Journal of Iranian Statistical Society
Volume:22 Issue: 1, Spring 2023

When the parametric model does not hold, and we cannot fit a parametric model to the data, the true density may be estimated non-parametrically, as in the case of a kernel estimate. The purpose of this paper is to present a comparison between parametric and non-parametric models. The parametric investigation contains Vuong's test, and tracking interval based on the known maximum likelihood estimation theory. The presented non-parametric analysis involves kernel density estimation. Modified differences of Kullback-Leibler criteria between two rival models and Vuong's test, have been considered. In this circumstance, we address the problem of cross-validation estimation of variance for Kullback-Leibler divergence between the true but unknown density and its kernel estimator. A simulation study and data analysis have shown that the parametric density is a more realistic estimate of the data generating density.

In this paper, we propose an alternative generalization of a recent test for univariate normality which is based on the empirical moment generating function to the multivariate case. We show, among other properties, that the proposed weighted L2-class of statistics is affine invariant and consistent. The empirical critical values of the proposed test are evaluated for different sample sizes, variable dimensions and values of the smoothing parameter through large scale simulations. The empirical power comparison of the test with a strong competitor shows that the test has a considerably high power performance, especially at large sample sizes as well as under heavy-tailed alternative distributions. The application of the statistic, together with its competitor, to six real-life datasets also supports the considerable good power performance of the proposed statistic as well as its ease of application.

We assume the Pareto distribution in the presence of outliers based on the Dixit model. We consider the estimation of the Bayesian Premium under squared error loss function (symmetric), linear exponential, and entropy loss functions (asymmetric), using informative and non-informative priors. We use the Lindley approximation and Markov Chain Monte Carlo methods such as the importance sampling procedure for deriving results. Finally, the results are analyzed using simulation studies.

This article deals with the problems of testing the hypothesis and interval estimation of the p-th quantile, ξ = μ+ησ1, where η=-log(1-p), (0<p<1) of the first population when samples are available from several exponential populations with a common location and possibly different scales. Several test procedures, such as tests using a generalized variable approach, tests based on parametric bootstrap method, and tests using a computational approach to test the null hypothesis against a suitable alternative, have been proposed. Besides several interval estimators for the quantile η, such as confidence intervals based on generalized variable approach, parametric bootstrap approach and Bayesian intervals using Markov chain Monte Carlo (MCMC) method have been suggested. The confidence intervals are compared through their coverage probabilities and average lengths, whereas the test statistics are compared in terms of powers and sizes numerically. The application of our model problem has been shown using real-life data sets, and conclusions have been made there.

The Pearson-type family densities are among the most important classes of distributions, also playing key roles in directional statistics. To model data scattered asymmetrically on non-Euclidean spaces, including spheres, the researchers confined themselves to extending particular distributions from the class of the Pearson-type family densities. Those specific distributions are symmetric, but their extended versions are usually heavy-tailed. This paper introduces alternative probability density functions in the class of Pearson-type distributions on the sphere with the spherical Student's t, Fisher, and Chi-square densities as the subfamilies. We show that it is intrinsically asymmetric by investigating various theoretical properties of this new subclass. Intensive simulation studies are conducted to explore various aspects of this subclass. Also, modeling two real-life data using the proposed densities and comparing the results with the fits arising from other common spherical distributions are considered.

A new technique is proposed for evaluating the statistical relationship between a quantitative variable Y and a dichotomous variable X assuming two values: X=0 and X=1. The technique is based on the division of the quantitative variable Y into strata by the moving average technique and computation of average values in the strata for the variables Y and X. Stratification turns the dichotomous variable X into a quantitative one. Once the variable X has been transformed in this way, the statistical relationship between Y and X may be analyzed by linear regression and by analysis of variance. Thus, the technique proposed expands the range of methods available for analyzing statistical relationships between quantitative and dichotomous variables. Specific examples are used to compare the moving average technique with the t-test for symmetric (normal) and asymmetric distributions of quantitative variable Y. It is shown that the statistical relationship between stratified Y and X can be strongly different for a symmetrically (normally) distributed variable Y.

In this paper we propose a zero-inflated version of the extended alternative hyper-Poisson distribution of Kumar and Nair (2013b) and investigate some of its important properties and applications. We derive expressions for its probability generating function, mean, variance, etc. along with recursion formulae for probabilities, raw moments and factorial moments. The estimation of the parameters of the distribution is also attempted and it has been fitted to certain real life data sets for highlighting its practical relevance. Further, generalized likelihood ratio test procedure is applied for examining the significance of the parameters of the model and a simulation study is conducted for assessing the performance of the maximum likelihood estimators of the parameters of the distribution.

The usual t-test or F-test can not be used to analyze unreplicated two-level factorial designs, since all the observations are used to estimate the factor effects and no observation is left to estimate the error variance. To overcome this difficulty, various procedures have been proposed in the literature and several simulation studies have been carried out to compare the performance of these methods. The results of these studies have been inconclusive, and no test is widely accepted as a “best” test. In this paper, we present results that show theoretically that no test has high power against all possible alternatives; i.e. no test can detect all patterns of active effects. Therefore, in the absence of any prior information concerning active and inactive effects, no test can be preferred to any other test based on power and the choice of a test should be based on other considerations, such as ease of use, control of individual or experimental error rate, the purpose of the experiment, etc.