On the distribution of the sum of independent uniform random variables

Motivated by an application in change point analysis, we derive a closed form for the density function of the sum of n independent, non-identically distributed, uniform random variables.     

More on the distribution of the sum of uniform random variables

The paper provides a simplified derivation of the density of the sum of independent non-identically distributed uniform random variables via an inverse Fourier transform. We also provide examples illustrating the quality of the Normal approximation and corresponding MATHEMATICA code.     

Approximated distributions of the weighted sum of correlated chi-squared random variables

We study the asymptotic behavior of the weighted sum of correlated chi-squared random variables. Both chi-squared and normal distributions are proved to approximate the exact distribution. These two approximations are established by matching the first two cumulants. Simulation comparison is made to study the performance of two approximations numerically. We find that the chi-squared approximation performs better than the normal one in the study.     

A comparison of efficient approximations for a weighted sum of chi-squared random variables

In many applications, the cumulative distribution function (cdf) \(F_{Q_N}\) of a positively weighted sum of N i.i.d. chi-squared random variables \(Q_N\) is required. Although there is no known closed-form solution for \(F_{Q_N}\), there are many good approximations. When computational efficiency is not an issue, Imhof’s method provides a good solution. However, when both the accuracy of the approximation and the speed of its computation are a concern, there is no clear preferred choice. Previous comparisons between approximate methods could be considered insufficient. Furthermore, in streaming data applications where the computation needs to be both sequential and efficient, only a few of the available methods may be suitable. Streaming data problems are becoming ubiquitous and provide the motivation for this paper. We develop a framework to enable a much more extensive comparison between approximate methods for computing the cdf of weighted sums of an arbitrary random variable. Utilising this framework, a new and comprehensive analysis of four efficient approximate methods for computing \(F_{Q_N}\) is performed. This analysis procedure is much more thorough and statistically valid than previous approaches described in the literature. A surprising result of this analysis is that the accuracy of these approximate methods increases with N.     

On the real representation of quaternion random variables

For the first time, the matrix-variate quaternion normal and quaternion Wishart distributions are derived from first principles, that is, from their real counterparts, exposing the relations between their respective densities and characteristic functions. Applications of this theory in hypothesis testing are presented, and the density function of Wilks’ statistic is derived for quaternion Wishart matrices.     

On inequalities in constrained random variables

A unified treatment allows an inequality due to Mallows and an inequality due to Slepian to be generalized     

On the invariance principle for exchangeable random variables

Summary Weak limit results for exchangeable arrays can be obtained via a martingale approach. This note explains why the conditions in the general martingale results simplify to ones on the fulln sum in the case of exchangeable random variables. Part of the work was completed while visiting the Department of Statistics, University of North Carolina, Chapel Hill.     

On the convolution of normal and t random variables

The need for the convolution of normal and Student's t random variables arises in many areas. Since the 1930s, various authors have attempted to derive closed-form expressions for the probability density function (pdf) of the convolution, but with little success. Here, general closed-form expressions are derived for the pdf.     

On simulating truncated skewed Cauchy random variables

Shirvani and Soltani suggested a method for characterizing and simulating one-sided truncated Cauchy random variables. For simulation, that involves solving a nonlinear transformation in each iteration. Here, we propose an alternative method to generate Cauchy random variables. Our method is more general (e.g., incorporates one-sided and two-sided truncations) and does not involve solving any equation. It can be implemented even using a hand calculator.     

On the exponential inequalities for widely orthant-dependent random variables

ABSTRACT

In this work, we establish some exponential inequalities for widely orthant-dependent random variables. We also obtain the convergence rate O(n^{? 1/2}ln?^1/2n) for the strong law of large numbers for widely orthant-dependent random variables.     

On the distribution of quadratic forms in normal random variables

This paper provides necessary and sufficient conditions for a quadratic form in singular normal random variables to be distributed as a given linear combination of independent noncentral chi-square variables. Using this result, an extension of Cochran's theorem to quadratic forms of noncentral chi-square variables is derived.     

On the strong convergence for weighted sums of random variables

A strong convergence result is obtained for weighted sums of identically distributed negatively associated random variables which have a suitable moment condition. This result improves the result of Cai (Metrika 68:323–331, 2008).     

Cumulants of random sum distributions

Consider a random variable S being the sum of a number N of independent and identically distributed random variables X_j (j = 1, 2, ...) where the number N is itself a non-negative integer-valued random variable independent of the X_j An explicit expression of the r-th cumulant of S is given in terms of the cumulants of N and X_j, Asymptotic properties of the distribution of S are also discussed.     

On the moment generating functions for integer valued random variables

Abstract

In this note, we give explicit expressions of moment generating functions for integer valued random variables in both univariate and multivariate cases, which extend the results obtained by Nadarajah and Mitov [Communications in Statistics–Theory and Methods, 32, 2003, 47–60] and more recently by Chakraborti, Jardim and Epprecht [The American Statistician, 2017], Kwong and Nadarajah [Communications in Statistics–Theory and Methods, 2017]. Some examples are also discussed.     

On the equivalence of conglomerability and disintegrability for unbounded random variables

Statistical Methods & Applications - We extend a result of Dubins (Ann Probab 3:89–99, 1975) from bounded to unbounded random variables. Dubins showed that a finitely additive expectation...     

On the normality of transformed beta and unit- gamma random variables

In this paper we consider properties of the logarithmic and Tukey's lambda-type transformations of random variables that follow beta or unit-gamma distributions. Beta distributions often arise as models for random proportions, and unit-gamma distributions, although not well- known, may serve the same purpose. The latter possess many properties similar to those of beta distributions. Some transformations of random variables that follow a beta distribution are considered by Johnson (1949) and Johnson and Kotz (1970,1973). These are used to obtain a -new"random variable that potentially approximately follows a normal distribution, so that practical analyses become possible. We study normality -related properties of the above transformations. This is done for the first time for unit-gamma distributions. Under the logarithmic transformation the beta and unit-gamma distributions become, respectively, the logarithmic F and generalized logistic distributions. The distributions of the transformed beta and unit-gamma distributions after application of Tukey's lambda-type transformations cannot be derived easily; however, we obtain the first four moments and expressions for the skewness and kudos is of the transformed variables. Values of skewness and kurtosis for a variety of different parameter values are calculated, and in consequence, the near (or not near) normality of the transformed variables is evaluated. Comments on the use of the various transformations are provided..     

On the convergence rate for arrays of row-wise NOD random variables

Abstract

In this article, the complete convergence results of weighted sums for arrays of rowwise negatively orthant dependent (NOD) random variables are investigated. Some sufficient conditions for complete convergence for arrays of rowwise NOD random variables are presented without assumption of identical distribution.