Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions<Superscript>∗</Superscript>

We improve bounds of accuracy of the normal approximation to the distribution of a sum of independent random variables under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. We extend these results to Poisson binomial, binomial, and Poisson random sums. Under the same conditions, we obtain bounds for the accuracy of approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, we construct these bounds for the accuracy of approximation of the distributions of geometric, negative binomial, and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma, and Student distributions, respectively.     

Approximating Probability Distributions Using Small Sample Spaces

We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group ?. The quality of the approximating distribution is characterized by a parameter ɛ which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximating distribution be of size polynomial in and 1/ɛ. Such approximations are useful in reducing or eliminating the use of randomness in certain randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range , we provide an efficient construction of a good approximation. The approximation constructed has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the approximating distribution as it does under the uniform distribution over . Our analysis is based on Weil's character sum estimates. We apply this result to the construction of a non-binary linear code where the alphabet symbols appear almost uniformly in each non-zero code-word. Received: September 22, 1990/Revised: First revision November 11, 1990; last revision November 10, 1997     

Translated Poisson Approximation for Markov Chains

The paper is concerned with approximating the distribution of a sum W of integer valued random variables Y _i , 1 ≤ i ≤ n, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y _i ’s between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good.     

Strong approximations of semimartingales by processes with independent increments

Summary Strong approximation theorems for continuous time semimartingales are obtained by combining some techniques of the general theory of stochastic processes with some of the direct approximation of dependent random variables by independent ones. Continuous processes with independent increments whose variance functions increase polynomially or exponentially are considered as approximating processes. The basic assumptions of the main results only contain rates of convergence for certain probabilities. In particular, moment assumptions are not required. Some almost sure invariance principles for partial sum processes with nonlinear growth of variance and for functionals of Markov processes are derived by applying the main results.     

Normally Distributed Activity Durations in PERT Networks

It is assumed that activity durations in a PERT network are independent and normal random variables. A simple method of obtaining lower and upper bounds for the expected project completion time is described. The tightness of the bounds is examined for some numerical examples. A problem of applying the method in the case of non-normal distributions of activity times is also discussed.     

Improved lower bounds on the total variation distance for the Poisson approximation

New lower bounds on the total variation distance between the distribution of a sum of independent Bernoulli random variables and the Poisson random variable (with the same mean) are derived via the Chen–Stein method. The new bounds rely on a non-trivial modification of the analysis by Barbour and Hall (1984) which surprisingly gives a significant improvement. A use of the new lower bounds is addressed.     

COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein’s identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.     

The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near‐exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent‐induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near‐exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method. Copyright © 2011 John Wiley & Sons, Ltd.     

Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

New bounds for the \(k\)th-order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature. We apply these bounds and local approach couplings to obtain an order \(n^{-(p-1)/2}\) bound, for smooth test functions, for the distance between the distribution of a standardised sum of independent and identically distributed random variables and the standard normal distribution when the first \(p\) moments of these distributions agree. We also obtain a bound on the convergence rate of a sequence of distributions to the normal distribution when the moment sequence converges to normal moments.     

Asymptotic Independence and Additive Functionals

A strong approximation result is proved for the partial sum process of i.i.d. sequence of vectors having dependent components, where the components of the approximating process are independent. This result is applied for additive functionals of random walks in one and two dimensions.     

Error bounds in approximations of random sums using gamma-type operators

In this work we deal with approximations of compound distributions, that is, distribution functions of random sums. More specifically, we obtain a discrete compound distribution by replacing each summand in the initial random sum by a discrete random variable whose probability mass function is related to a well-known inversion formula for Laplace transforms [cf. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. II, second edn. Wiley, New York]. Our aim is to show the advantages that this method has in the context of compound distributions. In particular we give accurate error bounds for the distance between the initial random sum and its approximation when the individual summands are mixtures of gamma distributions.     

Series representations for densities functions of a family of distributions—Application to sums of independent random variables

Series representations for several density functions are obtained as mixtures of generalized gamma distributions with discrete mass probability weights, by using the exponential expansion and the binomial theorem. Based on these results, approximations based on mixtures of generalized gamma distributions are proposed to approximate the distribution of the sum of independent random variables, which may not be identically distributed. The applicability of the proposed approximations are illustrated for the sum of independent Rayleigh random variables, the sum of independent gamma random variables, and the sum of independent Weibull random variables. Numerical studies are presented to assess the precision of these approximations.     

Fast calibrations of the forward search for testing multiple outliers in regression

The paper considers the problem of testing for multiple outliers in a regression model and provides fast approximations to the null distribution of the minimum deletion residual used as a test statistic. Since direct simulation of each combination of number of observations and number of parameters is too time consuming, methods using simple normal samples are described for approximating the pointwise distribution of the test statistic. One approximation is based on adjustments to the results of simple simulations. The other uses properties of order statistics from folded t distributions to move outside the significance levels available by simulation. Analyses of data with beta errors and of transformed data on survival times demonstrate the usefulness in graphical methods of the inclusion of our bounds.     

Heavy-traffic analysis of the sojourn time in a three node Jackson network with overtaking

We consider a Jackson network consisting of three first-in-first-out (FIFO)M/M/1 queues. When customers leave the first queue they can be routed to either the second or third queue. Thus, a customer that traverses the network by going from the first to the second to the third queue, can be overtaken by another customer that is routed from the first queue directly to the third. We study the distribution of the sojourn time of a customer through the three node network, in the heavy traffic limit. A three term heavy traffic asymptotic approximation to the sojourn time density is derived. The leading term shows that the nodes decouple in the heavy traffic limit. The next two terms, however, do show the dependence of the sojourn times at the individual nodes and give quantitative measures of the effects of overtaking.     

Transient flows in queueing systems via closure approximations

We present a method for obtaining approximations to the distribution of flow times of customers in arbitrary queueing systems. We first propose approximations for uni-variate and multi-variate distributions of non-negative random variables. Then using a closure approximation, we show that the distribution of flow time can be calculated recursively. Computational results for the single server, multi-server and tandem queues are encouraging, with less than 5%average error in the mean flow time in most cases. The average error in the variance of flow times is found to be less than 10% for the more regular distributions.     

线性模型中M检验原假设分布的随机加权逼近

在线性模型中M-方法可以用于线性假设检验, 其中M检验、Wald检验和Rao的计分型检验是最常用的检验准则. 但是在计算这些检验的临界值时都涉及到未知参数的估计. 在本文中我们利用随机加权的方法来逼近这些检验的原假设分布. 结果表明在原假设和局部对立假设之下随机加权统计量的渐近分布与原检验统计量在原假设之下的渐近分布相同. 因此我们不需要对冗余参数进行估计,利用随机加权的方法就可以得到这些检验的临界值. 而且在局部对立假设之下可以实现对功效的计算. 当取不同的误差分布和不同的随机权时, 我们对本文的方法进行了蒙特卡洛模拟. 结果表明用随机加权方法来逼近原假设分布是非常精确的.     

Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models

We consider a counting processes with independent inter-arrival times evaluated at a random end of observation time T, independent of the process. For instance, this situation can arise in a queueing model when we evaluate the number of arrivals after a random period which can depend on the process of service times. Provided that T has log-convex density, we give conditions for the inter-arrival times in the counting process so that the observed number of arrivals inherits this property. For exponential inter-arrival times (pure-birth processes) we provide necessary and sufficient conditions. As an application, we give conditions such that the stationary number of customers waiting in a queue is a log-convex random variable. We also study bounds in the approximation of log-convex discrete random variables by a geometric distribution.     

On the asymptotics of locally dependent point processes

We investigate a family of approximating processes that can capture the asymptotic behaviour of locally dependent point processes. We prove two theorems presented to accommodate respectively the positively and negatively related dependent structures. Three examples are given to illustrate that our approximating processes can circumvent the technical difficulties encountered in compound Poisson process approximation (see Barbour and Månsson (2002) [10]) and our approximation error bound decreases when the mean number of the random events increases, in contrast to the increasing of bounds for compound Poisson process approximation.     

On Mann–Whitney tests for comparing sojourn time distributions when the transition times are right censored

We consider the problem of comparing sojourn time distributions of a transient state in a general multistate system in two samples (groups) when the transition times are right censored. Using the reweighting principle, a two-sample Mann–Whitney type of $U$ -statistic is constructed that compares only the uncensored sojourn times from the two distributions. A second Mann–Whitney type of statistic is also constructed using a different reweighting that allows for comparisons when one of the two sojourn times is either uncensored or singly censored. Both these statistics are asymptotically unbiased, asymptotically normally distributed and reduce to the standard Mann–Whitney statistic when there is no censoring. A test of equality of sojourn time distributions in two independent samples is constructed by symmetrizing the second statistic. The testing methodology is illustrated using a data set on kidney disease patients.     

Semi-Markov Modelling for Multi-State Systems

In this work we focus on multi state systems that we model by means of semi-Markov processes. The sojourn times are seen to be independent not identically distributed random variables and assumed to belong to a general class of distributions that includes several popular reliability distributions like the exponential, Weibull, and Pareto. We obtain maximum likelihood estimators of the parameters of interest and investigate their asymptotic properties. Plug-in type estimators are furnished for various quantities related to the system under study.