Finite-sample distributions of the Wald,likelihood ratio,and Lagrange multiplier test statistics in the classical linear model

In this note, it is shown that the finite-sample distributions of the Wald, likelihood ratio, and Lagrange multiplier statistics in the classical linear regression model are members of the generalized beta model introduced by McDonald and Xu (1995a McDonald, J.B., Xu, Y.J. (1995a). A generalization of the beta distribution with applications. J. Econom. 66:133–152.[Crossref], [Web of Science ®] , [Google Scholar]). This is useful for examining the properties of these test statistics. For example, this characterization makes it easy to find distribution, quantile, and density functions for each test statistic, makes it clear why Wald tests may overreject the null hypothesis using asymptotic critical values, and formalizes the fact that the Lagrange multiplier statistic follows a distribution with bounded support.     

Reliability analysis of systems with components having two dependent subcomponents

In this article, a system that consists of n independent components each having two dependent subcomponents (A_i, B_i), i = 1, …, n is considered. The system is assumed to compose of components that have two correlated subcomponents (A_i, B_i), and functions iff both systems of subcomponents A₁, A₂, …, A_n and B₁, B₂, …, B_n work under certain structural rules. The expressions for reliability and mean time to failure of such systems are obtained. A sufficient condition to compare two systems of bivariate components in terms of stochastic ordering is also presented.     

Correlated random effects regression analysis for a log-normally distributed variable

In regression analysis, it is assumed that the response (or dependent variable) distribution is Normal, and errors are homoscedastic and uncorrelated. However, in practice, these assumptions are rarely satisfied by a real data set. To stabilize the heteroscedastic response variance, generally, log-transformation is suggested. Consequently, the response variable distribution approaches nearer to the Normal distribution. As a result, the model fit of the data is improved. Practically, a proper (seems to be suitable) transformation may not always stabilize the variance, and the response distribution may not reduce to Normal distribution. The present article assumes that the response distribution is log-normal with compound autocorrelated errors. Under these situations, estimation and testing of hypotheses regarding regression parameters have been derived. From a set of reduced data, we have derived the best linear unbiased estimators of all the regression coefficients, except the intercept which is often unimportant in practice. Unknown correlation parameters have been estimated. In this connection, we have derived a test rule for testing any set of linear hypotheses of the unknown regression coefficients. In addition, we have developed the confidence ellipsoids of a set of estimable functions of regression coefficients. For the fitted regression equation, an index of fit has been proposed. A simulated study illustrates the results derived in this report.     

A new family of multivariate slash distributions

In this article, a new form of multivariate slash distribution is introduced and some statistical properties are derived. In order to illustrate the advantage of this distribution over the existing generalized multivariate slash distribution in the literature, it is applied to a real data set.     

Objective Bayesian inference for the ratio of the scale parameters of two Weibull distributions

The Weibull distribution is widely used due to its versatility and relative simplicity. In our paper, the non informative priors for the ratio of the scale parameters of two Weibull models are provided. The asymptotic matching of coverage probabilities of Bayesian credible intervals is considered, with the corresponding frequentist coverage probabilities. We developed the various priors for the ratio of two scale parameters using the following matching criteria: quantile matching, matching of distribution function, highest posterior density matching, and inversion of test statistics. One particular prior, which meets all the matching criteria, is found. Next, we derive the reference priors for groups of ordering. We see that all the reference priors satisfy a first-order matching criterion and that the one-at-a-time reference prior is a second-order matching prior. A simulation study is performed and an example given.     

On some aspects of a generalized alternative zero-inflated logarithmic series distribution

Here, we consider a generalized form of the alternative zero-inflated logarithmic series distribution of Kumar and Riyaz (J. Statist. Comp. Simul., 2015) and study some of its important aspects. The parameters of the distribution are estimated by the method of maximum likelihood and some test procedures are developed for testing the significance of the additional parameter of the model. All these estimation and testing procedures are illustrated with the help of certain real life datasets. A simulation study is also carried out for assessing the performance of the estimators.     

A martingale-based test for independence of time to failure and cause of failure for competing risks models

In this article, we introduce a class of tests, using a martingale approach, for testing independence of failure time and cause of failure for competing risks data. Asymptotic distribution of the proposed test statistic is derived. The procedure is illustrated with a real-life data. A simulation study is carried out to assess the level and power of the test.     

A new stratified three-stage unrelated randomized response model for estimating a rare sensitive attribute based on the Poisson distribution

This article suggests an efficient method of estimating a rare sensitive attribute which is assumed following Poisson distribution by using three-stage unrelated randomized response model instead of the Land et al. model (2011 Land, M., S. Singh, and S. A. Sedory. 2011. Estimation of a rare sensitive attribute using poisson distribution. Statistics 46 (3):351–60. doi:10.1080/02331888.2010.524300.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) when the population consists of some different sized clusters and clusters selected by probability proportional to size(:pps) sampling. A rare sensitive parameter is estimated by using pps sampling and equal probability two-stage sampling when the parameter of a rare unrelated attribute is assumed to be known and unknown.

We extend this method to the case of stratified population by applying stratified pps sampling and stratified equal probability two-stage sampling. An empirical study is carried out to show the efficiency of the two proposed methods when the parameter of a rare unrelated attribute is assumed to be known and unknown.