Bivariate,multivariate, and matrix variate normal characterizations: A brief survey II

The paper entitled “Bivariate and Multivariate Normal Characterizations: A Brief Survey,” by Hamedani, which was published in 1992, covered the published characterizations of bivariate and multivariate normal (MVN) distributions from 1941 to 1991. The present work is a follow-up to the 1991/1992 survey which includes not only characterizations of the bivariate and MVN distributions, but also characterizations of the matrix variate normal distribution, which have appeared from 1991/1992 to the present.     

On the Distributions of Certain Spacings

A characterization of the uniform distribution based on distributions of spacings is presented which extends the existing result in this direction. Also, a result on the distribution of spacings for distributions close to the uniform one is discussed.     

An Application of EM Algorithm to a Change-Point Problem

We consider the problem of change-point in a classical framework while assuming a probability distribution for the change-point. An EM algorithm is proposed to estimate the distribution of the change-point. A change-point model for multiple profiles is also proposed, and EM algorithm is presented to estimate the model. Two examples of Illinois traffic data and Dow Jones Industrial Averages are used to demonstrate the proposed methods.     

On the mixtures of Weibull and Pareto (IV) distribution: An alternative to Pareto distribution

Finite mixture models have provided a reasonable tool to model various types of observed phenomena, specially those which are random in nature. In this article, a finite mixture of Weibull and Pareto (IV) distribution is considered and studied. Some structural properties of the resulting model are discussed including estimation of the model parameters via expectation maximization (EM) algorithm. A real-life data application exhibits the fact that in certain situations, this mixture model might be a better alternative than the rival popular models.     

Nonnormality of Linear Combinations of Normal Random Variables

Hamedani and Tata (1975) showed that the bivariate normal distribution is determined uniquely by any countably infinite collection of distinct linear combinations of the variables and by no finite number of them. It is shown here that this characterization of bivariate normal distribution cannot be extended to the multivariate case. More specifically, it is shown that the multivariate normality of subsets (r < n) of the normal variables X ₁, X ₂, …, X_n together with the normality of an infinite number of linear combinations of them do not guarantee the joint normality of these variables.     

A new extended generalized Gompertz distribution with statistical properties and simulations

Abstract

Statistical distributions are very useful in describing and predicting real world phenomena. In many applied areas there is a clear need for the extended forms of the well-known distributions. Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis. The choice of the best-suited statistical distribution for modeling data is very important.

In this article, we proposed an extended generalized Gompertz (EGGo) family of EGGo. Certain statistical properties of EGGo family including distribution shapes, hazard function, skewness, limit behavior, moments and order statistics are discussed. The flexibility of this family is assessed by its application to real data sets and comparison with other competing distributions. The maximum likelihood equations for estimating the parameters based on real data are given. The performances of the estimators such as maximum likelihood estimators, least squares estimators, weighted least squares estimators, Cramer-von-Mises estimators, Anderson-Darling estimators and right tailed Anderson-Darling estimators are discussed. The likelihood ratio test is derived to illustrate that the EGGo distribution is better than other nested models in fitting data set or not. We use R software for simulation in order to perform applications and test the validity of this model.     

Dealing with big data: comparing dimension reduction and shrinkage regression methods

In the past decades, the number of variables explaining observations in different practical applications increased gradually. This has led to heavy computational tasks, despite of widely using provisional variable selection methods in data processing. Therefore, more methodological techniques have appeared to reduce the number of explanatory variables without losing much of the information. In these techniques, two distinct approaches are apparent: ‘shrinkage regression’ and ‘sufficient dimension reduction’. Surprisingly, there has not been any communication or comparison between these two methodological categories, and it is not clear when each of these two approaches are appropriate. In this paper, we fill some of this gap by first reviewing each category in brief, paying special attention to the most commonly used methods in each category. We then compare commonly used methods from both categories based on their accuracy, computation time, and their ability to select effective variables. A simulation study on the performance of the methods in each category is generated as well. The selected methods are concurrently tested on two sets of real data which allows us to recommend conditions under which one approach is more appropriate to be applied to high-dimensional data.     

A new lifetime model with regression models,characterizations and applications

In this article, we introduce a new extension of Burr XII distribution called Topp Leone Generated Burr XII distribution. We derive some of its properties. Useful characterizations are presented. Simulation study is performed to assess the performance of the maximum likelihood estimators. Censored maximum likelihood estimation is presented in the general case of multi-censored data. The new location-scale regression model based on the proposed distribution is introduced. The usefulness of the proposed models is illustrated empirically by means of three real datasets.     

Sub-Independence: An Expository Perspective

Limit theorems as well as other well-known results in probability and statistics are often based on the distribution of the sums of independent random variables. The concept of sub-independence, which is much weaker than that of independence, is shown to be sufficient to yield the conclusions of these theorems and results. It also provides a measure of dissociation between two random variables which is much stronger than uncorrelatedness.