An Interesting Property of the Sample Mean under a First-Order Autoregressive Model

The unweighted sample mean is examined as an estimator of the population mean in a first-order autoregressive model. It is demonstrated that the precision of this estimator deteriorates as the number of equally spaced observations taken within a fixed time interval increases.     

On Necessary and Sufficient Conditions for Ordinary Least Squares Estimators to Be Best Linear Unbiased Estimators

Two often-quoted necessary and sufficient conditions for ordinary least squares estimators to be best linear unbiased estimators are described. Another necessary and sufficient condition is described, providing an additional tool for checking to see whether the covariance matrix of a given linear model is such that the ordinary least squares estimator is also the best linear unbiased estimator. The new condition is used to show that one of the two published conditions is only a sufficient condition.     

Free Knot Splines with RJMCMC in Survival Data Analysis

The B-spline representation is a common tool to improve the fitting of smooth nonlinear functions, it offers a fitting as a piecewise polynomial. The regions that define the pieces are separated by a sequence of knots. The main difficulty in this type of modeling is the choice of the number and the locations of these knots. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm provides a solution to simultaneously select these two parameters by considering the knots as free parameters. This algorithm belongs to the MCMC techniques that allow simulations from target distributions on spaces of varying dimension. The aim of the present investigation is to use this algorithm in the framework of the analysis of survival time, for the Cox model in particular. In fact, the relation between the hazard ratio function and the covariates being assumed to be log-linear, this assumption is too restrictive. Thus, we propose to use the RJMCMC algorithm to model the log hazard ratio function by a B-spline representation with an unknown number of knots at unknown locations. This method is illustrated with two real data sets: the Stanford heart transplant data and lung cancer survival data. Another application of the RJMCMC is selecting the significant covariates, and a simulation study is performed.     

Influence Diagnostics of Semiparametric Nonlinear Reproductive Dispersion Models

This article proposes a semiparametric nonlinear reproductive dispersion model (SNRDM) which is an extension of nonlinear reproductive dispersion model and semiparametric regression model. Maximum penalized likelihood estimators (MPLEs) of unknown parameters and nonparametric functions in SNRDMs are presented. Some novel diagnostic statistics such as Cook distance and difference deviance for parametric and nonparametric parts are developed to identify influence observations in SNRDMs on the basis of case-deletion method, and some formulae readily computed with the MPLEs algorithm for diagnostic measures are given. The equivalency of case-deletion models and mean-shift outlier models in SNRDM is investigated. A simulation study and a real example are used to illustrate the proposed diagnostic measures.     

Bayesian Analysis in Regression Models Using Pseudo-Likelihoods

This article deals with the issue of using a suitable pseudo-likelihood, instead of an integrated likelihood, when performing Bayesian inference about a scalar parameter of interest in the presence of nuisance parameters. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. Moreover, it is particularly useful when it is difficult, or even impractical, to write the full likelihood function.

We focus on Bayesian inference about a scalar regression coefficient in various regression models. First, in the context of non-normal regression-scale models, we give a theroetical result showing that there is no loss of information about the parameter of interest when using a posterior distribution derived from a pseudo-likelihood instead of the correct posterior distribution. Second, we present non trivial applications with high-dimensional, or even infinite-dimensional, nuisance parameters in the context of nonlinear normal heteroscedastic regression models, and of models for binary outcomes and count data, accounting also for possibile overdispersion. In all these situtations, we show that non Bayesian methods for eliminating nuisance parameters can be usefully incorporated into a one-parameter Bayesian analysis.     

Empirical Likelihood Inference for the Parameter in Additive Partially Linear EV Models

In this article, we consider empirical likelihood inference for the parameter in the additive partially linear models when the linear covariate is measured with error. By correcting for attenuation, a corrected-attenuation empirical log-likelihood ratio statistic for the unknown parameter β, which is of primary interest, is suggested. We show that the proposed statistic is asymptotically standard chi-square distribution without requiring the undersmoothing of the nonparametric components, and hence it can be directly used to construct the confidence region for the parameter β. Some simulations indicate that, in terms of comparison between coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the profile-based least-squares method. We also give the maximum empirical likelihood estimator (MELE) for the unknown parameter β, and prove the MELE is asymptotically normal under some mild conditions.     

Generalized Estimating Equations for Symmetric Distributions Observed with Admixture

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. Generalized estimating equations are constructed for the estimation of the mixture proportion and the location parameter of the primary component. Asymptotic normality of the estimates is demonstrated and the lower bound for the asymptotic covariance matrix is obtained. An adaptive estimation technique is proposed to obtain the estimates with nearly optimal asymptotic variances.     

Weighted Searching Probability for Classes of Equivalent Search Designs Comparison

Search design is searching and estimating for a few non zero effects in a large set of effects along with estimation of elements in a set of unknown parameters. In presence of noise, the probability of discrimination between the true non zero effect from an alternative one depends on the design and an unknown parameter, say ρ. We develop a new criterion for design comparison which is independent of ρ and for a family density weight function show that it discriminates and ranks the designs precisely. This criterion is invariance to the variable noise which may be present between designs due to noise factors. This allows us to extend the design comparison to classes of equivalent designs.     

Nonparametric Prediction Intervals for Future Order Statistics in a Proportional Hazard Model

Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F _i  = 1 ? (1 ? F)^{α _i} , where α _i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.     

Tests for Equality of Inflation Parameters of Two Zero-Inflated Power Series Distributions

In this article, we consider two independent zero-inflated power series distributions and provide likelihood ratio test for equality of inflation parameters of the same. As an illustration, testing equality of inflation parameters of two zero inflated Poisson distributions is provided. Further, simulation study to investigate power of likelihood ratio tests has been carried out.