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 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.     

Multi-Sample Likelihood Ratio Tests Based on Bipolar Watson Distributions Defined on the Hypersphere

We derive likelihood ratio tests for the equality of the directional parameters of k bipolar Watson distributions defined on the hypersphere with common concentration parameter. We analyze the power of these tests in the case of two distributions supposing in the alternative hypothesis two directional parameters forming an angle, which varies from 18° to 90°. We also compare the likelihood ratio tests with a high-concentration F-test.     

The Exact Likelihood Ratio Test for Equality of Two Normal Populations

Testing the equality of two independent normal populations is a perfect case of the two-sample problem, yet it is not treated in the main text of any textbook or handbook. In this article, we derive the exact distribution of the likelihood ratio test and implement this test with an R function. This article has supplementary materials online.     

Exact Likelihood Equations for Autoregression Models with Multivariate Elliptically Contoured Distributions

Abstract

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data.     

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 Likelihood Ratio Testing for Homogeneity of the Exponential Distribution

The aim of this article is to discuss homogeneity testing of the exponential distribution. We introduce the exact likelihood ratio test of homogeneity in the subpopulation model, ELR, and the exact likelihood ratio test of homogeneity against the two-components subpopulation alternative, ELR2. The ELR test is asymptotically optimal in the Bahadur sense when the alternative consists of sampling from a fixed number of components. Thus, in some setups the ELR is superior to frequently used tests for exponential homogeneity which are based on the EM algorithm (like the MLRT, ADDS, and D-tests). One important example of superiority of ELR and ELR2 tests is the case of lower contamination. We demonstrate this fact by both theoretical comparisons and simulations.     

Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations

It is well-known that maximum likelihood (ML) estimators of the two parameters in a gamma distribution do not have closed forms. This poses difficulties in some applications such as real-time signal processing using low-grade processors. The gamma distribution is a special case of a generalized gamma distribution. Surprisingly, two out of the three likelihood equations of the generalized gamma distribution can be used as estimating equations for the gamma distribution, based on which simple closed-form estimators for the two gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estimators. The study consolidates this conjecture by establishing the asymptotic behaviors of the new estimators. In addition, the closed-forms enable bias-corrections to these estimators. The bias-correction significantly improves the small-sample performance.     

A Note on the Poisson Likelihood Ratio Test Statistic for Kulldorff's Scan Methods

Methods based on scan statistics are widely used in health-related applications to detect clusters of disease. The most common methods are based on the Bernoulli and Poisson models. Kulldorff (1997 Kulldorff , M. ( 1997 ). A spatial scan statistic . Communications in Statistics—Theory and Methods 26 : 1481 – 1496 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) derived the likelihood ratio test statistic for his scan method for both of these models. His scan statistic is widely used with freely available software, SaTScan^? (see Kulldorff, 2005 Kulldorff , M. ( 2005 ). SaTScan: software for the spatial, temporal, and space-time scan statistics , version 5.1 [computer program]. Information Management Services 2005; Available: http://www.satscan.org/. Accessed July 2007 . [Google Scholar]). We provide an alternative derivation of the likelihood ratio test statistic in the Poisson case. Our derivation is simpler and more general in the sense that it applies when the incidences are not aggregated into subregional counts.     

Testing Homogeneity of Inverse Gaussian Scale Parameters Based on Generalized Likelihood Ratio

This article presents a new procedure for testing homogeneity of scale parameters from k independent inverse Gaussian populations. Based on the idea of generalized likelihood ratio method, a new generalized p-value is derived. Some simulation results are presented to compare the performance of the proposed method and existing methods. Numerical results show that the proposed test has good size and power performance.     

Likelihood Ratio Tests for High‐Dimensional Normal Distributions

In their recent work, Jiang and Yang studied six classical Likelihood Ratio Test statistics under high‐dimensional setting. Assuming that a random sample of size n is observed from a p‐dimensional normal population, they derive the central limit theorems (CLTs) when p and n are proportional to each other, which are different from the classical chi‐square limits as n goes to infinity, while p remains fixed. In this paper, by developing a new tool, we prove that the mentioned six CLTs hold in a more applicable setting: p goes to infinity, and p can be very close to n. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi‐square approximations and discussions are presented afterwards.     

Kolmogorov-Smirnov Test Statistic and Critical Values for the Erlang-3 and Erlang-4 Distributions

Following a procedure applied to the Erlang-2 distribution in a recent paper, an adjusted Kolmogorov-Smirnov statistic and critical values are developed for the Erlang-3 and -4 cases using data from Monte Carlo simulations. The test statistic produced features of compactness and ease of implementation. It is quite accurate for sample sizes as low as ten.     

Exact and Approximate Distributions of the Maximum Likelihood Estimate in a Simple Factor Analysis Model

We study a factor analysis model with two normally distributed observations and one factor. In the case when the errors have equal variance, the maximum likelihood estimate of the factor loading is given in closed form. Exact and approximate distributions of the maximum likelihood estimate are considered. The exact distribution function is given in a complex form that involves the incomplete Beta function. Approximations to the distribution function are given for the cases of large sample sizes and small error variances. The accuracy of the approximations is discussed     

A Likelihood Ratio Test for Idiosyncratic Unit Roots in the Exact Factor Model with Integrated Factors

We consider an exact factor model with unobservable common stochastic trends imposed by non-stationary factors, and study, by simulation, the power of the likelihood ratio test for unit roots in the idiosyncratic components. The power of the test is compared with the analogous Lagrange multiplier test and the Fisher-type test proposed by Bai and Ng. The results suggest that the benefit of the likelihood ratio test is in panels with a small cross-section.     

Use of the Likelihood Ratio Test on the Inverse Gaussian Distribution

In the following article, the likelihood ratio test is determined for four tests of hypotheses involving the inverse Gaussian distribution. For three of the hypotheses, the test produces the same statistic as the uniformly most powerful unbiased test.     

Bias Reduction for the Maximum Likelihood Estimator of the Parameters of the Generalized Rayleigh Family of Distributions

We derive analytic expressions for the biases, to O(n^{? 1}), of the maximum likelihood estimators of the parameters of the generalized Rayleigh distribution family. Using these expressions to bias-correct the estimators is found to be extremely effective in terms of bias reduction, and generally results in a small reduction in relative mean squared error. In general, the analytic bias-corrected estimators are also found to be superior to the alternative of bias-correction via the bootstrap.     

Application of the Generalized Likelihood Ratio Test for Detecting Changes in the Mean of Multivariate GARCH Processes

We derive several multivariate control charts to monitor the mean vector of multi-variate GARCH processes under the presence of changes, by means of maximizing the generalized likelihood ratio. This presentation is rounded up by a comparative performance study based on extensive Monte Carlo simulations. An empirical illustration shows how the obtained results can be applied to real data.     

One-Sample Likelihood Ratio Tests for Mixed Data

In this paper we present a procedure for finding the optimal order of a response polynomial. the procedure is based on the prediction distribution of future observations. The maximal length of the structural β - expectation tolerance region for each polynomial is calculated. The minimun of these maximal determines the optimal order of the response polynomial     

Simple and Exact Empirical Likelihood Ratio Tests for Normality Based on Moment Relations

The empirical likelihood (EL) technique is a powerful nonparametric method with wide theoretical and practical applications. In this article, we use the EL methodology in order to develop simple and efficient goodness-of-fit tests for normality based on the dependence between moments that characterizes normal distributions. The new empirical likelihood ratio (ELR) tests are exact and are shown to be very powerful decision rules based on small to moderate sample sizes. Asymptotic results related to the Type I error rates of the proposed tests are presented. We present a broad Monte Carlo comparison between different tests for normality, confirming the preference of the proposed method from a power perspective. A real data example is provided.