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 quick and easy approximation to the distribution of a sum of weighted chi-square variables

Various statistical inferences related to chi-square tests lead to the problem of evaluating the probability of a weighted sum of chi-square variables. Tables are available only for some special cases and the computation of the exact probabilities is very complicated due to the well-known problems of numerical integration. This paper presents the theoretical approach of an approximation and an easily implementable algorithm.     

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.     

An approximation to the convolution of gamma distributions

In general, the exact distribution of a convolution of independent gamma random variables is quite complicated and does not admit a closed form. Of all the distributions proposed, the gamma-series representation of Moschopoulos (1985 Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics 37Part A:541–544. [Google Scholar]) is relatively simple to implement but for particular combinations of scale and/or shape parameters the computation of the weights of the series can result in complications with too much time consuming to allow a large-scale application. Recently, a compact random parameter representation of the convolution has been proposed by Vellaisamy and Upadhye (2009 Vellaisamy, P., Upadhye, N. S. (2009). On the sums of compound negative binomial and gamma random variables. Journal of Applied Probability 46:272–283.[Crossref], [Web of Science ®] , [Google Scholar]) and it allows to give an exact interpretation to the weights of the series. They describe an infinite discrete probability distribution. This result suggested to approximate Moschopoulos’s expression looking for an approximating theoretical discrete distribution for the weights of the series. More precisely, we propose a general negative binomial distribution. The result is an “excellent” approximation, fast and simple to implement for any parameter combination.     

Exact distribution of the convolution of negative binomial random variables

In this note, we derive the exact distribution of S by using the method of generating function and BELL polynomials, where S = X₁ + X₂ + ??? + X_n, and each X_i follows the negative binomial distribution with arbitrary parameters. As a particular case, we also obtain the exact distribution of the convolution of geometric random variables.     

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.     

The higher order large-deviation approximation for the distribution of the sum of independent discreterandom variables

For a sum of not identic ally but independently distributed discrete random variables, its higher order large-deviation approximation in given. They are compared with the normal and Edge-worth type approximations in various cases. Consequently, the large-deviation approximations give sufficiently accurate results.     

A bootstrap approximation to the joint distribution of sum and maximum of a stationary sequence

This paper establishes the asymptotic validity for the moving block bootstrap as an approximation to the joint distribution of the sum and the maximum of a stationary sequence. An application is made to statistical inference for a positive time series where an extreme value statistic and sample mean provide the maximum likelihood estimates for the model parameters. A simulation study illustrates small sample size behavior of the bootstrap approximation.     

Order statistics of independent identically distributed variables when the sum is known

In this paper we consider the situation where we know the sum of n independent observations from the same probability distribution. We investigate how to empirically determine the marginal probability distributions of the different order statistics conditional upon knowing the sum. This research was motivated by explorations in process improvement where we know the total expected value or variance of a key measure of an n-step process and would like to estimate the proportion of the expected value or variance that is contributed by the most important step (i.e. the single step having the largest expected value or variance), the two most important steps, etc. Both graphical and tabular results are presented for exponential, gamma and normal distributions.     

A new family of lifetime models

We introduce a new family of distributions by adding a parameter to the Marshall–Olkin family of distributions. Some properties of the new family of distributions are derived. A particular case of the family, a three-parameter generalization of the exponential distribution, is given special attention. The shape properties, moments, distributions of the order statistics, entropies and estimation procedures are derived. An application to a real data set is discussed.     

An approximation to the distribution of the product of two dependent correlation coefficients

Many methodological studies depend on the product of two dependent correlation coefficients. However, the behavior of the distribution of the product of two dependent correlation coefficients is not well known. The distribution of sets of correlation coefficients has been well studied, but not the distribution of the product of two dependent correlation coefficients. The present study derives an approximation to the distribution of the product of two dependent correlation coefficients with a closed form, resulting in a Pearson Type I distribution. A simulation study is also conducted to assess the accuracy of the approximation.     

An approximation to the exact distribution of the wilcoxon-mann-whitney rank sum test statistic

An approximation to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) and the Siegel-Tukey test based on a linear combination of the two-sample t-test applied to ranks and the normal approximation is compared with the usual normal approximation. The normal approximation results in a conservative test in the tails while the linear combination of the test statistics provides a test that has a very high percentage of agreement with tables of the exact distribution. Sample sizes 3≤m, n≤50 were considered.     

Nonnull distribution of Wilks' statistic for Manova in the complex case

In this paper, the exact distribution of Wilks' likelihood ratio criterion, A, for MANOVA, in the complex case when the alternate hypothesis is of unit rank (i.e. the linear case) has been derived and the explicit expressions for the same for p = 2 and 3 (where p is the number of variates) and general f₁ (the error degrees of freedom) and f₂ (the hypothesis degrees of freedom), are given. For an unrestricted number of variables, a general form of the density and the distribution of A in this case, is also given. It has been shown that the total integral of the series obtained by taking a few terms only, rapidly approaches the theoretical value one as more terms are taken into account, and some percentage points have also been computed.     

On asymptotic approximation of inverse moments for a class of nonnegative random variables

Sung [On inverse moments for a class of nonnegative random variables. J Inequal Appl. 2010;2010:1–13. Article ID 823767, doi:10.1155/2010/823767] obtained the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite second moments and satisfying a Rosenthal-type inequality. In the paper, we further study the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite first moments, which generalizes and improves the corresponding ones of Wu et al. [Asymptotic approximation of inverse moments of nonnegative random variables. Statist Probab Lett. 2009;79:1366–1371], Wang et al. [Exponential inequalities and inverse moment for NOD sequence. Statist Probab Lett. 2010;80:452–461; On complete convergence for weighted sums of ? mixing random variables. J Inequal Appl. 2010;2010:1–13, Article ID 372390, doi:10.1155/2010/372390], Sung (2010) and Hu et al. [A note on the inverse moment for the nonnegative random variables. Commun Statist Theory Methods. 2012. Article ID 673677, doi:10.1080/03610926.2012.673677].     

Testing for a trend in exponential means: A two-moment approximation to the null distribution of the likelihood-ratio statistic

The likelihood-ratio test statistic for testing homogeneity of exponential means with an ordered alternative has a rather complex null distribution. Expressions for the mean and variance of its null distribution are derived, and the accuracy of a two-moment chi-squared approximation is studied. The coefficients needed to implement the approximation are tabled. The application of these results in testing for a constant versus a nondecreasing intensity in a nonhomogeneous Poisson process is also discussed.     

A weighted Poisson distribution and its application to cure rate models

The family of weighted Poisson distributions offers great flexibility in modeling discrete data due to its potential to capture over/under-dispersion by an appropriate selection of the weight function. In this paper, we introduce a flexible weighted Poisson distribution and further study its properties by using it in the context of cure rate modeling under a competing cause scenario. A special case of the new distribution is the COM-Poisson distribution which in turn encompasses the Bernoulli, Poisson, and geometric distributions; hence, many of the well-studied cure rate models may be seen as special cases of the proposed model. We focus on the estimation, through the maximum likelihood method, of the cured proportion and the properties of the failure time of the susceptibles/non cured individuals; a profile likelihood approach is also adopted for estimating the parameters of the weighted Poisson distribution. A Monte Carlo simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data set.     

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.     

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.     

Approximations to the distribution of weighted combination of independent probabilities

Two simple approximations are proposed for the distribution of the weighted combina-tion of n independent probabilities. The approximations are compared with other avail-able approximations. It is shown that one of the proposed approximations is better than the other approximations.