A consistent correlation-type goodness-of-fit test;with application to the two-parameter weibull distribution

In this article the properties of a general univariate JiT-sample rank tests for complete block designs are investigated. Especially, the asymptotic distribution of the test .statistic under H0 and under contiguous alternatives is derived. Some asymptotic relative'PITMAN efficiencies are computed.

AMSX 1980 subject classifications: Primary 62G10; secondary 62K10     

Testing the equality of variance-covariance matrices the robust way

The robust statistic T² _Dproposed by Tiku and Singh (1982) for testing the equality of mean vectors of two mu1 t ivariate populations is modified to test the equality of variance-covariance matrices.     

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.     

Comparison of powers for the sphericity tests using both the asymptotic distribution and the bootstrap

The asymptotic distributions of two tests for sphericity:the locally most powerful invariant test and the likelihood ratio test are derived under the general alternaties ∑?σ² I. The powers of these two tests are then compared when the data are from a trivariate normal population. The bootstrap method is also used to obtain the powers and the powers obtained by this method agree with those from the asymptotic distributions.     

On the role of minque in testing of hypotheses under hiked linear models

Testing of hypotheses under balanced ANOVA models is fairly simple and generally based on the usual ANOVA sums of squares. Difficulties may arise in special cases when these sums of squares do not form a complete sufficient statistic. There is a huge literature on this subject which was recently surveyed in Seifert's contribution to the book of Mumak (1904). But there are only a few results about unbalanced models. In such models the consideration of likelihood ratios leads to more complex sums of squares known from MINQUE theory.

Uniform optimality of testsusually reduces to local optimality. Here we prespnt a small review of methods proposed for testing of hypotheses in unbalanced models. where MINQUEI playb a major role. We discuss the use of iterated MINQUE for the construction of asymptotically optimal tests described in Humak (1984) and approximate tests based on locally uncorrelated linear combinations of MINQUE estimators by Seifert (1985), We show that the latter tests coincide with robust locally optimal invariant tests proposeci by Kariya and Sinha and Das and Sinha, if the number of variance components is two. Explicit expressions for corresponding tests are given for the unbalanced two-way cross classification random model, which covers some other models as special cases. A simulation study under lines the relevance of MINQUE for testing of hypotheses problems.     

Chi-squared tests in kth-moment sufficient dimension reduction

We present a novel approach to sufficient dimension reduction for the conditional kth moments in regression. The approach provides a computationally feasible test for the dimension of the central kth-moment subspace. In addition, we can test predictor effects without assuming any models. All test statistics proposed in the novel approach have asymptotic chi-squared distributions.     

A Comparative Study of Some Modified Chi-Squared Tests

Some recent results in the theory and applications of modified chi-squared goodness-of-fit tests are briefly discussed. It seems that for the first time power of modified chi-squared type tests for the logistic and three-parameter Weibull distributions based on moment type estimators is studied. Power of different modified tests against some alternatives for equiprobable fixed or random grouping intervals, and for Neyman–Pearson classes is investigated. It is shown that power of test statistic essentially depends on the quantity of Fisher's sample information this statistic uses. Some recommendations on implementing modified chi-squared type tests are given.     

A Nonparametric Test for Symmetry Based on Freeman and Halton's Ideas on Contingency Tables

In this article, we propose a nonparametric method to test for symmetry in bivariate data. By using the extension of Fisher's exact treatment for 2 × 2 contingency tables proposed by Freeman and Halton (1951 Freeman , G. H. , Halton , J. H. ( 1951 ). Note on an exact treatment of contingency tables, goodness of fit and other problems of significance . Biometrika 38 : 141 – 149 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), we can test the hypothesis of equal distribution for two samples of integer valued variables. Then, by counting the number of observations belonging to each cell of a symmetric, appropriately built grid, we can produce the two samples of integers required to use this test for equal distribution. The resulting test for symmetry is potentially extendible to higher dimensions. A simulation study is performed to compare with some known tests (Bowker, 1948 Bowker , A. H. ( 1948 ). A test for symmetry in contingency tables . Journal of the American Statistical Association 43 : 572 – 574 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Hollander, 1971 Hollander , M. ( 1971 ). A nonparametric test for bivariate symmetry . Biometrika 58-1 : 203 – 212 .[Crossref], [Web of Science ®] , [Google Scholar]; and its improvement given in Krampe and Kuhnt, 2007 Krampe , A. , Kuhnt , S. ( 2007 ). Bowker's test for symmetry and modifications within the algebraic framework . Computational Statistics and Data Analysis 51 : 4124 – 4142 .[Crossref], [Web of Science ®] , [Google Scholar]). Our proposal represents a competitive option as a test for symmetry.     

A Statistical Reanalysis of the Classical Rutherford's Experiment

Modified chi-squared and some newly developed tests for the Poisson, binomial, and an approximated Feller's distribution are discussed. A reanalysis of the classical Rutherford's experimental data on alpha decay is done. Previous analyses of the data were not correct from the point of view of the theory of statistical testing. Tests used show that the data contradict to both Poisson and binomial distribution and do not contradict to a precise “binomial” approximation of Feller's distribution that takes into account a counter's dead time. This gives a plausible statistically correct confirmation of the well-established exponential law of radioactive decay.     

Unbiased and upper critical values of mean trace of multivariate beta for testing difference of two covariance matrices or several mean vectors

For the invariant unbiased level-α test of equality of two co-variance matrices, the quantities b and B satisfying the equations P(b≤T≤B) = 1-α, E(T|b≤T≤B) = E(T), where T is the mean trace of a multivariate beta, are required. Five and one per cent values of B are tabulated for m = 2,3(2)11,16; b can be obtained from B. Upper five and one per cent values of T are also included, as these are required for the locally most powerful invariant test of nullity of any source of difference in several mean vectors and the locally most powerful invariant one-sided test of equality of two covariance matrices. Lower critical values may be obtained from upper critical values.