Taka-aki Shiraishi

Asymptotic equivalence of statistical inference based on aligned ranks and on within-block ranks

Abstract Rank tests for the null hypothesis of no treatment effect and rank-estimators of treatment effects, based on aligned ranks and on within-block ranks, are proposed for multiresponse experiments in two-way layouts without interaction, having one or more observations in each cell. Large sample properties of the tests and the estimators as cell sizes tend to infinity are investigated. It is shown that the aligned rank tests are asymptotically power-equivalent to the Friedman-type tests (within-block rank tests) and that the two R-estimators have asymptotically the same normal distribution. Further for the univariate case, it is found that the asymptotic relative efficiency (ARE) of the proposed rank test (R-estimator) with respect to the parametric F-test (parametric estimator) is equivalent to the classical ARE-result of the two-sample rank test with respect to the t-test and additionally asymptotically maximin power tests and minimax variance estimators due to Huber (1981) can be drawn.

Rank analogues of the likelihood ratio test for an ordered alternative in a two-way layout

SummaryDistribution-free tests for no treatment effect against the simple order alternative in a two-way layout with equal number of observations per cell are considered. The nonparametric test statistics are constructed by the rank analogues of the likelihood ratio test statistic assuming normality (i) based on within-block rankings and (ii) based on combined rankings of all the observations after alignment within each block. The exact distributions are given and large sample properties are investigated. The asymptotic power of the test (i) as the number of observations per eell tends to infinity can be satisfied enough, and in the case that the number of blocks tends to infinity, the asymptotic power of the test (ii) is almost higher than that of the test (i). Also these rank tests are compared with linear rank tests and it is shown that these proposed tests are robust by a table.

R-Estimators and confidence regions for treatment effects in multi-response experiments

A class of rank estimators of treatment effects in a two-way MANOVA model without interaction is considered. Asymptotic distributions of the R-estimators as the number of blocks tends to infinity are derived. Based on the R-estimators and their asymptotic distributions, confidence regions for the treatment effects are proposed. The asymptotic relative efficiencies of the nonparametric procedures relative to the classical procedures based on unbiased estimators are investigated.

Multivariate multi-sample rank tests for location-scale alternatives

A distribution-free test problem for the null hypothesis of equality of multivariate distributions versus location-scale alternatives in a multivariate multi-sample data setting is considered. Asymptotic distributions of random vectors of multivariate linear rank statistics are respectively investigated under the null hypothesis and under a contiguous sequence of the location-scale alternatives and, based on these asymptotic distributions, rank tests which have high asymptotic power are proposed. Each of the proposed test statistics is defined in terms of a sum of multivariate rank test statistics against location alternatives and statistics against scale alternatives, similar to the approach used by Lepage (1971). The limiting distributions are respectively X 2 - and noncentral x 2-distributions under the null hypothesis and under the alternatives. It is shown that the proposed tests are asymptotically power-equivalent to rank tests proposed by shen (1986). Methods of assigning optium scores and the comp...

R-estimators and confidence regions in one-way MANOVA

Abstract R-estimators and confidence regions for additive treatment constants, grand mean and population means in the one-way MANOVA model are proposed and asymptotic distributions of these procedures are investigated. The asymptotic relative efficiencies (AREs) of the proposed procedures relative to classical procedures based on least squares estimators (LSE) are studied. The R-methods are illustrated with a real data set. By numerical studies, it is domonstrated that the variation of the R-procedures due to the influence of outliers is less then that of the LSE.

The asymptotic power of rank tests under scale-al ternatives including contaminated distributions

SummaryThe alternative hypothesis of translated scale for the classical non-parametric hypothesis of equality of two distribution functions in the two-sample problem is extended to a scale-alternative including contamination. The asymptotic power of rank tests and the two-sampleF-test under contiguous sequences of the alternatives is derived and asymptotic relative efficiency of these rank tests with respect to theF-test is investigated. It is found that some of the rank tests have reasonably high asymptotic powers satisfied enough.

Local powers of two-sample and multi-sample rank tests for Lehmann's contaminated alternative

SummaryLocal powers of two- andk-sample rank tests under alternatives of contaminated distributions are investigated. It is shown that the rank tests based on normal scores and Wilcoxon scores are superior to thet-test or theF-test for many choices of alternatives of contaminated distributions and that the values of the asymptotic relative efficiency of the rank test based on Wilcoxon scores with respect to the normal scores are about one in all the investigated cases.

Semi-aligned rank tests

SummaryThe distribution-free test based on semi-aligned rankings for no treatment effects in a two-way layout, with unequal number of replications in each cell is considered. The asymptotic χ-square distribution of the test statistic under the null hypothesis is derived. The Pitman asymptotic relative efficiency of the test (i) based on semi-aligned rankings with respect to the test (ii) based on within-block rankings, is shown to be larger than one as the number of blocks tends to infinity. Also the asymptotic properties of linear rank statistics (i) and (ii) are investigated and the asymptotic relative efficiency of the test (i) with respect to the test (ii) is again shown to be larger than one.

R-estimators and confidence regions for main effects in a two–factor manova

A class of rank estimators of relative main effects in a two–factor MANOVA is constructed. Asymptotic distributions of the R–estimators as cell sizes tend to infinity are drawn. Based on the R–estimators and on the asymptotic distributions, confidence regions for the relative main effects and estimators of contrasts among the main effects are proposed and discussed. For proportional frequencies and a univariate case, it is seen that asymptotic relative efficiencies (AREs) of the nonparametric procedures relative to classical unbiased estimators agree with the classical ARE-results of two-sample rank tests relative to the ttest. Additionally for the univariate case, robust choices of score functions due to Huber (1964) are studied.

An Asymptotic Acceptance of Aligned Rank Tests under Alternatives of Contaminated Distributions in a Randomized-Blocks Design

Abstract Asymptotic noncentral X 2 distributions of aligned rank test statistics under a contiguous sequence of alternatives of contaminated distributions are obtained as the number of blocks tends to infinity in a randomized-blocks design with one observation per cell. I show that the asymptotic relative efficiency of the aligned rank test, with respect to the likelihood ratio F test under the sequence of these alternatives, is one in the case of normal scores and is nearly equal to one in the case of linear scores when a distribution of a null hypothesis is normal. As numerical results, the asymptotic powers of the aligned rank tests are superior to those of the Friedman-type tests and the Anderson test.