Daya Krishna Nagar

This article defines an inverted Topp–Leone distribution. Several mathematical properties and maximum likelihood estimation of parameters of this distribution are considered. The shape of the distribution for different sets of parameters is discussed. Several mathematical properties such as the cumulative distribution function, mode, moment-generating function, survival function, hazard rate function, stress-strength reliability R, moments, Rényi entropy, Shannon entropy, Fisher information matrix, and partial ordering associated with this distribution, have been derived. Distributions of the sum and quotient of two independent inverted Topp–Leone variables have also been obtained.

The exact percentage points of the likelihood ratio statistic for testing the hypothesis that two p-variate normal distributions are independent and their covariance matrices are equal have been computed for p = 2,3,4 and 5.

In this article, we derive a closed-form expression for computing the probabilities of p-dimensional rectangles by means of a multivariate skew-normal distribution. We use a stochastic representation of the multivariate skew-normal/independent distributions to derive expressions that relate their probability density functions to the expected values of positive random variables. We also obtain an analogous expression for probabilities of p-dimensional rectangles for these distributions. Based on this, we propose a procedure based on Monte Carlo integration to evaluate the probabilities of p-dimensional rectangles through multivariate skew-normal/independent distributions. We use these findings to evaluate the probability density functions of a truncated version of this class of distributions, for which we also suggest a scheme to generate random vectors by using a stochastic representation involving a truncated multivariate skew-normal random vector. Finally, we derive distributional properties involving affine transformations and marginalization. We illustrate graphically several of our methodologies and results derived in this article.

We introduce a quantile-based multivariate log-normal distribution, providing a new multivariate skewed distribution with positive support. The parameters of this distribution are interpretable in terms of quantiles of marginal distributions and associations between pairs of variables, a desirable feature for statistical modeling purposes. We derive statistical properties of the quantile-based multivariate log-normal distribution involving the transformations, closed-form expressions for the mixed moments, expected value, covariance matrix, mode, Shannon entropy, and Kullback–Leibler divergence. We also present results on marginalization, conditioning, and independence. Additionally, we discuss parameter estimation and verify its performance through simulation studies. We evaluate the model fitting based on Mahalanobis-type distances. An application to children data is presented.

The multivariate skew-t distribution plays an important role in statistics since it combines skewness with heavy tails, a very common feature in real-world data. A generalization of this distribution is the truncated multivariate skew-t distribution which contains the truncated multivariate t distribution and the truncated multivariate skew-normal distribution as special cases. In this article, we study several distributional properties of the truncated multivariate skew-t distribution involving affine transformations, marginalization, and conditioning. The generation of random samples from this distribution is described.