Extensions of multiple linear regression

Many statistical methods are linked together through their connection with weighted least squares and hence regression. This article reviews these connections, emphasising the iteratively weighted least squares algorithm.     

Projection Mean–Variance Bounds on Expectations of kth Record Values from Restricted Families

We present sharp mean–variance bounds for expectations of kth record values based on distributions coming from restricted families of distributions. These families are defined in terms of convex or star ordering with respect to generalized Pareto distribution. The bounds for expectations of kth record values from DD, DFR, DDA, and DFRA families are special cases of our results. The bounds are derived by application of the projection method.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

Two-Stage Cluster Sampling with Ranked Set Sampling in the Secondary Sampling Frame

Motivated by a real-life problem, we develop a Two-Stage Cluster Sampling with Ranked Set Sampling (TSCRSS) design in the second stage for which we derive an unbiased estimator of population mean and its variance. An unbiased estimator of the variance of mean estimator is also derived. It is proved that the TSCRSS is more efficient—in the sense of having smaller variance—than the conventional two-stage cluster simple random sampling in which the second-stage sampling is with replacement. Using a simulation study on a real-life population, we show that the TSCRSS is more efficient than the conventional two-stage cluster sampling when simple random sampling without replacement is used in both stages.     

Accelerated Burn-In Procedures and System Maintenance Policies

Burn-in is a widely used engineering method which is adopted to eliminate defective items before they are shipped to customers or put into field operation. In order to shorten the burn-in process, burn-in is most often accomplished in an accelerated environment. However, there have been few probabilistic or stochastic models for the burn-in procedures in accelerated environment. In this article, under a new stochastic model for accelerated burn-in procedure, the problems of determining both optimal accelerated burn-in time and optimal replacement policy are considered. Components are burned-in under an accelerated environment, then those surviving the burn-in procedure are put into field operation and they are maintained under a replacement policy. The properties of the optimal accelerated burn-in time and optimal replacement policy are obtained and a numerical example which illustrates the usage of obtained results will be presented.     

Maximum Likelihood Estimation in Compound Geometric Failure Model with Changing Parameters From Type-I Two-Stage Progressively Censored and Group Censored Samples

Boardman and Kendell (1970 Boardman , T. J. , Kendell , P. J. ( 1970 ). Estimation in compound failure models . Technometrics 12 : 891 – 908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered the problem of estimation with respect to Type-I censoring when an item is subjected to only one of the two causes of failure assuming exponential model. Patel and Gajjar (1992 Patel , M. N. , Gajjar , A. V. ( 1992 ). Maximum likelihood estimation in compound exponential failure model with changing failure rates from Type-I progressively censored and group censored samples . Commun. Statist. Theor. Meth. 21 ( 10 ): 2899 – 2908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered extension of the Boardman and Kendell's results in case of two-stage progressive censoring. Here we have considered geometric competing risk failure model with two independent causes of failures. Maximum likelihood estimation of the parameters is carried out using Type-I two-stage progressively censored and group censored samples. Asymptotic standard errors of the estimators are obtained for both the cases. Two illustrative examples are cited for ungroup and group competing risk models.     

Bayes Estimation of Change Point in Non Standard Mixture of Left-Truncated Exponential With Unknown Proportions

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,…, X _n were observed from the mixture of a degenerate and left-truncated exponential (LTE) distribution, with reliability R _1τ at time τ and minimum life length η with unknown proportion p ₁ and θ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in reliability R _2τ at time τ and unknown proportion p ₂ and θ₂. This distribution occurs in many practical situations, for instance; life of a unit may have a LTE distribution but some of the units fail instantaneously. Apart from mixture distributions, the phenomenon of change point is also observed in several situations in life testing and reliability estimation problems. It may happen that at some point of time instability in the sequence of failure times is observed. The problem of study is: When and where this change has started occurring. This is called change point inference problem. The estimators of m, R ₁(t ₀), R ₂(t ₀), p ₁, and p ₂ are derived under asymmetric loss functions namely Linex loss & general entropy loss functions. Both the non informative and informative prior are considered. The effects of prior consideration on Bayes estimates of change point are also studied.     

Estimated Generalized Estimating Equation for Correlated Failure Time Data with Auxiliary Covariates

In this article, we study a marginal hazard model with common baseline hazard for correlated failure time data. We assume that the true covariate is measured precisely in a subset of the whole study cohort, whereas an auxiliary information for the true covariate is available for the whole cohort. We first estimate the relative risk function empirically. Then we obtain the estimator for the regression parameter by replacing the relative risk function with its estimator in a generalized estimating equation (GEE) proposed by Cai (1992 Cai , J. ( 1992 ). Generalized estimation equations for censored multivariate failure time data. Ph.D. thesis, University of Washington, Seattle, Washington . [Google Scholar]). A key feature of this method is that it is nonparametric with respect to the association between the missing covariate and the observed auxiliary covariate. The proposed estimator is shown to be consistent and asymptotically normal. Furthermore, we present a corrected Breslow-type estimator for the cumulative hazard function. Simulation studies are conducted to evaluate the proposed method.     

Asymptotic Inference with Incomplete Data

This article discusses asymptotic theory for the maximum likelihood estimator based on incomplete data. Although much literature has implicitly assumed the basic properties of the estimator, such as consistency and asymptotic normality, it is hard to find their precise and comprehensive proofs. In this article, we first show that under MAR an estimator based on the likelihood function ignoring the missing-data mechanism is strongly consistent. The estimator is then shown to be asymptotically normal. When the data are NMAR and when the data are MAR without parameter distinctness, the consistency and the asymptotic normality are shown. Several examples are provided.