Semiparametric median residual life model and inference

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada     

Confidence intervals for the scale parameter of exponential distribution based on Type II doubly censored samples

This paper deals with the problem of interval estimation of the scale parameter in the two-parameter exponential distribution subject to Type II double censoring. Base on a Type II doubly censored sample, we construct a class of interval estimators of the scale parameter which are better than the shortest length affine equivariant interval both in coverage probability and in length. The procedure can be repeated to make further improvement. The extension of the method leads to a smoothly improved confidence interval which improves the interval length with probability one. All improved intervals belong to the class of scale equivariant intervals.     

Efficient estimation for the proportional hazards model with competing risks and current status data

The proportional hazards model is the most commonly used model in regression analysis of failure time data and has been discussed by many authors under various situations (Kalbfleisch & Prentice, 2002. The Statistical Analysis of Failure Time Data, Wiley, New York). This paper considers the fitting of the model to current status data when there exist competing risks, which often occurs in, for example, medical studies. The maximum likelihood estimates of the unknown parameters are derived and their consistency and convergence rate are established. Also we show that the estimates of regression coefficients are efficient and have asymptotically normal distributions. Simulation studies are conducted to assess the finite sample properties of the estimates and an illustrative example is provided. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Asymptotic properties of nonparametric M-estimation for mixing functional data

We investigate the asymptotic behavior of a nonparametric M-estimator of a regression function for stationary dependent processes, where the explanatory variables take values in some abstract functional space. Under some regularity conditions, we give the weak and strong consistency of the estimator as well as its asymptotic normality. We also give two examples of functional processes that satisfy the mixing conditions assumed in this paper. Furthermore, a simulated example is presented to examine the finite sample performance of the proposed estimator.     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Multivariate meta-analysis for data consortia,individual patient meta-analysis,and pooling projects

We discuss maximum likelihood and estimating equations methods for combining results from multiple studies in pooling projects and data consortia using a meta-analysis model, when the multivariate estimates with their covariance matrices are available. The estimates to be combined are typically regression slopes, often from relative risk models in biomedical and epidemiologic applications. We generalize the existing univariate meta-analysis model and investigate the efficiency advantages of the multivariate methods, relative to the univariate ones. We generalize a popular univariate test for between-studies homogeneity to a multivariate test. The methods are applied to a pooled analysis of type of carotenoids in relation to lung cancer incidence from seven prospective studies. In these data, the expected gain in efficiency was evident, sometimes to a large extent. Finally, we study the finite sample properties of the estimators and compare the multivariate ones to their univariate counterparts.     

Weak convergence of bounded influence regression estimates with applications to repeated significance testing

We obtained weak convergence results for bounded influence regression M-estimates and apply the results to sequential clinical trials, with special reference to repeated significance tests in the two-sample problem with covariates.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

On the consistency of MLE in finite mixture models of exponential families

Finite mixtures of densities from an exponential family are frequently used in the statistical analysis of data. Modelling by finite mixtures of densities from different exponential families provide more flexibility in the fittings, and get better results. However, in mixture problems, the log-likelihood function very often does not have an upper bound and therefore a global maximum does not always exist. Redner and Walker (1984. Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev. 26, 195–239) provide conditions to assure the existence, consistency and asymptotic normality of the maximum likelihood estimator.     

Semiparametric analysis of long-range dependence in nonlinear regression

This paper establishes consistency and asymptotic distribution theory for the least squares estimate of a vector parameter of non-linear regression with long-range dependent noise. A covariance-based estimate of the memory parameter is proposed. The consistency of the estimate is established.     

Smoothed empirical likelihood confidence intervals for the relative distribution with left‐truncated and right‐censored data

The study of differences among groups is an interesting statistical topic in many applied fields. It is very common in this context to have data that are subject to mechanisms of loss of information, such as censoring and truncation. In the setting of a two‐sample problem with data subject to left truncation and right censoring, we develop an empirical likelihood method to do inference for the relative distribution. We obtain a nonparametric generalization of Wilks' theorem and construct nonparametric pointwise confidence intervals for the relative distribution. Finally, we analyse the coverage probability and length of these confidence intervals through a simulation study and illustrate their use with a real data set on gastric cancer. The Canadian Journal of Statistics 38: 453–473; 2010 © 2010 Statistical Society of Canada     

Limit theory of generalized order statistics

Generalized order statistics (gos) were introduced by Kamps [1995. A Concept of Generalized Order Statistics. Teubner, Stuttgart] to unify several models of ordered random variables (rv's), e.g., (ordinary) order statistics (oos), records, sequential order statistics (sos). In a wide subclass of gos that includes oos and sos, the possible limit distribution functions (df's) of the maximum gos are obtained in Nasri-Roudsari [1996. Extreme value theory of generalized order statistics. J. Statist. Plann. Inference 55, 281–297]. In this paper, for this subclass, necessary and sufficient conditions of weak convergence, as well as the form of the possible limit df's of extreme, intermediate and central gos are derived. These results are extended to a wider subclass.     

Does bias reduction with external estimator of second order parameter work for endpoint?

Bias reduction estimation for tail index has been studied in the literature. One method is to reduce bias with an external estimator of the second order regular variation parameter; see Gomes and Martins [2002. Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31]. It is known that negative extreme value index implies that the underlying distribution has a finite right endpoint. As far as we know, there exists no bias reduction estimator for the endpoint of a distribution. In this paper, we study the bias reduction method with an external estimator of the second order parameter for both the negative extreme value index and endpoint simultaneously. Surprisingly, we find that this bias reduction method for negative extreme value index requires a larger order of sample fraction than that for positive extreme value index. This finding implies that this bias reduction method for endpoint is less attractive than that for positive extreme value index. Nevertheless, our simulation study prefers the proposed bias reduction estimators to the biased estimators in Hall [1982. On estimating the endpoint of a distribution. Ann. Statist. 10, 556–568].     

Moderate deviations for parameter estimation in some time inhomogeneous diffusions

We study moderate deviations for the maximum likelihood estimation of some inhomogeneous diffusions. The moderate deviation principle with explicit rate functions is obtained. Moreover, we apply our result to the parameter estimation in α$\alpha$-Wiener bridges.     

Pseudo MLE for semiparametric transformation model with doubly truncated data

In this article, we consider the efficient estimation of the semiparametric transformation model with doubly truncated data. We propose a two-step approach for obtaining the pseudo maximum likelihood estimators (PMLE) of regression parameters. In the first step, the truncation time distribution is estimated by the nonparametric maximum likelihood estimator (Shen, 2010a) when the distribution function $K$ of the truncation time is unspecified or by the conditional maximum likelihood estimator (Bilker and Wang, 1996) when $K$ is parameterized. In the second step, using the pseudo complete-data likelihood function with the estimated distribution of truncation time, we propose expectation–maximization algorithms for obtaining the PMLE. We establish the consistency of the PMLE. The simulation study indicates that the PMLE performs well in finite samples. The proposed method is illustrated using an AIDS data set.     

Robust estimators and designs for field experiments

We consider the construction of designs for test-control field experiments, with particular attention being paid to the effects of spatial correlation between adjoining plots. In contrast to previous approaches, in which very specific correlation structures were modelled, we explicitly allow a degree of uncertainty on the part of the experimenter. While fitting a particular correlation structure—and variance structure and regression response—the experimenter is thought to be seeking protection against other possible structures in full neighbourhoods of these particular choices. Robustness, in a minimax sense, is obtained through a modification of the kriging estimation procedure, and through the assignment of treatments to field plots.     

The Likelihood Ratio Test for the Presence of Immunes in a Censored Sample

Models for populations with immune or cured individuals but with others subject to failure are important in many areas, such as medical statistics and criminology. One method of analysis of data from such populations involves estimating an immune proportion 1 ? p and the parameter(s) of a failure distribution for those individuals subject to failure. We use the exponential distribution with parameter λ for the latter and a mixture of this distribution with a mass 1 ? p at infinity to model the complete data. This paper develops the asymptotic theory of a test for whether an immune proportion is indeed present in the population, i.e., for H ₀:p = 1. This involves testing at the boundary of the parameter space for p. We use a likelihood ratio test for H ₀. and prove that minus twice the logarithm of the likelihood ratio has as an asymptotic distribution, not the chi-square distribution, but a 50–50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. The result is proved under an independent censoring assumption with very mild restrictions.