Levels of significance of some two-sample tests wkem observations are from compound normal distributions

This paper presents an investigation of the behavior of the levels of significance of the two-sample t and its related tests and the Mann-Whitney test when the samples are randomly drawn from mixtures of two normal populations (compound normals) and when the sample sizes are small (combined sample sizes ? 15). The use of the compound normal allows for investigation when the underlying populations are unequal, nonnormal, heterogeneous in variances, unimodal or bimodal, possessing smaller than normal kurtosis or containing contamination. The exact distribution of the t and its related tests are given. However, they are not readily amenable to calculations. Most of the numerical results presented were obtained by simulations     

Tail Behavior of the General Error Distribution

An inequality on the tail behavior of the general error distribution and an asymptotic Mills-type ratio are established. Two applications are provided. The first application considers the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the general error and normal distributions. The second application establishes the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the general error distribution.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

An alternative version of zero-inflated logarithmic series distribution and some of its applications

An important drawback of the standard logarithmic series distribution (LSD) in several practical applications is that it excludes the zero observation from its support. The LSD with non-negative support is not much studied in the literature. Recently Kumar and Riyaz [On the zero-inflated LSD and its modification. Statistica (accepted for publication). 2013] considered a distribution in this respect namely ‘zero-inflated logarithmic series distribution (ZILSD)’. Through this paper we propose an alternative form of the ZILSD and study some of its properties. We obtain expressions for its probability-generating function, mean and variance, and develop certain recurrence relations for its probabilities, raw moments and factorial moments. The parameters of the model are estimated by the method of moments and the method of maximum likelihood, and certain test procedures are considered for testing the significance of the additional parameter of the distribution. The distribution has been fitted to certain real-life data sets for illustrating its usefulness compared with certain existing models available in the literature. Further, a simulation study is conducted for assessing the performance of the estimators.     

On some test statistics for testing homogeneity of variances: a comparative study

In this paper, we have reviewed 25 test procedures that are widely reported in the literature for testing the hypothesis of homogeneity of variances under various experimental conditions. Since a theoretical comparison was not possible, a simulation study has been conducted to compare the performance of the test statistics in terms of robustness and empirical power. Monte Carlo simulation was performed for various symmetric and skewed distributions, number of groups, sample size per group, degree of group size inequalities, and degree of variance heterogeneity. Using simulation results and based on the robustness and power of the tests, some promising test statistics are recommended for practitioners.     

Estimation of the parameters of Downton's bivariate exponential distribution using ranked set sampling scheme

The parameters of Downton's bivariate exponential distribution are estimated based on a ranked set sample. Parametric and nonparametric methods are considered. The suggested estimators are compared to the corresponding ones based on simple random sampling. It turns out that some of the suggested estimators are significantly more efficient than the ones based on simple random sampling.