Distributions of exact tests in the exponential family

The aim of this paper is to give some results on the exact density of the I-divergence in the exponential family with gamma distributed observations. It is shown in particular that the I-divergence can be decomposed as a sum of two independent variables with known distributions. Since the considered I-divergence is related to the likelihood ratio statistics, we apply the method to compute the exact distribution of the likelihood ratio tests and discuss the optimality of such exact tests. One of these tests is the exact LR test of the model which is asymptotically optimal in the Bahadur sense. Numerical examples are provided to illustrate the methods discussed. Received: January 2002 Acknowledgements. I am grateful to Prof. Andrej Pázman for helpful discussions during the setup and the preparation of the paper and to the referees for constructive comments on earlier versions of the paper. Research is supported by the VEGA grant (Slovak Grant Agency) No 1/7295/20     

Optimal tests and estimators for truncated exponential families

Blackwell-Rao-Lehmann-Scheffe' theory is used to derive the minimum variance unbiased estimators for the functions of scale and truncation parameters as well as the reliability function of the truncated exponential family distribution. Uniformly most powerful unbiased tests of hypotheses are formulated. Finally, a particular model of this family, viz., the truncated exponential model is discussed.     

Simultaneous estimation of hazard rates of several exponential populations

Suppose independent random samples are drawn from k (2) populations with a common location parameter and unequal scale parameters. We consider the problem of estimating simultaneously the hazard rates of these populations. The analogues of the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU) and the best scale equivariant (BSE) estimators for the one population case are improved using Rao‐Blackwellization. The improved version of the BSE estimator is shown to be the best among these estimators. Finally, a class of estimators that dominates this improved estimator is obtained using the differential inequality approach.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

Some tests of analytical multiobjective decision-making methods

Many analytical methods have been developed for multiobjective decision problems. All attempt to achieve the same goal. This paper reports on tests of five decision rules and seven weighting methods applied by three different panels to determine whether or not they produce different results and to infer, where possible, the nature and origin of the differences.Different methods do produce different results. Lacking a definition of “goodness,” however, it is not clear how much these differences matter. Nevertheless, some methods have obvious problems, either in theoretical validity or in application, and they should be avoided.     

Some robust exact results on sample autocorrelations and tests of randomness

Several exact results on the second moments of sample autocorrelations, for both Gaussian and non-Gaussian series, are presented. General formulae for the means, variances and covariances of sample autocorrelations are given for the case where the variables in a sequence are exchangeable. Bounds for the variances and covariances of sample autocorrelations from an arbitrary random sequence are derived. Exact and explicit formulae for the variances and covariances of sample autocorrelations from a Gaussian white noise are given. It is observed that the latter results hold for all spherically symmetric distributions. A simulation experiment, with Gaussian series, indicates that normalizing each sample autocorrelation with its exact mean and variance, instead of the usual approximate moments, can improve considerably the accuracy of the asymptotic N(0,1) distribution to obtain critical values for tests of randomness. The exact second moments of rank autocorrelations are also studied.     

Some tests of alternative urban population density functions

The purpose of this study is to test the usefulness of some alternative gross population density functions for urbanized areas. Two evaluation criteria are used; maximum explanatory power in standard regression analysis and accuracy in predicting total population in the urbanized area. It is concluded that the explanatory power of the negative exponential function can be improved upon in some cases by adding a quadratic distance term, but that population can be predicted more accurately if the quadratic term is omitted. Also, it is found that constraining the negative exponential function to predict population exactly reduces explanatory power by an insignificant amount.     

Some remarks on the average sample number of sequential and non-sequential tests

Summary We prove some inequalities concerning the (mean) number of observations needed to attain a prescribed error sum and apply them to some examples.     

Detecting a shift in location: Some robust tests

The test statistic W, suggested by Hawkins (1977) and Worsley (1979) for testing a sequence of observations for a shift in location, is sensitive to the assumption that the distribution of the population being sampled is normal. A modification of this statistic is proposed which is robust against ‘heavy tailed’ distributions. We also study the performance of several alternative testing procedures by means of a simulation experiment. The Farley-Hinich test (1975) and the Homogeneity test discussed in Brown, Durbin and Evans (1975) perform well in the Simulation experiment against ‘heavy tailed’ distributions. Some applications to economic data are also discussed in the paper.     

Some tests for the constancy of regressions under heteroscedasticity

For testing the equality of coefficients of a linear regression model under heteroscedasticity, we suggest an F criterion conditioned on the posterior mean of the ratio of standard deviations of error terms in two subsamples. For pairable subsamples, and exact F test is derived. Sampling experiments show that the Chow test differs substantially from the nominal significance level when the two subsample sizes are unequal, and that the F test conditioned on the posterior mean is superior to other tests when sample sizes are small.     

Locally most powerful sequential tests for processes of the exponential class with stationary and independent increments

Sequential tests that are locally most powerful (LMP) for certain one-sided problems in the case of processes of the exponential class with stationary and independent increments are discussed. After having proved the existence of an LMP sequential test in the above setting each LMP test is shown to be a Wald SPRT for a family of paired (conjugate) simple hypotheses.     

Some rank tests for one–sided many–one comparisons

We compare some nonparametric tests for the (/+ 1)–sample problem with additive effects under the constraint that in every sample the treatment effect is not less than that in the first sample, i.e. of some control. The behavior of the Pitman efficiency of the respective tests (essentially tests of a Kruskal–Wallis–, Wilcoxon–, Fligner–Wolfe–, Steel–, and Nemenyi–type) is discussed which turns out to depend on the level and power of the tests as well as on the directions, from which the alternative tends to the hypothesis. It will be shown that none of the tests under consideration is uniformly superior to the others.     

Matrix exponential GARCH

I propose a new multivariate GARCH specification that maintains positive definiteness of the conditional covariance matrix. The idea is to specify the dynamics in the matrix logarithm of the conditional covariance. Because the matrix exponential transformation ensures positive definiteness, the dynamics can be specified without the positive definiteness constraint. This affords a variety of specifications and, in particular, we can specify element-by-element the dynamics of the matrix logarithm. I discuss specifications with leverage effects, estimation with multivariate Gaussian and t-distributions, and diagnostics that evaluate the appropriateness of the matrix exponential specification.     

Some simple structure significance tests for exploratory component analysis with market survey data

Several uncertainties continue to hinder researchers using exploratory factor analysis. Among other decisions, the choice of the number of factors to retain and the interpretability of the rotated solution remain problematical. In this paper, we suggest a simple structure criterion as a general framework for choosing the number of factors and for interpreting the rotated factor structure matrix. The proposed methodology is demonstrated with demographic data from a consumer panel. The results show the importance of increasing the information yield from exploratory factor analysis.     

On the exponential law

A characterization of the exponential law is given using its property thatE[(X–a)^r|X>a] is constant for alla0,r being a positive integer. Then this result is applied to the order statistics and some further characterizations are found.     

A characterization of exponential distributions based on order statistics

This note contains a characterization of exponential distributions based on the properties of linear transformations of order statistics. This is a certain converse of a well known theorem of Rényi about the distribution of linear combinations of order statistics from exponential distributions. Some statistical applications of the result are indicated.