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.     

Two normal approximations to the binomial distribution

Two new normal approximations are proposed for the cumulative binomial distribution when the mean-is reasonably large. Their adequacy is compared with that of certain well-known approximations- The first is recommended for its simplicity and accuracy relative to the standard and Gram-Charlier approximations. The second Is shown to be more accurate than all known approximations for a certain range of the probability of success.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

The likelihood ratio test for simple tree order: A useful asymptotic expansion

We derive new approximations for the likelihood ratio statistics that are used in testing hypotheses involving simple tree order. They are based on asymptotics when the number of populations tends to infinity and variances are equal.As an application quantiles are obtained that are much easier to calculate than exact critical values or the gamma-approximation usually proposed in literature. Our quantitles also provide often better approximations than those known in literature.     

Densities and distribution functions of the ratio of two linear functions and the product of Gamma variates: A saddlepoint approach

Saddlepoint approximations for the densities and the distribution functions of the ratio of two linear functions of gamma random variables and the product of gamma random variables are derived. Ratios of linear functions with positive and negative weights and non identical gamma variables are considered. The saddlepoint approximations are very accurate in the tails as in the center of the distribution. Extensive simulation studies are used to evaluate the accuracy of the proposed methods.     

Smooth functional tempering for nonlinear differential equation models

Differential equations are used in modeling diverse system behaviors in a wide variety of sciences. Methods for estimating the differential equation parameters traditionally depend on the inclusion of initial system states and numerically solving the equations. This paper presents Smooth Functional Tempering, a new population Markov Chain Monte Carlo approach for posterior estimation of parameters. The proposed method borrows insights from parallel tempering and model based smoothing to define a sequence of approximations to the posterior. The tempered approximations depend on relaxations of the solution to the differential equation model, reducing the need for estimating the initial system states and obtaining a numerical differential equation solution. Rather than tempering via approximations to the posterior that are more heavily rooted in the prior, this new method tempers towards data features. Using our proposed approach, we observed faster convergence and robustness to both initial values and prior distributions that do not reflect the features of the data. Two variations of the method are proposed and their performance is examined through simulation studies and a real application to the chemical reaction dynamics of producing nylon.     

A semiparametric maximum likelihood ratio test for the change point in copula models

In the present paper, a semiparametric maximum-likelihood-type test statistic is proposed and proved to have the same limit null distribution as the classical parametric likelihood one. Under some mild conditions, the limiting law of the proposed test statistic, suitably normalized and centralized, is shown to be double exponential, under the null hypothesis of no change in the parameter of copula models. We also discuss the Gaussian-type approximations for the semiparametric likelihood ratio. The asymptotic distribution of the proposed statistic under specified alternatives is shown to be normal, and an approximation to the power function is given. Simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the double exponential and Gaussian-type approximations.     

General saddlepoint approximations and normalizing transformations for multivariate statistics

The use of general saddlepoint approximations is investigated for the problem of approximating the tail probabilities of statistics in multivariate analysis. A method based on normalizing transformations is proposed to prevent po¬tential deficiencies in general saddlepoint approximations. The efficiency of the proposed method is illustrated through examples of the sample correlation     

An approach for finding fully Bayesian optimal designs using normal-based approximations to loss functions

The generation of decision-theoretic Bayesian optimal designs is complicated by the significant computational challenge of minimising an analytically intractable expected loss function over a, potentially, high-dimensional design space. A new general approach for approximately finding Bayesian optimal designs is proposed which uses computationally efficient normal-based approximations to posterior summaries to aid in approximating the expected loss. This new approach is demonstrated on illustrative, yet challenging, examples including hierarchical models for blocked experiments, and experimental aims of parameter estimation and model discrimination. Where possible, the results of the proposed methodology are compared, both in terms of performance and computing time, to results from using computationally more expensive, but potentially more accurate, Monte Carlo approximations. Moreover, the methodology is also applied to problems where the use of Monte Carlo approximations is computationally infeasible.     

Testing equality of ratios from two finite populations

Approximations are given for the bias and variance of both the regression and ratio estimator when sampling from a finite population, and simulation results are given indicating the accuracy of the approximations and the bias of the estimated approximations. A different estimator for the variance of the regression estimator is recommended. Test procedures are proposed for testing the hypothesis of equality of ratios from two finite populations, the procedures depending upon the types of populations being sampled. Simulation results indicating the effectiveness of the test procedures in controlling their size are given.     

Empirical likelihood for parameters in an additive partially linear errors-in-variables model with longitudinal data

Empirical likelihood inferences for the parameter component in an additive partially linear errors-in-variables model with longitudinal data are investigated in this article. A corrected-attenuation block empirical likelihood procedure is used to estimate the regression coefficients, a corrected-attenuation block empirical log-likelihood ratio statistic is suggested and its asymptotic distribution is obtained. Compared with the method based on normal approximations, our proposed method does not require any consistent estimator for the asymptotic variance and bias. Simulation studies indicate that our proposed method performs better than the method based on normal approximations in terms of relatively higher coverage probabilities and smaller confidence regions. Furthermore, an example of an air pollution and health data set is used to illustrate the performance of the proposed method.     

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.     

Selection of a better component in bivariate exponential models

Summary Selection procedures of the better component in bivariate exponential (BVE) models are proposed. In this paper, we consider onlyBVE models proposed by Freund (1961) Marshall-Olkin (1967) and Block-Basu (1974). The probabilities of correct selection for the proposed procedures are compared by using the normal approximations. A numerical study on the determination of asymptotic relative efficiency (ARE) of the proposed procedures are presented.     

Approximations for Multiple Scan Statistics on the Circle

Multiple scan statistic is usually used by epidemiologist to test the uniformity or clustering of data. In this article, we extend the work of Lin (1999) to give a general expression for the moments of multiple scan statistic on a circle, and use these moments to approximate its distribution using Markov chain and compound Poisson approximations proposed by Huffer and Lin (1997a) and Lin (1993). Numerical results are presented to evaluate the performance of these approximations.     

Density approximations and VaR computation for compound Poisson-lognormal distributions

Parametric approximations of the compound Poisson-lognormal distribution are developed and used to compute Value-at-Risk (VaR). As guidelines for finding an approximation, the skewness–kurtosis space and the tail behavior are considered. The Generalized Beta distribution of the second kind (GB2) and a mixture of lognormals are found to provide a good fit. In certain cases, the GB2 can be estimated by moment-matching, thus providing a simulation-free procedure for VaR computation. For confidence levels larger than 99%, extreme value theory approaches are developed. According to extensive Monte Carlo evidence, the proposed approximations are more efficient than crude Monte Carlo.     

Percentiles of sums of heavy-tailed random variables: beyond the single-loss approximation

A perturbative approach is used to derive approximations of arbitrary order to estimate high percentiles of sums of positive independent random variables that exhibit heavy tails. Closed-form expressions for the successive approximations are obtained both when the number of terms in the sum is deterministic and when it is random. The zeroth order approximation is the percentile of the maximum term in the sum. Higher orders in the perturbative series involve the right-truncated moments of the individual random variables that appear in the sum. These censored moments are always finite. As a result, and in contrast to previous approximations proposed in the literature, the perturbative series has the same form regardless of whether these random variables have a finite mean or not. For high percentiles, and specially for heavier tails, the quality of the estimate improves as more terms are included in the series, up to a certain order. Beyond that order the convergence of the series deteriorates. Nevertheless, the approximations obtained by truncating the perturbative series at intermediate orders are remarkably accurate for a variety of distributions in a wide range of parameters.     

Modeling Association Between Two or More Categorical Variables that Allow for Multiple Category Choices

Multiple-response (or pick any/c) categorical variables summarize responses to survey questions that ask “pick any” from a set of item responses. Extensions to loglinear model methodology are proposed to model associations between these variables across all their items simultaneously. Because individual item responses to a multiple-response categorical variable are likely to be correlated, the usual chi-square distributional approximations for model-comparison statistics are not appropriate. Adjusted statistics and a new bootstrap procedure are developed to facilitate distributional approximations. Odds ratio and standardized Pearson residual measures are also developed to estimate specific associations and examine deviations from a specified model.     

Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables

Abstract

Several approximations of copulas have been proposed in the literature. By using empirical versions of checker-type copulas approximations, we propose non parametric estimators of the copula. Under some conditions, the proposed estimators are copulas and their main advantage is that they can be sampled from easily. One possible application is the estimation of quantiles of sums of dependent random variables from a small sample of the multivariate law and a full knowledge of the marginal laws. We show that estimations may be improved by including in an easy way in the approximated copula some additional information on the law of a sub-vector for example. Our approach is illustrated by numerical examples.     

Approximations for the doubly noncentral-F distribution

The approximate normality of the cube root of the noncentral chi-square observed by Aty (1954) and an Edgeworth-series expansion are used to derive an approximation for the doubly noncentral-F distribution. Another approximation in terms of a noncentral-F distribution is also proposed. Both these approximations are seen to compare favorably with some earlier approximations due to Das Gupta (1968) and Tiku (1972). The problem of approximating the cumulants of the doubly noncentral-F variable, which is pivotal in Tiku’s approximation, is examined and use of a noncentral-F distribution is seen to provide a good solution for it. A FORTRAN routine for the Edgeworth-series approximation is given.     

First moment approximations for order statistics from the extreme value distribution

Approximate expressions of the first moment of the order statistics of standard extreme value distributions are proposed. We compare different previously given approximations with the exact values. The results show that the approximation given here fits the exact values better than previously given models.