Extension of a Two-Stage Conditionally Unbiased Estimator of the Selected Population to the Bivariate Normal Case

A useful parameterization of the exponential failure model with imperfect signalling, under random censoring scheme, is considered to accommodate covariates. Simple sufficient conditions for the existence, uniqueness, consistency, and asymptotic normality of maximum likelihood estimators for the parameters in these models are given. The results are then applied to derive the asymptotic properties of the likelihood ratio test for a difference between failure signalling proportions between groups in a ‘one-way’ classification.     

Testing Equality of Order Constrained Inverse Gaussian Means

The inverse Gaussian family of non negative, skewed random variables is analytically simple, and its inference theory is well known to be analogous to the normal theory in numerous ways. Hence, it is widely used for modeling non negative positively skewed data. In this note, we consider the problem of testing homogeneity of order restricted means of several inverse Gaussian populations with a common unknown scale parameter using an approach based on the classical methods, such as Fisher's, for combining independent tests. Unlike the likelihood approach which can only be readily applied to a limited number of restrictions and the settings of equal sample sizes, this approach is applicable to problems involving a broad variety of order restrictions and arbitrary sample size settings, and most importantly, no new null distributions are needed. An empirical power study shows that, in case of the simple order, the test based on Fisher's combination method compares reasonably with the corresponding likelihood ratio procedure.     

Asymptotic Tail Properties of Student's t-Distribution

We investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of Student's t-distribution with ν > 0 degrees of freedom (t _ν for short) for ν tending to infinity when the argument x = x _ν of the pdf (cdf) depends on ν and tends to ± ∞ (?∞). To this end, we consider the ratio of the pdf's (cdf's) of the t _ν- and the standard normal distribution. Depending on the choice of the argument x _ν, the pdf-ratio (cdf-ratio) tends to 1, a fixed value greater than 1, or to ∞. As a byproduct, we obtain a result for Mill' ratio when x _ν → ?∞.     

Discriminating Between the Log-Normal and Log-Logistic Distributions

Log-normal and log-logistic distributions are often used to analyze lifetime data. For certain ranges of the parameters, the shape of the probability density functions or the hazard functions can be very similar in nature. It might be very difficult to discriminate between the two distribution functions. In this article, we consider the discrimination procedure between the two distribution functions. We use the ratio of maximized likelihood for discrimination purposes. The asymptotic properties of the proposed criterion are investigated. It is observed that the asymptotic distributions are independent of the unknown parameters. The asymptotic distributions are used to determine the minimum sample size needed to discriminate between these two distribution functions for a user specified probability of correct selection. We perform some simulation experiments to see how the asymptotic results work for small sizes. For illustrative purpose, two data sets are analyzed.     

A Robust Score Test for Testing Several Coefficients of Variation with Unknown Underlying Distributions

A parametric robust test is proposed for comparing several coefficients of variation. This test is derived by properly correcting the normal likelihood function according to the technique suggested by Royall and Tsou. The proposed test statistic is asymptotically valid for general random variables, as long as their underlying distributions have finite fourth moments.

Simulation studies and real data analyses are provided to demonstrate the effectiveness of the novel robust procedure.     

Performing Legitimate Parametric Regression Analysis without Knowing the True Underlying Random Mechanisms

Real data are rarely normally distributed. Nonetheless, regression analysis is routinely done under the assumption of normality. Such a practice generally results in invalid statistical inferences once normality is false.

This article shows how one could carry out corrected normal regression and gamma regression analysis, which provides asymptotically valid inferences without the knowledge of the true underlying distributions. No additional programming is necessary in order to implement the proposed novel regression method. Outputs provided by existing statistical software suffice.     

Bayes Estimation of Weibull Distribution Parameters Using Ranked Set Sampling

Estimation of the parameters of Weibull distribution is considered using different methods of estimation based on different sampling schemes namely, Simple Random Sample (SRS), Ranked Set Sample (RSS), and Modified Ranked Set Sample (MRSS). Methods of estimation used are Maximum Likelihood (ML), Method of moments (Mom), and Bayes. Comparison between estimators is made through simulation via their Biases, Relative Efficiency (RE), and Pitman Nearness Probability (PN). Estimators based on RSS and MRSS have many advantages over those that are based on SRS.     

Empirical Likelihood Method for General Additive-Multiplicative Hazard Models

An empirical likelihood-based inferential procedure is developed for a class of general additive-multiplicative hazard models. The proposed log-empirical likelihood ratio test statistic for the parameter vector is shown to have a chi-squared limiting distribution. The result can be used to make inference about the entire parameter vector as well as any linear combination of it. The asymptotic power of the proposed test statistic under contiguous alternatives is discussed. The method is illustrated by extensive simulation studies and a real example.     

Ordering Comparison of Logarithmic Series Random Variables with their Mixtures

The problem of comparing some known distributions in various types of stochastic orderings has been of interest to many authors. In particular, several authors have been recently concerned with the comparison of Poisson, binomial, and negative binomial distributions with their respective mixtures. Incidentally, these distributions are among the four well-known distributions of the family of generalized power series distributions (GPSD's). The remaining distribution is the logarithmic series distribution. In this paper, we shall be concerned with comparing this remaining distribution of the class GPSD with its mixture in terms of various types of stochastic orderings such as the simple stochastic, likelihood ratio, uniformly more variable, convex, hazard rate and expectation orderings. Derivation of the results in this case prove to be computationally trickier than the other three. The special case when the means of the two distributions are the same is also discussed. Finally, an illustrative explicit example is provided.