Complete convergence for weighted sums of extended negatively dependent random variables

In this article, the complete convergence for weighted sums of extended negatively dependent (END, in short) random variables without identical distribution is investigated. In addition, the complete moment convergence for weighted sums of END random variables is also obtained. As an application, the Baum–Katz type result for END random variables is established. The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.     

The Representation of Hypergeometric Random Variables using Independent Bernoulli Random Variables

In this paper, we show that a hypergeometric random variable can be represented as a sum of independent Bernoulli random variables that are, except in degenerate cases, not identically distributed. In the proof, we use the factorial moment generating function. An asymptotic result on the probabilities of the Bernoulli random variables in the sum is also presented. Numerical examples are used to illustrate the results.     

On complete and complete moment convergence for weighted sums of ANA random variables and applications

In this article, the complete convergence and complete moment convergence for weighted sums of asymptotically negatively associated (ANA, for short) random variables are studied. Several sufficient conditions of the complete convergence and complete moment convergence for weighted sums of ANA random variables are presented. As an application, the complete consistency for the weighted estimator in a nonparametric regression model based on ANA random errors is established by using the complete convergence that we established. We also give a simulation to verify the validity of the theoretical result.     

Correction

In many probability and mathematical statistics courses the probability generating function (PGF) is typically overlooked in favor of the more utilized moment generating function. However, for certain types of random variables, the PGF may be more appealing. For example, sums of independent, non-negative, integer-valued random variables with finite support are easily studied via the PGF. In particular, the exact distribution of the sum can easily be calculated. Several illustrative classroom examples, with varying degrees of difficulty, are presented. All of the examples have been implemented using the R statistical software package.     

The exponentiated exponential distribution: a survey

The exponentiated exponential distribution, a most attractive generalization of the exponential distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Rényi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic distribution of the extreme order statistics, reliability, distribution of the sum of exponentiated exponential random variables, distribution of the product of exponentiated exponential random variables and the distribution of the ratio of exponentiated exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated exponential distribution and encourage further research.     

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).     

Properties of doubly-truncated gamma variables

The truncated gamma distribution has been widely studied, primarily in life-testing and reliability settings. Most work has assumed an upper bound on the support of the random variable, i.e. the space of the distribution is (0,u). We consider a doubly-truncated gamma random variable restricted by both a lower (l) and upper (u) truncation point, both of which are considered known. We provide simple forms for the density, cumulative distribution function (CDF), moment generating function, cumulant generating function, characteristic function, and moments. We extend the results to describe the density, CDF, and moments of a doubly-truncated noncentral chi-square variable.     

Complete moment convergence for weighted sums of extended negatively dependent random variables

Complete moment convergence for weighted sums of sequence of extended negatively dependent (END) random variables is discussed. Some new sufficient and necessary conditions of complete moment convergence for weighted sums of END random variables are obtained, which improve and extend some well-known results in the literature.     

Convergence for weighted sums of widely orthant dependent random variables

ABSTRACT

In this article, a complete convergence result and a complete moment convergence result are obtained for the weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results are also obtained under the extended negatively orthant dependent setup. In particular, the complete convergence result generalizes and improves the related known works in the literature.     

A Proposal for a New Bound for Discrete Distributions

In this work we re-examine some classical bounds for non negative integer-valued random variables by means of information theoretic or maxentropic techniques using fractional moments as constraints. The proposed new bound, no more analytically expressible in terms of moments or moment generating function (mgf), is built by mixing classical bounds and the Maximum Entropy (ME) approximant of the underlying distribution; such a new bound is able to exploit optimally all the information content provided by the sequence of given moments or by the mgf. Particular care will be devoted to obtain fractional moments from the available information given in terms of integer moments and/or moment generating function. Numerical examples show clearly that the bound improvement involving the ME approximant based on fractional moments is not trivial.     

Complete convergence and complete moment convergence for widely orthant-dependent random variables

In this paper, we first establish the complete convergence for weighted sums of widely orthant-dependent (WOD, in short) random variables by using the Rosenthal type maximal inequality. Based on the complete convergence, we further study the complete moment convergence for weighted sums of arrays of rowwise WOD random variables which is stochastically dominated by a random variable X. The results obtained in the paper generalize the corresponding ones for some dependent random variables.     

A limit theorem on moment convergence of centered spectral statistics of random matrices

ABSTRACT

The eigenvalues of a random matrix are a sequence of specific dependent random variables, the limiting properties of which are one of interesting topics in probability theory. The aim of the article is to extend some probability limiting properties of i.i.d. random variables in the context of the complete moment convergence to the centered spectral statistics of random matrices. Some precise asymptotic results related to the complete convergence of p-order conditional moment of Wigner matrices and sample covariance matrices are obtained. The proofs mainly depend on the central limit theorem and large deviation inequalities of spectral statistics.     

Large deviation local limit theorems for ratio statistics

Let {T_n, n ≥ 1} be an arbitrary sequence of nonlattice random variables and let {S_n, n ≥ 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic R_n = T_n/S_n based on simple conditions on the moment generating functions of T_n and S_n. When S_n = re, our main result reduces to that of Chaganty and Sethura-man[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when T_n and S_n are both lattice random variables. We call our theorems large deviation local limit theorems for R_n, since the conditions of our theorems imply that R_n → c in probability for some constant c. We present some examples to illustrate our theorems.     

Complete Convergence for Weighted Sums and Arrays of Rowwise Extended Negatively Dependent Random Variables

In this article, we study the complete convergence for weighted sums of extended negatively dependent random variables and row sums of arrays of rowwise extended negatively dependent random variables. We apply two methods to prove the results: the first of is based on exponential bounds and second is based on the generalization of the classical moment inequality for extended negatively dependent random variables.     

Multiple change-point estimation with U-statistics

We consider a multiple change-point problem: a finite sequence of independent random variables consists of segments given by a known number of the so-called change-points such that the underlying distribution differs from segment to segment. The task is to estimate these change-points under no further assumptions on the within-segment distributions. In this completely nonparametric framework the proposed estimator is defined as the maximizing point of weighted multivariate U-statistic processes. Under mild moment conditions we prove almost sure convergence and the rate of convergence.     

On the rate of convergence in the strong law of large numbers for negatively orthant-dependent random variables

ABSTRACT

In this paper, we study the complete convergence and complete moment convergence for negatively orthant-dependent random variables. Especially, we obtain the Hsu–Robbins-type theorem for negatively orthant-dependent random variables. Our results generalize the corresponding ones for independent random variables.     

On the convergence of moving average processes under dependent conditions

This paper considers a moving average process for a sequence of negatively associated random variables. It discusses the complete convergence of such a moving average process under suitable conditions. These results generalize and complement earlier results on independent random variables. Also, a conjecture for the case of a sequence of independent and identically distributed random variables is resolved and its moment condition weakened.     

An Alpha Half Normal Slash Distribution for Analyzing Non Negative Data

In this paper, we study a new class of slash distribution. We define the distribution through means of a stochastic representation as the mixture of an alpha half normal random variable with respect to the power of a uniform random variable. Properties involving moments and moment generating function are derived. The usefulness and flexibility of the proposed distribution is illustrated through a real application by maximum likelihood procedure.     

Limit theorems for random maximum of independent and non-identically distributed random vectors

In this paper, we study the weak convergence of the random maximum of independent and non-identical random vectors. When the random sample size is assumed to be independent of the basic variables and its distribution function is assumed to converge weakly to a non-degenerate limit, the necessary and sufficient conditions for the weak convergence of the random maximum are derived. An illustrative example is given.     

Positive dependence of random variables with a common marginal distribution

In this paper we review some notions of positive dependence of random variables with a common univariate marginal distribution and describe the related moment and probability inequalities. We first present a comparison between i.i.d. random variables and exchangeable random variables via an application of de Finetti's theorem, then describe some useful probability inequalities via partial orderings of the strength of their positive dependence. Finally, we state a result for random variables which are not necessarily exchangeable. Special applications to the multivariate normal distribution will be discussed, and the results involve only the correlation matrix of the distribution.