Improving survey-weighted least squares regression

The weighted least squares (WLS) estimator is often employed in linear regression using complex survey data to deal with the bias in ordinary least squares (OLS) arising from informative sampling. In this paper a 'quasi-Aitken WLS' (QWLS) estimator is proposed. QWLS modifies WLS in the same way that Cragg's quasi-Aitken estimator modifies OLS. It weights by the usual inverse sample inclusion probability weights multiplied by a parameterized function of covariates, where the parameters are chosen to minimize a variance criterion. The resulting estimator is consistent for the superpopulation regression coefficient under fairly mild conditions and has a smaller asymptotic variance than WLS.     

The Sample Selection Model from a Method of Moments Perspective

It is shown how the usual two-step estimator for the standard sample selection model can be seen as a method of moments estimator. Standard GMM theory can be brought to bear on this model, greatly simplifying the derivation of the asymptotic properties of this model. Using this setup, the asymptotic variance is derived in detail and a consistent estimator of it is obtained that is guaranteed to be positive definite, in contrast with the estimator given in the literature. It is demonstrated how the MM approach easily accommodates variations on the estimator, like the two-step IV estimator that handles endogenous regressors, and a two-step GLS estimator. Furthermore, it is shown that from the MM formulation, it is straightforward to derive various specification tests, in particular tests for selection bias, equivalence with the censored regression model, normality, homoskedasticity, and exogeneity.     

Mixed Discrete and Continuous Cox Regression Model

The Cox (1972) regression model is extended to include discrete and mixed continuous/discrete failure time data by retaining the multiplicative hazard rate form of the absolutely continuous model. Application of martingale arguments to the regression parameter estimating function show the Breslow (1974) estimator to be consistent and asymptotically Gaussian under this model. A computationally convenient estimator of the variance of the score function can be developed, again using martingale arguments. This estimator reduces to the usual hypergeometric form in the special case of testing equality of several survival curves, and it leads more generally to a convenient consistent variance estimator for the regression parameter. A small simulation study is carried out to study the regression parameter estimator and its variance estimator under the discrete Cox model special case and an application to a bladder cancer recurrence dataset is provided.     

Estimation of treatment effects based on possibly misspecified Cox regression

In randomized clinical trials, a treatment effect on a time-to-event endpoint is often estimated by the Cox proportional hazards model. The maximum partial likelihood estimator does not make sense if the proportional hazard assumption is violated. Xu and O'Quigley (Biostatistics 1:423-439, 2000) proposed an estimating equation, which provides an interpretable estimator for the treatment effect under model misspecification. Namely it provides a consistent estimator for the log-hazard ratio among the treatment groups if the model is correctly specified, and it is interpreted as an average log-hazard ratio over time even if misspecified. However, the method requires the assumption that censoring is independent of treatment group, which is more restricted than that for the maximum partial likelihood estimator and is often violated in practice. In this paper, we propose an alternative estimating equation. Our method provides an estimator of the same property as that of Xu and O'Quigley under the usual assumption for the maximum partial likelihood estimation. We show that our estimator is consistent and asymptotically normal, and derive a consistent estimator of the asymptotic variance. If the proportional hazards assumption holds, the efficiency of the estimator can be improved by applying the covariate adjustment method based on the semiparametric theory proposed by Lu and Tsiatis (Biometrika 95:679-694, 2008).     

On weak consistency in linear models with equi-correlated random errors

A new estimator in linear models with equi-correlated random errors is postulated. Consistency properties of the proposed estimator and the ordinary least squares estimator are studied. It is shown that the new estimator has smaller variance than the usual least squares estimator under some mild conditions. In addition, it is observed that the new estimator tends to be weakly consistent in many cases where the usual least squares estimator is not.     

Errors-in-variables regression using estimated latent variables

This note considers a method for estimating regression parameters from the data containing measurement errors using some natural estimates of the unobserved explanatory variables. It is shown that the resulting estimator is consistent not only in the usual linear regression model but also in the probit model and regression models with censoship or truncation. However, it fails to be consistent in nonlinear regression models except for special cases.     

Errors-in-variables regression using estimated latent variables

This note considers a method for estimating regression parameters from the data containing measurement errors using some natural estimates of the unobserved explanatory variables. It is shown that the resulting estimator is consistent not only in the usual linear regression model but also in the probit model and regression models with censoship or truncation. However, it fails to be consistent in nonlinear regression models except for special cases.     

Pseudo‐R2 statistics under complex sampling

Model summaries based on the ratio of fitted and null likelihoods have been proposed for generalised linear models, reducing to the familiar R² coefficient of determination in the Gaussian model with identity link. In this note I show how to define the Cox–Snell and Nagelkerke summaries under arbitrary probability sampling designs, giving a design‐consistent estimator of the population model summary. It is also shown that for logistic regression models under case–control sampling the usual Cox–Snell and Nagelkerke R² are not design‐consistent, but are systematically larger than would be obtained with a cross‐sectional or cohort sample from the same population, even in settings where the weighted and unweighted logistic regression estimators are similar or identical. Implementation of the new estimators is straightforward and code is provided in R.     

An Empirical Likelihood Estimate of the Finite Population Correlation Coefficient

In this article, the problem of the estimation of finite population correlation coefficient is considered using the empirical likelihood method. A new estimator that makes the use of both the known mean and variance of an auxiliary variable is proposed. The percent relative bias and percent relative efficiency of the proposed new estimator with respect to the usual estimator of the correlation coefficient is investigated through extensive simulation study for values of the correlation coefficient from ?0.90 to +0.90. The proposed estimator is found to perform better than the simple correlation coefficient from both the bias and relative efficiency points of views, for the population, considered in the investigation. At the end, the proposed estimator has been extended to complex survey designs. Supplementary materials for this article are available online.     

Estimation of population coefficient of variation in simple and stratified random sampling under two-phase sampling scheme when using two auxiliary variables

We propose an improved class of exponential ratio type estimators for coefficient of variation (CV) of a finite population in simple and stratified random sampling using two auxiliary variables under two-phase sampling scheme. We examine the properties of the proposed estimators based on first order of approximation. The proposed class of estimators is more efficient than the usual sample CV estimator, ratio estimator, exponential ratio estimator, usual difference estimator and modified difference type estimator. We also use real data sets for numerical comparisons.     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.     

On the stability of the jackknife variance estimator in ratio estimation

The stability of a slightly modified version of the usual jackknife variance estimator is evaluated exactly in small samples under a suitable linear regression model and compared with that of two different linearization variance estimators. Depending on the degree of heteroscedasticity of the error variance in the model, the stability of the jackknife variance estimator is found to be somewhat comparable to that of one or the other of the linearization variance estimators under conditions especially favorable to ratio estimation (i.e., regression approximately through the origin with a relatively small coefficient of variation in the x population). When these conditions do not hold, however, the jackknife variance estimator is found to be less stable than either of the linearization variance estimators.     

Pooling reliability functions

In this article large sample pooling procedures for reliability functions of an exponential life testing model is considered. Asymptotic properties of shrinkage estimation procedure subsequent to preliminary tests are developed. It is shown that the proposed estimator possesses substantially snakker asymptotic mean squared error than the usual estimator in a region of the lparameter space. Relative efficiencies of the purposed estimators to the usual estimators are obtained and recommendations of the level of the preliminary tests are provided. Relative dominance picture of the estimators is presented. It is shown that the proposed estimator provides a wider dominance range over usual estimator than the usual preliminary test estimator. More importantly, the size of the preliminary test is meaningful. Simulation studies is also carried out to appraise the performance of the estimators when samples are small.     

Sieved Maximum Likelihood Estimation in Wicksell's Problem and Related Deconvolution Problems

It is shown that the classical Wicksell problem is related to a deconvolution problem where the convolution kernel is unbounded, convex and decreasing on (0, ∞). For that type of deconvolution problems, the usual non-parametric maximum likelihood estimator of the distribution function is shown not to exist. A sieved maximum likelihood estimator is defined, and some algorithms are described that can be used to compute this estimator. Moreover, this estimator is proved to be strongly consistent.     

Efficiency of an adaptive estimator of a log-zero-poisson parameter

An estimator, λ is proposed for the parameter λ of the log-zero-Poisson distribution. While it is not a consistent estimator of λ in the usual statistical sense, it is shown to be quite close to the maximum likelihood estimates for many of the 35 sets of data on which it is tried. Since obtaining maximum likelihood estimates is extremely difficult for this and other contagious distributions, this estimate will act at least as an initial estimate in solving the likelihood equations iteratively. A lesson learned from this experience is that in the area of contagious distributions, variability is so large that attention should be focused directly on the mean squared error and not on consistency or unbiasedness, whether for small samples or for the asymptotic case. Sample sizes for some of the data considered in the paper are in hundreds. The fact that the estimator which is not a consistent estimator of λ is closer to the maximum likeli-hood estimator than the consistent moment estimator shows that the variability is large enough to not permit consistency to materialize even for such large sample sizes usually available in actual practice.     

A new heteroskedasticity-consistent covariance matrix estimator for the linear regression model

The assumption that all random errors in the linear regression model share the same variance (homoskedasticity) is often violated in practice. The ordinary least squares estimator of the vector of regression parameters remains unbiased, consistent and asymptotically normal under unequal error variances. Many practitioners then choose to base their inferences on such an estimator. The usual practice is to couple it with an asymptotically valid estimation of its covariance matrix, and then carry out hypothesis tests that are valid under heteroskedasticity of unknown form. We use numerical integration methods to compute the exact null distributions of some quasi-t test statistics, and propose a new covariance matrix estimator. The numerical results favor testing inference based on the estimator we propose.     

Whither jackknifing in stein-rule estimation

In multi-parameter ( multivariate ) estimation, the Stein rule provides minimax and admissible estimators , compromising generally on their unbiasedness. On the other hand, the primary aim of jack-knifing is to reduce the bias of an estimator ( without necessarily compromising on its efficacy ), and, at the same time, jackknifing provides an estimator of the sampling variance of the estimator as well. In shrinkage estimation ( where minimization of a suitably defined risk function is the basic goal ), one may wonder how far the bias-reduction objective of jackknifing incorporates the dual objective of minimaxity ( or admissibility ) and estimating the risk of the estimator ? A critical appraisal of this basic role of jackknifing in shrinkage estimation is made here. Restricted, semi-restricted and the usual versions of jackknifed shrinkage estimates are considered and their performance characteristics are studied . It is shown that for Pitman-type ( local ) alternatives, usually, jackkntfing fails to provide a consistent estimator of the ( asymptotic ) risk of the shrinkage estimator, and a degenerate asymptotic situation arises for the usual fixed alternative case.     

Improved pretest nonparametric estimation in a multivariate regression model

Shrinkage pretest nonparametric estimation of the location parameter vector in a multivariate regression model is considered when nonsample information (NSI) about the regression parameters is available. By using the quadratic risk criterion, the dominance of the pretest estimators over the usual estimators has been investigated. We demonstrate analytically and computationally that the proposed improved pretest estimator establishes a wider dominance range for the parameter under consideration than that of the usual pretest estimator in which it is superior over the unrestricted estimator.     

Asymptotic distribution of regression correlation coefficient for Poisson regression model

This study treats an asymptotic distribution for measures of predictive power for generalized linear models (GLMs). We focus on the regression correlation coefficient (RCC) that is one of the measures of predictive power. The RCC, proposed by Zheng and Agresti is a population value and a generalization of the population value for the coefficient of determination. Therefore, the RCC is easy to interpret and familiar. Recently, Takahashi and Kurosawa provided an explicit form of the RCC and proposed a new RCC estimator for a Poisson regression model. They also showed the validity of the new estimator compared with other estimators. This study discusses the new statistical properties of the RCC for the Poisson regression model. Furthermore, we show an asymptotic normality of the RCC estimator.     

Local likelihood with time-varying additive hazards model

The authors propose the local likelihood method for the time-varying coefficient additive hazards model. They use the Newton-Raphson algorithm to maximize the likelihood into which a local polynomial expansion has been incorporated. They establish the asymptotic properties for the time-varying coefficient estimators and derive explicit expressions for the variance and bias. The authors present simulation results describing the performance of their approach for finite sample sizes. Their numerical comparisons show the stability and efficiency of the local maximum likelihood estimator. They finally illustrate their proposal with data from a laryngeal cancer clinical study.