Search Results (5)

The a priori procedure (APP) is concerned with determining appropriate sample sizes to ensure that sample statistics to be obtained are likely to be good estimators of corresponding population parameters. Previous researchers have shown how to compute a priori confidence interval means or locations for normal and skew-normal distributions. However, two critical limitations persist in the literature: (1) While numerous skewed models have been proposed, the APP equations for location parameters have only been formally established for the basic skew-normal distributions. (2) Even within this fundamental framework, the APPs for sample size determinations in estimating locations are constructed on samples of specifically dependent observations having multivariate skew-normal distributions jointly. Our work addresses these limitations by extending a priori reasoning to the more comprehensive unified skew-normal (SUN) distribution. The SUN family not only encompasses multiple existing skew-normal models as special cases but also enables broader practical applications through its capacity to model mixed skewness patterns and diverse tail behaviors. In this paper, we establish APP equations for determining the required sample sizes and set up confidence intervals for the location parameter in the one-sample case, as well as for the difference in locations in matched pairs and two independent samples, assuming independent observations from the SUN family. This extension addresses a critical gap in the literature and offers a valuable contribution to the field. Simulation studies support the equations presented, and two applications involve real data sets for illustrations of our main results. Full article

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. Full 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. Full article

Inventory-pooling (IP) is an effective tool to mitigate demand uncertainty and variability, to reduce operational costs, and consequently to increase the profit. The major assumptions of the previous works in literature on IP include the following: (1) Independents demand, which satisfy the typical normal independent and identically distributed (iid) random variables; (2) dependents (correlated) symmetric demands, which follows to a multivariate normal distribution. The effect of the dependent asymmetric demand is not yet studied. The aim of this paper is to consider this more realistic case. Indeed, the contribution of this paper is twofold. Firstly, it analyzes both the sensitivity of dependence structure and the levels of skewness of distributions on IP policies in terms of optimal total cost and demand satisfaction constraint. Secondly, both symmetric and asymmetric demand distributions are modeled using various beta distribution and the dependance between demands are modeled using various copulas. A newsvendor problem inspired by the literature, with two decentralized locations and two centralized locations, is considered the empirical study. For each dependance situation, three IP models are considered: inventory centralization, regular transshipments, and independent systems. The results suggest divergences in the decisions in about 9% of cases. Bad choice of marginal distributions given that the copula is appropriate can lead to divergences that vary between 2.2% and 4%, depending on whether the demand distributions are symmetric or asymmetric. Full article

Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distribution of the product for two log-normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas, and estimate some common measures including Kendall’s coefficient, mean, median, standard deviation, skewness, and kurtosis for the distributions. We found that different types of copulas affect the behavior of distributions differently. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. Our results are the foundation of any further study that relies on the density and cumulative probability functions of product for two or more random variables. Thus, the theory developed in this paper is useful for academics, practitioners, and policy makers. Full article