Search Results (6)

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

The aim of this paper is to discuss both higher-order asymptotic expansions and skewed approximations for the Bayesian discrepancy measure used in testing precise statistical hypotheses. In particular, we derive results on third-order asymptotic approximations and skewed approximations for univariate posterior distributions, including cases with nuisance parameters, demonstrating improved accuracy in capturing posterior shape with little additional computational cost over simple first-order approximations. For third-order approximations, connections to frequentist inference via matching priors are highlighted. Moreover, the definition of the Bayesian discrepancy measure and the proposed methodology are extended to the multivariate setting, employing tractable skew-normal posterior approximations obtained via derivative matching at the mode. Accurate multivariate approximations for the Bayesian discrepancy measure are then derived by defining credible regions based on an optimal transport map that transforms the skew-normal approximation to a standard multivariate normal distribution. The performance and practical benefits of these higher-order and skewed approximations are illustrated through two examples. Full article

This paper is concerned with the multivariate extended skew-normal [MESN] and multivariate extended skew-Student [MEST] distributions, that is, distributions in which the location parameters of the underlying truncated distributions are not zero. The extra parameter leads to greater variability in the moments and critical values, thus providing greater flexibility for empirical work. It is reported in this paper that various theoretical properties of the extended distributions, notably the limiting forms as the magnitude of the extension parameter, denoted $\tau$ in this paper, increases without limit. In particular, it is shown that as $\tau$, the limiting forms of the MESN and MEST distributions are different. The effect of the difference is exemplified by a study of stockmarket crashes. A second example is a short study of the extent to which the extended skew-normal distribution can be approximated by the skew-Student. Full article

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented. Full article

The univariate power-normal distribution is quite useful for modeling many types of real data. On the other hand, multivariate extensions of this univariate distribution are not common in the statistic literature, mainly skewed multivariate extensions that can be bimodal, for example. In this paper, based on the univariate power-normal distribution, we extend the univariate power-normal distribution to the multivariate setup. Structural properties of the new multivariate distributions are established. We consider the maximum likelihood method to estimate the unknown parameters, and the observed and expected Fisher information matrices are also derived. Monte Carlo simulation results indicate that the maximum likelihood approach is quite effective to estimate the model parameters. An empirical application of the proposed multivariate distribution to real data is provided for illustrative purposes. Full article

Process capability indices (PCIs) have always been used to improve the quality of products and services. Traditional PCIs are based on the assumption that the data obtained from the quality characteristic (QC) under consideration are normally distributed. However, most data on manufacturing processes violate this assumption. Furthermore, the products and services of the manufacturing industry usually have more than one QC; these QCs are functionally correlated and, thus, should be evaluated together to evaluate the overall quality of a product. This study investigates and extends the existing multivariate non-normal PCIs. First, a multivariate non-normal PCI model from the literature is modeled and validated. An algorithm to generate non-normal multivariate data with the desired correlations is also modeled. Then, this model is extended using two different approaches that depend on the well-known Box–Cox and Johnson transformations. The skewness reduction is further improved by applying heuristics algorithms. These two approaches outperform the investigated model from the literature because they can provide more precise results regardless of the skewness type. The comparison is made based on the generated data and a case study from the literature. Full article