Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation

Abstract

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n(n > 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n ? 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.     

Better approximate confidence intervals for a binomial parameter

This paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x² distribution. The confidence limits of method IV are then provided by the Wilson-Hilferty approximation of the x². Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson-Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.     

Continuity corrected approximations for and ‘exact’ inference with Pearson's X2

The classical adjustments for the inadequacy of the asymptotic distribution of Pearson's X² statistic, when some cells are sparse or the cell expectations are small, use continuity corrections and exact moments; the recent approach is to use computer based ‘exact inference’. In this paper we observe that the original exact test due to Freeman and Halton (Biometrika 38 (1951), 141–149) and its computer implementation are theoretically unsound. Furthermore, the corrected algorithmic version for the exact p-value in StatXact is practically useful in very few cases, and the results of its present version which includes Monte Carlo estimates can be highly variable. We then derive asymptotic expansions for the moments of the null distribution of Pearson's X², introduce a new method of correcting for discreteness and finite range of Pearson's X² as an alternative to the classical continuity correction, and use them to construct new and improved approximations for the null distribution. We also offer diagnostic criteria applicable to the tables for selecting an appropriate approximation. The exact methods and the competing approximations are studied and compared using thirteen test cases from the literature. It is concluded that the accuracy of the appropriate approximation is comparable with the truly exact method whenever it is available. The use of approximations is therefore preferable if the truly exact computer intensive solutions are unavailable or infeasible.     

A new procedure for testing equivalence in comparative bioavailability and other clinical trials

The problem of testing for equivalence in clinical trials is restated here in terms of the proper clinical hypotheses and a simple classical frequentist significance test based on the central t distribution is derived. This method is then shown to be more powerful than the methods based on usual (shortest) and symmetric confidence intervals.

We begin by considering a noncentral t statistic and then consider three approximations to it. A simulation is used to compare actual test sizes to the nominal values in crossover and completely randomized designs. A central t approximation was the best. The power calculation is then shown to be based on a central t distribution, and a method is developed for obtaining the sample size required to obtain a specified power. For the approximations, a simulation compares actual powers to those obtained for the t distribution and confirms that the theoretical results are close to the actual powers.     

Bivariate symmetry tests for complete and competing risks data: a saddlepoint approach

A class of bivariate symmetry tests for complete data and competing risks data is considered. Saddlepoint approximation for the exact p-values of the underlying permutation distribution of these tests is derived. Several simulation studies are conducted to evaluate the performance of the saddlepoint approximation and the asymptotic approximation. The saddlepoint approximation was found to be highly accurate and superior to the asymptotic approximations in replicating the exact permutation significance.     

A simple approximation relation between the symmetric generalized logistic and the students t distributions

In this paper, we obtain a new approximation of the Student's t distribution by using the symmetric generalized logistic (SGL) distribution function. The error of this approximation is shown to be 0(1/n² )where nis the degrees of freedom of thetdistribution. In comparison to similar approximations by George and Ojo and George et al. (1986), this new approximation is much simpler and more accurate. It is also shown that under some conditions, the tdistribution is a good approximation of the SGL distribution. Therefore, the complicated expressions for the cumulants and moments of the SGL can be approximated by those of the t, distribution. Finally, numerical results are given.     

Improved Measures of the Spread of Data for Some Unknown Complex Distributions Using Saddlepoint Approximations

Measures of the spread of data for random sums arise frequently in many problems and have a wide range of applications in real life, such as in the insurance field (e.g., the total claim size in a portfolio). The exact distribution of random sums is extremely difficult to determine, and normal approximation usually performs very badly for this complex distributions. A better method of approximating a random-sum distribution involves the use of saddlepoint approximations.

Saddlepoint approximations are powerful tools for providing accurate expressions for distribution functions that are not known in closed form. This method not only yields an accurate approximation near the center of the distribution but also controls the relative error in the far tail of the distribution.

In this article, we discuss approximations to the unknown complex random-sum Poisson–Erlang random variable, which has a continuous distribution, and the random-sum Poisson-negative binomial random variable, which has a discrete distribution. We show that the saddlepoint approximation method is not only quick, dependable, stable, and accurate enough for general statistical inference but is also applicable without deep knowledge of probability theory. Numerical examples of application of the saddlepoint approximation method to continuous and discrete random-sum Poisson distributions are presented.     

Rejoinder

Abstract

In this article several formulae for the approximation of the critical values for tests on the actual values of the process capability indices CPL, CPU, and C_pk are provided. These formulae are based on different approximations of the percentiles of the noncentral t distribution and their performance is evaluated by comparing the values assessed through them from the exact critical values, for several significance levels, test values, and sample sizes. As supported by the obtained results, some of the presented techniques constitute valuable tools in situations where the exact critical values of the tests are not available, since one may approximate them readily and rather accurately through them.     

On the exact distribution of the sum of the largest n−k out of n normal random variables with differing mean values

Motivated by an application in Electrical Engineering, we derive the exact distribution of the sum of the largest n?k out of n normally distributed random variables, with differing mean values. Comparisons are made with two normal approximations to this distribution—one arising from the asymptotic negligibility of the omitted order statistics and one from the theory of L-statistics. The latter approximation is found to be in excellent agreement with the exact distribution.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

Range-preserving unbiased estimators in the poisson case

Abstract

The Kruskal–Wallis test is a popular nonparametric test for comparing k independent samples. In this article we propose a new algorithm to compute the exact null distribution of the Kruskal–Wallis test. Generating the exact null distribution of the Kruskal–Wallis test is needed to compare several approximation methods. The 5% cut-off points of the exact null distribution which StatXact cannot produce are obtained as by-products. We also investigate graphically a reason that the exact and approximate distributions differ, and hope that it will be a useful tutorial tool to teach about the Kruskal–Wallis test in undergraduate course.     

Exact-and approximate bayes credibility intervals for the one-way analysis of variance model

In most hierarchical Bayes cases the posterior distributions are difficult to derive and cannot be obtained in closed form. In some special cases, however, it is possible to obtain the exact moments of the posterior distributions.

By applying these moments and Pearson curves or Cornish-Fisher expansions to real problems, good approximations of the exact posterior distributions of individual parameter values as well as linear combinations of parameter values could easily be obtained.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

The Cornish-Fisher expansion for a class of statistics in first order autoregression

ABSTRACT

This article is concerned with the derivation and study of the Cornish-Fisher expansion for a wide class of estimators of the parameter in the first order autoregressive process. Second and third order Cornish-Fisher approximations to the quantile of the distribution of the corresponding asymptotically normal standardized statistic are stated explicitly and their accuracy is examined, both theoretically and numerically, by comparing them with the exact value of the quantile obtained by Monte Carlo simulation.     

A note on the most powerful invariant test of Rayleigh against exponential distribution

Abstract

The problem of testing Rayleigh distribution against exponentiality, based on a random sample of observations is considered. This problem arises in survival analysis, when testing a linearly increasing hazard function against a constant hazard function. It is shown that for this problem the most powerful invariant test is equivalent to the “ratio of maximized likelihoods” (RML) test. However, since the two families are separate, the RML test statistic does not have the usual asymptotic chi-square distribution. Normal and saddlepoint approximations to the distribution of the RML test statistic are derived. Simulations show that saddlepoint approximation is more accurate than the normal approximation, especially for tail probabilities that are the main values of interest in hypothesis testing.     

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.     

Exact calculation of power and sample size in bioequivalence studies using two one‐sided tests

The number of subjects in a pharmacokinetic two‐period two‐treatment crossover bioequivalence study is typically small, most often less than 60. The most common approach to testing for bioequivalence is the two one‐sided tests procedure. No explicit mathematical formula for the power function in the context of the two one‐sided tests procedure exists in the statistical literature, although the exact power based on Owen's special case of bivariate noncentral t‐distribution has been tabulated and graphed. Several approximations have previously been published for the probability of rejection in the two one‐sided tests procedure for crossover bioequivalence studies. These approximations and associated sample size formulas are reviewed in this article and compared for various parameter combinations with exact power formulas derived here, which are computed analytically as univariate integrals and which have been validated by Monte Carlo simulations. The exact formulas for power and sample size are shown to improve markedly in realistic parameter settings over the previous approximations. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic convergence in distribution for spearman's rank correlation coefficient

The Edgeworth expansion for the distribution function of Spearman's rank correlation coefficient may be used to show that the rates of convergence for the normal and Pearson type II approximations are l/nand l/n² respectively. Using the Edgeworth expansion up to terms involving the sixth moment of the exact distribution allows an approximation with an error of order l/n³.     

Linear approximate ML estimation in scaled Type I generalized logistic distribution based on Type-II censored samples

The scaled (two-parameter) Type I generalized logistic distribution (GLD) is considered with the known shape parameter. The ML method does not yield an explicit estimator for the scale parameter even in complete samples. In this article, we therefore construct a new linear estimator for scale parameter, based on complete and doubly Type-II censored samples, by making linear approximations to the intractable terms of the likelihood equation using least-squares (LS) method, a new approach of linearization. We call this as linear approximate maximum likelihood estimator (LAMLE). We also construct LAMLE based on Taylor series method of linear approximation and found that this estimator is slightly biased than that based on the LS method. A Monte Carlo simulation is used to investigate the performance of LAMLE and found that it is almost as efficient as MLE, though biased than MLE. We also compare unbiased LAMLE with BLUE based on the exact variances of the estimators and interestingly this new unbiased LAMLE is found just as efficient as the BLUE in both complete and Type-II censored samples. Since MLE is known as asymptotically unbiased, in large samples we compare unbiased LAMLE with MLE and found that this estimator is almost as efficient as MLE. We have also discussed interval estimation of the scale parameter from complete and Type-II censored samples. Finally, we present some numerical examples to illustrate the construction of the new estimators developed here.     

A new variables sampling plan for normally distributed lots with unknown standard deviation and double specification limits

We deal with single sampling by variables with two-way-protection in case of normally distributed characteristics with unknown variance. Givenp ₁(AQL),p ₂ (LQ) and α, β (risks of errors of the first and the second kind), there are two well-known methods of determining the corresponding sampling plans. Both methods are based on an approximation of the OC. Therefore these plans are only approximations, the true risks α and β are not known exactly. In section II we present a new sampling scheme based on an estimatorp for the percent defectivep. We give an exact formula for the OC. Thus we are able to determine these plans exactly without any approximations.