposed by Tsiatis. In this article, we revisit such relationship in censored data with covariates missing by design. By exploring a similar relationship between our proposed class of Buckley–James type estimating functions to the class of rank‐based estimating functions recently generalized by Nan, Kalbfleisch and Yu, we establish asymptotic properties of our proposed estimators. We also conduct numerical studies to compare asymptotic efficiencies from various estimators.     

Performance of the stein-type two-parameter estimator in multiple linear regression model

In this article, we consider the Stein-type approach to the estimation of the regression parameter in a multiple regression model under a multicollinearity situation. The Stein-type two-parameter estimator is proposed when it is suspected that the regression parameter may be restricted to a subspace. The bias and the quadratic risk of the proposed estimator are derived and compared with the two-parameter estimator (TPE), the restricted TPE and the preliminary test TPE. The conditions of superiority of the proposed estimator are obtained. Finally, a real data example is provided to illustrate some of the theoretical results.     

A unified approach to estimation of nonlinear mixed effects and Berkson measurement error models

Mixed effects models and Berkson measurement error models are widely used. They share features which the author uses to develop a unified estimation framework. He deals with models in which the random effects (or measurement errors) have a general parametric distribution, whereas the random regression coefficients (or unobserved predictor variables) and error terms have nonparametric distributions. He proposes a second-order least squares estimator and a simulation-based estimator based on the first two moments of the conditional response variable given the observed covariates. He shows that both estimators are consistent and asymptotically normally distributed under fairly general conditions. The author also reports Monte Carlo simulation studies showing that the proposed estimators perform satisfactorily for relatively small sample sizes. Compared to the likelihood approach, the proposed methods are computationally feasible and do not rely on the normality assumption for random effects or other variables in the model.     

Modified and Restricted r-k Class Estimators

In this article, we introduce the modified r-k class estimator and the restricted r-k class estimator. We compare the performances of the new estimators to the r-k class estimator with respect to the matrix mean square error (MSE) criterion. As a special case of the restricted r-k class estimator, we obtain the restricted principal components regression (RPCR) estimator. Finally, we conduct a Monte Carlo simulation study and a numerical example to investigate the performances of the proposed estimators by the scalar mean square error (mse) criterion.