Approximation of power in multivariate analysis

We consider the calculation of power functions in classical multivariate analysis. In this context, power can be expressed in terms of tail probabilities of certain noncentral distributions. The necessary noncentral distribution theory was developed between the 1940s and 1970s by a number of authors. However, tractable methods for calculating the relevant probabilities have been lacking. In this paper we present simple yet extremely accurate saddlepoint approximations to power functions associated with the following classical test statistics: the likelihood ratio statistic for testing the general linear hypothesis in MANOVA; the likelihood ratio statistic for testing block independence; and Bartlett's modified likelihood ratio statistic for testing equality of covariance matrices.     

Phi-Divergence Statistics for Testing Linear Hypotheses in Logistic Regression Models

In this paper we introduce and study two new families of statistics for the problem of testing linear combinations of the parameters in logistic regression models. These families are based on the phi-divergence measures. One of them includes the classical likelihood ratio statistic and the other the classical Pearson's statistic for this problem. It is interesting to note that the vector of unknown parameters, in the two new families of phi-divergence statistics considered in this paper, is estimated using the minimum phi-divergence estimator instead of the maximum likelihood estimator. Minimum phi-divergence estimators are a natural extension of the maximum likelihood estimator.     

A Simple Derivation of r* for Curved Exponential Families

For curved exponential families we consider modified likelihood ratio statistics of the form r_L=r+ log( u/r)/r , where r is the signed root of the likelihood ratio statistic. We are testing a one-dimensional hypothesis, but in order to specify approximate ancillary statistics we consider the test as one in a series of tests. By requiring asymptotic independence and asymptotic normality of the test statistics in a large deviation region there is a particular choice of the statistic u which suggests itself. The derivation of this result is quite simple, only involving a standard saddlepoint approximation followed by a transformation. We give explicit formulas for the statistic u , and include a discussion of the case where some coordinates of the underlying variable are lattice.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Testtesting for the equality of scale parameters of k(> 2) exponential populations based on complete and type ii censored samples

This paper is concerned with testing the equality of scale parameters of K(> 2) two-parameter exponential distributions in presence of unspecified location parameters based on complete and type II censored samples. We develop a marginal likelihood ratio statistic, a quadratic statistic (Qu) (Nelson, 1982) based on maximum marginal likelihood estimates of the scale parameters under the null and the alternative hypotheses, a C(a) statistic (CPL) (Neyman, 1959) based on the profile likelihood estimate of the scale parameter under the null hypothesis and an extremal scale parameter ratio statistic (ESP) (McCool, 1979). We show that the marginal likelihood ratio statistic is equivalent to the modified Bartlett test statistic. We use Bartlett's small sample correction to the marginal likelihood ratio statistic and call it the modified marginal likelihood ratio statistic (MLB). We then compare the four statistics, MLBi Qut CPL and ESP in terms of size and power by using Monte Carlo simulation experiments. For the variety of sample sizes and censoring combinations and nominal levels considered the statistic MLB holds nominal level most accurately and based on empirically calculated critical values, this statistic performs best or as good as others in most situations. Two examples are given.     

A comparison of tests of homogeneity for sparse contingency tables

Five tests of homogeneity for a 2x(k+l) contingency table are compared using Monte Carlo techniques. For these studiesit is assumed that k becomes large in such a way that thecontingency table is sparse for 2xk of the cells, but the sample size in two of the cells remains large. The test statistics studied are: the chi-square approximation to the Pearson test statistic, the chi-square approximation to the likelihood ratio statistic, the normal approximation to Zelterman's (1984)the normal approximation to Pearson's chi-square, and the normal approximation to the likelihood ratio statistic. For the range of parameters studied the chi-square approximation to Pearson's statistic performs consistently well with regard to its size and power.     

Iso‐chi‐squared testing of 2 × k: ordered tables

The Pearson chi‐squared statistic for testing the equality of two multinomial populations when the categories are nominal is much less appropriate for ordinal categories. Test statistics typically used in this context are based on scorings of the ordinal levels, but the results of these tests are highly dependent on the choice of scores. The authors propose a test which naturally modifies the Pearson chi‐squared statistic to incorporate the ordinal information. The proposed test statistic does not depend on the scores and under the null hypothesis of equality of populations, it is asymptotically equivalent to the likelihood ratio test against the alternative of two‐sided likelihood ratio ordering.     

Tests of Symmetry in Three-dimensional Contingency Tables Based on Phi-divergence Statistics

In this paper we introduce a family of test statistics for testing complete symmetry in three-dimensional contingency tables based on phi- divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistics as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to show that some new statistics offer an attractive alternative to the classical Pearson and likelihood ratio test statistics for this problem of complete symmetry.     

Hypothesis testing for the dispersion parameter of the hyper-Poisson regression model

The Poisson distribution is widely used to deal with count data, however, it is unable to capture the dispersion problems. The hyper-Poisson distribution is a particular case of the extended Conway–Maxwell distribution which takes into account the dispersion phenomena of the count data. The main motivation to consider this model is the possibility to link the mean to the regressor variables in very natural way to solve testing problems. So, this paper will be focalized in the gradient statistics to detect dispersions and to compare with the classical likelihood ratio statistic. Two illustrative applications are considered.     

Testing equality of correlation coefficients for paired binary data from multiple groups

Paired binary data arise naturally when paired body parts are investigated in clinical trials. One of the widely used models for dealing with this kind of data is the equal correlation coefficients model. Before using this model, it is necessary to test whether the correlation coefficients in each group are actually equal. In this paper, three test statistics (likelihood ratio test, Wald-type test, and Score test) are derived for this purpose. The simulation results show that the Score test statistic maintains type I error rate and has satisfactory power, and therefore is recommended among the three methods. The likelihood ratio test is over conservative in most cases, and the Wald-type statistic is not robust with respect to empirical type I error. Three real examples, including a multi-centre Phase II double-blind placebo randomized controlled trial, are given to illustrate the three proposed test statistics.     

On unification of the asymptotic theory of nonlinear econometric models

After reading a few articles in the nonlinear econonetric literature one begins to notice that each discussion follows roughly the same lines as the classical treatment of maximum likelihood estimation. There are some technical problems having to do with simultaneously conditioning on the exogenous variables and subjecting the true parameter to a Pittman drift which prevent the use of the classical methods of proof but the basic impression of similarity is correct . An estimator – be it nonlinear least squares, three – stage nonlinear least squares, or whatever – is the solution of an optimization problem. And the objective function of the optimization problem can be treated as if it were the likelihood to derive the Wald test statistic, the likelihood ratio test statistic , and Rao's efficient score statistic. Their asymptotic null and non – null distributions can be found using arguments fairly similar to the classical maximum likelihood arguments. In this article we exploit these observations and unify much of the nonlinear econometric literature. That which escapes this unificationis that which has an objective function which is not twice continuously differentiable with respect to the parameters – minimum absolute deviations regression for example.

The model which generates the data need not bethe same as the model which was presumed to define the optimization problem. Thus, these results can be used to obtain the asymptotic behavior of inference procedures under specification error We think that this will prove to be the nost useful feature of the paper. For example, it i s not necessary toresortto Monte Carlo simulat ionto determine i f a Translog estimate of an elasticity of sub stitution obtained by nonlinear three-stage least squares is robust against a CES truestate of nature. The asymptotic approximations we give here w ill provide an analytic answer to the question, sufficiently accurate for most purposes.     

Power-Divergence Test Statistics for Testing Linear by Linear Association

Power-divergence test statistics have been considered to test linear by linear association for two-way contingency tables. These test statistics have been compared based on designed simulation study and asymptotic results for 2 × 2, 2 × 3, and 3 × 3 tables. According to the results, there are test statistics with better properties than the well-known likelihood ratio test statistic for small and moderate samples.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

Tests for symmetry about an unknown value based on the empirical distribution function

A class of Kolmogorov-Smirnov and Cramér-von Mises type statistics for testing symmetry about an unknown value is described. These statistics are not distribution-free, however, and, indeed, are not readily amenable to calculation. A linear rank statistic analog of the first component of the Cramér-von Mises type statistic is investigated. Asymptotic non-null properties of these procedures in the normal case are studied, and an efficiency comparison of the Cramér-vonMises statistic, the linear rank statistic analog, the modified Wil-coxon statistic, and the likelihood ratio test is reported.     

Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics

We consider the problem of detecting a ‘bump’ in the intensity of a Poisson process or in a density. We analyze two types of likelihood ratio‐based statistics, which allow for exact finite sample inference and asymptotically optimal detection: The maximum of the penalized square root of log likelihood ratios (‘penalized scan’) evaluated over a certain sparse set of intervals and a certain average of log likelihood ratios (‘condensed average likelihood ratio’). We show that penalizing the square root of the log likelihood ratio — rather than the log likelihood ratio itself — leads to a simple penalty term that yields optimal power. The thus derived penalty may prove useful for other problems that involve a Brownian bridge in the limit. The second key tool is an approximating set of intervals that is rich enough to allow for optimal detection, but which is also sparse enough to allow justifying the validity of the penalization scheme simply via the union bound. This results in a considerable simplification in the theoretical treatment compared with the usual approach for this type of penalization technique, which requires establishing an exponential inequality for the variation of the test statistic. Another advantage of using the sparse approximating set is that it allows fast computation in nearly linear time. We present a simulation study that illustrates the superior performance of the penalized scan and of the condensed average likelihood ratio compared with the standard scan statistic.     

A likelihood ratio like permutation test for one way designs

We consider permutation tests based on a likelihood ratio like statistic for the one way or k sample design used in an example in Kolassa and Robinson [(2011), ‘Saddlepoint Approximations for Likelihood Ratio Like Statistics with Applications to Permutation Tests’, Annals of Statistics, 39, 3357–3368]. We give explicitly the region in which the statistic exists, obtaining results which permit calculation of the statistic on the boundary of this region. Numerical examples are given to illustrate improvement in the power of the tests compared to the classical statistics for long-tailed error distributions and no loss of power for normal error distributions.     

Divergences and duality for estimation and test under moment condition models

We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood approach. We treat the problems of existence and characterization of the divergence projections of probability distributions on sets of signed finite measures. We give a precise characterization of duality, for the proposed class of estimates and test statistics, which is used to derive their limiting distributions (including the EL estimate and the EL ratio statistic) both under the null hypotheses and under alternatives or misspecification. An approximation to the power function is deduced as well as the sample size which ensures a desired power for a given alternative.     

Testing homogeneity in a mixture of von mises distributions with a structural parameter

The modified likelihood ratio statistic can be used to test the homogeneity in a variety of mixture models. Here, the authors propose the use of the modified and the iterative modified likelihood ratio for testing homogeneity against a two‐component von Mises mixture with a structural parameter. They derive the limiting distributions of the test statistics and propose methods to improve the accuracy of the asymptotic approximation in finite samples. Their simulations show that the tests maintain their nominal level and that they have adequate power. Data on movements of turtles are used as an illustration     

Comparison of two weibull distributions under random censoring

The two-sample problem for comparing Weibull scale parameters is studied for randomly censored data. Three different test statistics are considered and their asymptotic properties are established under a sequence of local alternatives, It is shown that both the test statistic based on the mlefs (maximum likelihood estimators) and the likelihood ratio test are asymptotically optimum. The third statistic based only on the number of failures is not, Asymptotic relative efficiency of this statistic is obtained and its numerical values are computed for uniform and Weibull censoring, Effects of uniform random censoring on the censoring level of the experiment are illus¬trated, A direct proof for the joint asymptotic normality of the mlefs of the shape and the scale parameters is also given     

On testing homogeneity of variances for nonnormal models using entropy

This paper deals with testing equality of variances of observations in the different treatment groups assuming treatment effects are fixed. We study the distribution of a test statistic which is known to perform comparably well with other statistics for the same purpose under normality. The statistic we consider is based on Shannon’s entropy for a distribution function. We will derive the asymptotic expansion for the distribution of the test statistic based on Shannon’s entropy under nonnormality and numerically examine its performance in comparison with the modified likelihood ratio criteria for normal and some nonnormal populations.