Moment-Based Approximations of Probability Mass Functions with Applications Involving Order Statistics

Provost, Serge B.; Jiang, Min; Ha, Hyung-Tae

doi:10.1080/03610920902835052

Cited by 8 publications

(8 citation statements)

References 8 publications

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“…results of the time-varying queues with periodic Poisson arrivals by Lemoine [36] only present the moments of some statistics such as workload and waiting time), the perfect sampling method provides direct solutions for the probability mass of the queue length and tail probabilities in a simulation based way without having to approximate based upon moment-based methods (c.f. Provost et al [46]). …”

Section: Problems To Be Solvedmentioning

confidence: 99%

Perfect and nearly perfect sampling of work-conserving queues

2015

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We present sampling-based methods to treat work-conserving queueing systems. A variety of models are studied. Besides the First Come First Served (FCFS) queues, many efforts are putted on the accumulating priority queue (APQ), where a customer accumulates priority linearly while waiting. APQs have Poisson arrivals, multi-class customers with corresponding service durations, and single or multiple servers.Perfect sampling is an approach to draw a sample directly from the steady-state distribution of a Markov chain without explicitly solving for it. Statistical inference can be conducted without initialization bias. If an error can be tolerated within some limit, i.e. the total variation distance between the simulated draw and the stationary distribution can be bounded by a specified number, then we get a so called "nearly" perfect sampling.Coupling from the past (CFTP) is one approach to perfect sampling, but it usually requires a bounded state space. One strategy for perfect sampling on unbounded state spaces relies on construction of a reversible dominating process. If only the dominating property is guaranteed, then regenerative method (RM) becomes an alternative choice. In the case where neither the reversibility nor dominance hold, a nearly perfect sampling method will be proposed. It is a variant of dominated CFTP that we call the CFTP Block Absorption (CFTP-BA) method.Time-varying queues with periodic Poisson arrivals are being considered in this thesis. It has been shown that a particular limiting distribution can be obtained for each point in time in the periodic cycle. Because there are no analytical solutions in closed forms, we explore perfect (or nearly perfect) sampling of these systems.

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Section: Problems To Be Solvedmentioning

confidence: 99%

Perfect and nearly perfect sampling of work-conserving queues

2015

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“…Several methods have been proposed to determine the optimal degree, see for instance Ha and Provost (2007) and Provost et al (2009). A suitable degree for a density approximation can be determined by a de visu inspection of the density plots of approximants of successive degrees, as explained in Ha and Provost (2007).…”

Section: The Gaussian-polynomial Approximationmentioning

confidence: 99%

“…More specifically, one would be satisfied that a density approximant of degree d is adequate if no noticeable differences between approximants of successive orders are observed. Provost et al (2009) proposed making use of measures of discrepancy such as the integrated squared differences between density or distribution approximants of successive degrees. Entropy minimization could also be considered as an alternative for optimal degree selection.…”

Section: The Gaussian-polynomial Approximationmentioning

confidence: 99%

Numerical Comparisons for the Null Distribution of the Bagai Statistic

2012

Communications for Statistical Applications and Methods

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Bagai et al. (1989) proposed a distribution-free test for stochastic ordering in the competing risk model, and recently Murakami (2009) utilized a standard saddlepoint approximation to provide tail probabilities for the Bagai statistic under finite sample sizes. In the present paper, we consider the Gaussian-polynomial approximation proposed in Ha and Provost (2007) and compare it to the saddlepoint approximation in terms of approximating the percentiles of the Bagai statistic. We make numerical comparisons of these approximations for moderate sample sizes as was done in Murakami (2009). From the numerical results, it was observed that the Gaussianpolynomial approximation provides comparable or greater accuracy in the tail probabilities than the saddlepoint approximation. Unlike saddlepoint approximation, the Gaussian-polynomial approximation provides a simple explicit representation of the approximated density function. We also discuss the details of computations.

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“…and Provost et al (2009) used a beta-polynomial approximant to the target distribution. In this paper, we utilize a special type of general semi parametric approach (the normal-polynomial approximation) proposed in Ha and Provost (2007).…”

Section: Ansari-bradley Statisticmentioning

confidence: 99%

“…Its main concept and computing procedures were proposed in Ha and Provost (2007). While Provost et al (2009) explained that beta-polynomial approximation provides high accuracy for Ansari-Bradley statistic, this paper aims to show that normal-polynomial approximation is also very flexible and fast to adapt the features of the target distributions. We observed that the normal polynomial approximant requires only four moments in most cases to provide accuracy.…”

Section: Introductionmentioning

confidence: 99%

Polynomially Adjusted Normal Approximation to the Null Distribution of Ansari-Bradley Statistic

Ha¹,

Yang²

2011

Korean Journal of Applied Statistics

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The approximation for the distribution functions of nonparametric test statistics is a significant step in statistical inference. A rank sum test for dispersions proposed by Ansari and Bradley (1960), which is widely used to distinguish the variation between two populations, has been considered as one of the most popular nonparametric statistics. In this paper, the statistical tables for the null distribution of the nonparametric Ansari-Bradley statistic is produced by use of polynomially adjusted normal approximation as a semi parametric density approximation technique. Polynomial adjustment can significantly improve approximation precision from normal approximation. The normal-polynomial density approximation for Ansari-Bradley statistic under finite sample sizes is utilized to provide the statistical table for various combination of its sample sizes. In order to find the optimal degree of polynomial adjustment of the proposed technique, the sum of squared probability mass function(PMF) difference between the exact distribution and its approximant is measured. It was observed that the approximation utilizing only two more moments of Ansari-Bradley statistic (in addition to the first two moments for normal approximation provide) more accurate approximations for various combinations of parameters. For instance, four degree polynomially adjusted normal approximant is about 117 times more accurate than normal approximation with respect to the sum of the squared PMF difference.

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Moment-Based Approximations of Probability Mass Functions with Applications Involving Order Statistics

Cited by 8 publications

References 8 publications

Perfect and nearly perfect sampling of work-conserving queues

Perfect and nearly perfect sampling of work-conserving queues

Numerical Comparisons for the Null Distribution of the Bagai Statistic

Polynomially Adjusted Normal Approximation to the Null Distribution of Ansari-Bradley Statistic

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