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 Multi-Sample Block-Scalar Sphericity Test: Exact and Near-Exact Distributions for Its Likelihood Ratio Test Statistic

In this article the authors show how by adequately decomposing the null hypothesis of the multi-sample block-scalar sphericity test it is possible to obtain the likelihood ratio test statistic as well as a different look over its exact distribution. This enables the construction of well-performing near-exact approximations for the distribution of the test statistic, whose exact distribution is quite elaborate and non-manageable. The near-exact distributions obtained are manageable and perform much better than the available asymptotic distributions, even for small sample sizes, and they show a good asymptotic behavior for increasing sample sizes as well as for increasing number of variables and/or populations involved.     

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.     

A Family of Near-Exact Distributions Based on Truncations of the Exact Distribution for the Generalized Wilks Lambda Statistic

For the case where at least two sets have an odd number of variables we do not have the exact distribution of the generalized Wilks Lambda statistic in a manageable form, adequate for manipulation. In this article, we develop a family of very accurate near-exact distributions for this statistic for the case where two or three sets have an odd number of variables. We first express the exact characteristic function of the logarithm of the statistic in the form of the characteristic function of an infinite mixture of Generalized Integer Gamma distributions. Then, based on truncations of this exact characteristic function, we obtain a family of near-exact distributions, which, by construction, match the first two exact moments. These near-exact distributions display an asymptotic behaviour for increasing number of variables involved. The corresponding cumulative distribution functions are obtained in a concise and manageable form, relatively easy to implement computationally, allowing for the computation of virtually exact quantiles. We undertake a comparative study for small sample sizes, using two proximity measures based on the Berry-Esseen bounds, to assess the performance of the near-exact distributions for different numbers of sets of variables and different numbers of variables in each set.     

On two asymptotic normal distributions for the generalized wilks lambda statistic

In this paper we.present a Normal asymptotic distribution for the logarithm of the generalized Wilks Lambda statistic based on an asymptotic distribution for the determinant of a Wishart matrix. This distribution is obtained through the combined use of Taylor expansions of random variables whose exponentials have chi-square distributions and the Lindeberg-Feller version of the Central Limit Theorem, Another asymptotic Normal distribution for the logarithm of the generalized Wilks Lambda statistic for the case when at most one of the sets has an odd number of variables is derived directly from the exact distribution. Both distributions are non-degenerate and non-singular. The first Normal distribution compares favorably with other known approximations and asymptotic distributions namely for large numbers of variables and small sample sizes, while the second Normal distribution, which has a more restricted application, compares in most cases highly favorably with other known asymptotic distributions and approximations. Finally, a method to compute approximate quantiles which lay very close and converge steadily to the exact ones is presented.     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

A general approximation to quantiles

For many continuous distributions, a closed-form expression for their quantiles does not exist. Numerical approximations for their quantiles are developed on a distribution-by-distribution basis. This work develops a general approximation for quantiles using the Taylor expansion. Our method only requires that the distribution has a continuous probability density function and its derivatives can be derived to a certain order (usually 3 or 4). We demonstrate our unified approach by approximating the quantiles of the normal, exponential, and chi-square distributions. The approximation works well for these distributions.     

Hypothesis testing for quantiles of two-parameter binary models

A procedure is developed to test the equality of the quantiles from k populations, assuming the responses follow a two-parameter binary model.The method utilizes the asymptotic distribution of the maximum likelihood estimators.The exact distribution of the test statistic is discussed in general.This exact distribution is generated for the logit model in order to investgate the convergence properties of the asymptotic procedure.     

General Saddlepoint Approximation for Testing Separate Location-Scale Families of Hypotheses

For testing separate families of hypotheses, the likelihood ratio test does not have the usual asymptotic properties. This paper considers the asymptotic distribution of the ratio of maximized likelihoods (RML) statistic in the special case of testing separate scale or location-scale families of distributions. We derive saddlepoint approximations to the density and tail probabilities of the log of the RML statistic. These approximations are based on the expansion of the log of the RML statistic up to the second order, which is shown not to depend on the location and scale parameters. The resulting approximations are applied in several cases, including normal versus Laplace, normal versus Cauchy, and Weibull versus log-normal. Our results show that the saddlepoint approximations are satisfactory, even for fairly small sample sizes, and are more accurate than normal approximations and Edgeworth approximations, especially for tail probabilities that are the values of main interest in hypothesis testing problems.     

Saddlepoint Approximation for Sample Quantiles with Some Applications

In this article, we use the integral form of the binomial distribution to derive saddlepoint approximations for sample quantiles. As an application, we present the calculation of the tail probability of the empirical log-likelihood ratio statistic for quantiles. Simulation results are also given to show that our approximations are extremely accurate.     

Asymptotic distributions of the sphericity test in a complex multivariate normal distribution

In this paper, asymptotic expansions of the null and non-null distributions of the sphericity test criterion in the case of a complex multivariate normal distribution are obtained for the first time in terms of beta distributions. In the null case, it is found that the accuracy of the approximation by taking the first term alone in the asymptotic series is sufficient for practical purposes. In fact for p - 2. the asymptotic expansion reduces to the first term which is also the exact distribution in this case. Applications of the results to the area of inferences on multivariate time series are also given.     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.     

Distributions of lrt associated with two-parameter exponential distributions

In this paper, an exact distribution of the likelihood ratio criterion for testing the equality of p two-parameter exponential distributions is obtained for unequal sample sizes in a computational form. A useful asymptotic expansion of the distribution is also obtained up to the order of n^-4 with the second term of the order of n^-3 and so can be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n where n is the combined sample size. In fact the first term alone which is a single beta distribution provides a powerful approximation for moderately large values of n.     

Asymptotic behaviour of M‐estimators in AR(p) models under nonstandard conditions

The authors derive the limiting distribution of M‐estimators in AR(p) models under nonstandard conditions, allowing for discontinuities in score and density functions. Unlike usual regularity assumptions, these conditions are satisfied in the context of L₁‐estimation and autoregression quantiles. The asymptotic distributions of the resulting estimators, however, are not generally Gaussian. Moreover, their bootstrap approximations are consistent along very specific sequences of bootstrap sample sizes only.     

Asymptotic approximations for the distributions of the <Emphasis Type="Italic">K</Emphasis>ø-divergence goodness-of-fit statistics

Kø-divergence’s statistic family for goodness-of-fit, under the null hypothesis, has an asymptotic chi-squared distribution; however, for small samples, the chi-squared approximation in some cases does not well agree with the exact distribution. In this paper, a closer approximation to the exact distribution is obtained by extracting the ø-dependent second order component from the distribution. Moreover, numerical results are presented for moderate sample sizes with moderate number of cells.     

NONNULL DISTRIBUTION OF LIKELIHOOD RATIO CRITERION FOR TESTING MULTISAMPLE SPHERICITY IN THE COMPLEX CASE

In this paper, nonnull moments of the likelihood ratio statistic for testing multisample sphericity in the complex case have been derived in series involving zonal polynomials. The nonnull asymptotic distribution of the statistic is also derived for certain alternatives.     

Empirical likelihood intervals for the population mean and quantiles based on balanced ranked set samples

We consider nonparametric interval estimation for the population mean and quantiles based on a ranked set sample. The asymptotic distributions of the empirical log likelihood ratio statistic for the mean and quantiles are derived. Interval estimates of the population mean and quantiles are obtained by inverting the likelihood ratio statistic. Simulations are carried out to investigate and compare the performance of the empirical likelihood intervals with other known intervals.     

Confidence and Likelihood*

Confidence intervals for a single parameter are spanned by quantiles of a confidence distribution, and one‐sided p‐values are cumulative confidences. Confidence distributions are thus a unifying format for representing frequentist inference for a single parameter. The confidence distribution, which depends on data, is exact (unbiased) when its cumulative distribution function evaluated at the true parameter is uniformly distributed over the unit interval. A new version of the Neyman–Pearson lemma is given, showing that the confidence distribution based on the natural statistic in exponential models with continuous data is less dispersed than all other confidence distributions, regardless of how dispersion is measured. Approximations are necessary for discrete data, and also in many models with nuisance parameters. Approximate pivots might then be useful. A pivot based on a scalar statistic determines a likelihood in the parameter of interest along with a confidence distribution. This proper likelihood is reduced of all nuisance parameters, and is appropriate for meta‐analysis and updating of information. The reduced likelihood is generally different from the confidence density. Confidence distributions and reduced likelihoods are rooted in Fisher–Neyman statistics. This frequentist methodology has many of the Bayesian attractions, and the two approaches are briefly compared. Concepts, methods and techniques of this brand of Fisher–Neyman statistics are presented. Asymptotics and bootstrapping are used to find pivots and their distributions, and hence reduced likelihoods and confidence distributions. A simple form of inverting bootstrap distributions to approximate pivots of the abc type is proposed. Our material is illustrated in a number of examples and in an application to multiple capture data for bowhead whales.     

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.     

Computing Critical Values of Exact Tests by Incorporating Monte Carlo Simulations Combined with Statistical Tables

Various exact tests for statistical inference are available for powerful and accurate decision rules provided that corresponding critical values are tabulated or evaluated via Monte Carlo methods. This article introduces a novel hybrid method for computing p‐values of exact tests by combining Monte Carlo simulations and statistical tables generated a priori. To use the data from Monte Carlo generations and tabulated critical values jointly, we employ kernel density estimation within Bayesian‐type procedures. The p‐values are linked to the posterior means of quantiles. In this framework, we present relevant information from the Monte Carlo experiments via likelihood‐type functions, whereas tabulated critical values are used to reflect prior distributions. The local maximum likelihood technique is employed to compute functional forms of prior distributions from statistical tables. Empirical likelihood functions are proposed to replace parametric likelihood functions within the structure of the posterior mean calculations to provide a Bayesian‐type procedure with a distribution‐free set of assumptions. We derive the asymptotic properties of the proposed nonparametric posterior means of quantiles process. Using the theoretical propositions, we calculate the minimum number of needed Monte Carlo resamples for desired level of accuracy on the basis of distances between actual data characteristics (e.g. sample sizes) and characteristics of data used to present corresponding critical values in a table. The proposed approach makes practical applications of exact tests simple and rapid. Implementations of the proposed technique are easily carried out via the recently developed STATA and R statistical packages.