On the distribution of the likelihood ratio test of equality of normal populations

It has recently been shown by Perlman (1980) that when testing the equality of several normal distributions it is the likelihood ratio test which is unbiased rather than a test based on a modified statistic in common use. This paper gives expansions for the null distribution of the likelihood ratio statistic as well as for the nonnull distribution in a special case.     

On the distribution of lrt for testing the equality of exponential distributions

In this paper, an asymptotic expansion of the distribution' of the likelihood ratio criterion for testing the equality of p one-parameter exponential distributions is obtained for unequal sample sizes. The expansion is obtained up to the order of n^-3 with the second term of the order of n^-2 so that the first term of this expansion alone should provide an excellent approximation to the distribution for moderately large values of n, where n is the combined sample size.     

A modified likelihood ratio test for the mean direction in the von mises distribution

The likelihood ratio test (LRT) for the mean direction in the von Mises distribution is modified for possessing a common asymptotic distribution both for large sample size and for large concentration parameter. The test statistic of the modified LRT is compared with the F distribution but not with the chi-square distribution usually employed, Good performances of the modified LRT are shown by analytical studies and Monte Carlo simulation studies, A notable advantage of the test is that it takes part in the unified likelihood inference procedures including both the marginal MLE and the marginal LRT for the concentration parameter.     

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 multivariate von mises distribution with applications to bioinformatics

Motivated by problems of modelling torsional angles in molecules, Singh, Hnizdo & Demchuk (2002) proposed a bivariate circular model which is a natural torus analogue of the bivariate normal distribution and a natural extension of the univariate von Mises distribution to the bivariate case. The authors present here a multivariate extension of the bivariate model of Singh, Hnizdo & Demchuk (2002). They study the conditional distributions and investigate the shapes of marginal distributions for a special case. The methods of moments and pseudo‐likelihood are considered for the estimation of parameters of the new distribution. The authors investigate the efficiency of the pseudo‐likelihood approach in three dimensions. They illustrate their methods with protein data of conformational angles     

Asymptotic non-null distribution of a test connected with exponential populations

In this paper asymptotic expansions of the non-null distribution of the likelihood ratio criterion for testing the equality of several one parameter exponential distributions are obtained under local alternatives. These expansions are in terms of Chi-square distributions.     

Empirical likelihood ratio-based goodness-of-fit test for the logistic distribution

The logistic distribution has been used to model growth curves in survival analysis and biological studies. In this article, we propose a goodness-of-fit test for the logistic distribution based on the empirical likelihood ratio. The test is constructed based on the methodology introduced by Vexler and Gurevich [17 A. Vexler and G. Gurevich, Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy, Comput. Stat. Data Anal. 54 (2010), pp. 531–545. doi: 10.1016/j.csda.2009.09.025[Crossref], [Web of Science ®] , [Google Scholar]]. In order to compute the test statistic, parameters of the distribution are estimated by the method of maximum likelihood. Power comparisons of the proposed test with some known competing tests are carried out via simulations. Finally, an illustrative example is presented and analyzed.     

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.     

Jackknife empirical likelihood method for testing the equality of two variances

In this paper, we propose a nonparametric method based on jackknife empirical likelihood ratio to test the equality of two variances. The asymptotic distribution of the test statistic has been shown to follow χ² distribution with the degree of freedom 1. Simulations have been conducted to show the type I error and the power compared to Levene's test and F test under different distribution settings. The proposed method has been applied to a real data set to illustrate the testing procedure.     

On estimating the common mean in two normal distributions after a preliminary test for equality of variances

Given two random samples of equal size from two normal distributions with common mean but possibly different variances, we examine the sampling performance of the pre-test estimator for the common mean after a preliminary test for equality of variances. It is shown that when the alternative in the pretest is one-sided, the Graybill-Deal estimator is dominated by the pre-test estimator if the critical value is chosen appropriately. It is also shown that all estimators, the grand mean, the Graybill-Deal estimator and the pre-test estimator, are admissible when the alternative in the pre-test is two-sided. The optimal critical values in the two-sided pre-test are sought based on the minimax regret and the minimum average risk criteria, and it is shown that the Graybill-Deal estimator is most preferable under the minimum average risk criterion when the alternative in the pre-test is two-sided.     

Testing for the mean of the inverse gaussian distribution and adaptive estimation of parameters

This paper provides the modified likelihood ratio criterion for testing whether the mean of the inverse Gaussian distribution can be set to unity giving rise to Standard form of the Wald distribution. Estimates of probability of correct selection has been obtained on the basis of a Monte Carlo study of 1,000 samples. Finally a set of adaptive estimators for the parameters are proposed and studied on the basis of data generated from the two distributions.     

The mean of the conditional,truncated multinormal distribution

An expression is derived for the mean of the conditional, truncated multinormal distribution in the general case, and for the situation where the conditioning variates have iden¬tical correlation. This equicorrelated case occurs in some models of duration and in panel data. An example is taken from economics, where a cross section of firms consider the adoption of several technological innovations. The results of the paper are used to estimate the firm-specific.     

Testing the equality of multinomial populations ordered by increasing convexity under the alternative

The authors derive the asymptotic null distribution of the likelihood ratio statistic for testing equality of multinomial populations whose parameters are ordered by increasing convexity under the alternative. They also show how to compute critical values for the test.     

The efficiency of a test based on the asymptotic distribution of the mle for a linear functional relationship

A rigorous derivation is given of the asymptotic normality of the MLE of a linear functional relationship. Using these results, it is shown that the test proposed by VILLEGAS (1964) has Pitman efficiency zero w.r.t, a test based on the asymptotic distribution of the MLE.     

A new goodness of fit test for the equality of multinomial cell probabilities versus trend alternatives

In this paper, a new test statistic is presented for testing the null hypothesis of equal multinomial cell probabilities versus various trend alternatives. Exact asymptotic critical values are obtained, The power of the test is compared with several other statistics considered by Choulakian et al (1995), The test is shown to have better power for certain trend alternatives.     

Locally optimal test for simultaneously testing the mean and variance of a normal distribution

A generalization of the locally most powerful unbiased (LMPU) test for the single parameter case to the k-parameter case was proposed by SenGupta and Vermeire (1986). In particular we defined a locally most mean power unbiased (LMMPU) test based on the mean curvature of the power hypersurface. Compared to the type C tests of Neyman and Pearson and the type D tests (Isaacson, 1951), LMMPU tests possess better theoretical properties and enjoy ease of construction of critical regions. In this paper we present an interesting example of a two-parameter univariate normal population for which Isaacson (1951, p. 233) was unsuccessful in finding a type D test. For the case of one observation, we prove that no Type D region exists but the LMMPU test is obtained - it is an example of a test with singular Hessian matrix for its power but is nevertheless a strictly locally unbiased (LU) test.     

Multivariate Saddlepoint test for the wrapped normal model

In this article we show the effectiveness and the accuracy of the test statistic based on the expnnent of the saddlepoint approximation for the density of M-estimators, proposed by Robinson, Ronchetti and Young (1999), for testing simultaneous hypotheses on the mean and on the variance of a wrapped normal distribution. We base this test statistic on the trigonometric method of moments estimator proposed by Gatto and Jammalamadaka (l999b), which admits the M-estimator representation necessary for this test. This test statistic has an approximate chi-squared distribution, asympiotically up to the second order, and the high accuracy of this approximation is shown by numerical simulations.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.