Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.     

A directional look at F‐tests

Directional testing of vector parameters, based on higher order approximations of likelihood theory, can ensure extremely accurate inference, even in high‐dimensional settings where standard first order likelihood results can perform poorly. Here we explore examples of directional inference where the calculations can be simplified, and prove that in several classical situations, the directional test reproduces exact results based on F‐tests. These findings give a new interpretation of some classical results and support the use of directional testing in general models, where exact solutions are typically not available. The Canadian Journal of Statistics 47: 619–627; 2019 © 2019 Statistical Society of Canada     

Optimum invariant tests for random manova models

Consider the canonical-form MANOVA setup with X: n × p = (+ E, X_i n_i × p, i = 1, 2, 3, M_i: n_i × p, i = 1, 2, n₁ + n₂ + n₃) p, where E is a normally distributed error matrix with mean zero and dispersion I_n (> 0 (positive definite). Assume (in contrast with the usual case) that M₁i is normal with mean zero and dispersion I_n1) and M₂2 is either fixed or random normal with mean zero and different dispersion matrix I_n2 (being unknown. It is also assumed that M₁ E, and M₂ (if random) are all independent. For testing H₀) = 0 versus H₁: (> 0, it is shown that when either n₂ = 0 or M₂ is fixed if n₂ > 0, the trace test of Pillai (1955) is uniformly most powerful invariant (UMPI) if min(n₁, p)= 1 and locally best invariant (LBI) if min(n₁ p) > 1 underthe action of the full linear group Gl (p). When p > 1, the LBI test is also derived under a somewhat smaller group G_T(p) of p × p lower triangular matrices with positive diagonal elements. However, such results do not hold if n₂ > 0 and M₂ is random. The null, nonnull, and optimality robustness of Pillai's trace test under Gl(p) for suitable deviations from normality is pointed out.     

Locally optimal invariant tests for change in level

The classical change point problem is considered, from the invariance point of view. Locally optimal invariant tests are derived for the change in level, when the initial level and the common variance are assumed to be unknown. The tests derived by Chernoff and Zacks (1964) and Gardner (1969), for the change in level, when variance is known, are shown to be locally optimal invariant tests.     

Admissible tests for the mean with additional information

Let X be a po-normal random vector with unknown µ and unknown covariance matrix ∑ and let X be partitioned as X = (X^′ ₍₁₎, …, X^′ _(r))′ where X_(j)is a subvector of X with dimension p_jsuch that ∑^r _j=1Pj = P0. Some admissible tests are derived for testing H₀: μ = 0 versus H₁: μ ¦0 based on a sample drawn from the whole vector X of dimension p and r additional samples drawn from X₍₁₎, X₍₂₎, …, X(r) respectively, All (r+1) samples are assumed to be independent. The distribution of some of the tests' statistics involved are also derived.     

Locally best tests for exponential distributions with interval-censored data

In this article, we derive a locally best test for testing the mean of exponential distributions with interval-censored samples. This locally best test possesses certain optimality. It is of unbiasedness and equivalent to a likelihood ratio test in some circumstances, and it is also a Bayes test for some loss function. For the locally best test, the associated critical values and powers at a nominal level of significance are provided. For a large sample size case, asymptotic critical values and powers are also calculated and tabulated. Moreover, based on the locally best test, a curtailed test is proposed. This curtailed test is equivalent to the locally best test on the acceptance or rejection of the null hypothesis. A Monte Carlo simulation is carried out to illustrate the performance of the curtailed test compared with the locally best test. Numerical results show that the experimental duration time of the curtailed test is substantially smaller than that of the locally best test.     

Efficiency of t-Test and Hotelling's T 2-Test After Box-Cox Transformation

Early investigations of the effects of non-normality indicated that skewness has a greater effect on the distribution of t-statistic than does kurtosis. When the distribution is skewed, the actual p-values can be larger than the values calculated from the t-tables. Transformation of data to normality has shown good results in the case of univariate t-test. In order to reduce the effect of skewness of the distribution on normal-based t-test, one can transform the data and perform the t-test on the transformed scale. This method is not only a remedy for satisfying the distributional assumption, but it also turns out that one can achieve greater efficiency of the test. We investigate the efficiency of tests after a Box-Cox transformation. In particular, we consider the one sample test of location and study the gains in efficiency for one-sample t-test following a Box-Cox transformation. Under some conditions, we prove that the asymptotic relative efficiency of transformed t-test and Hotelling's T ²-test of multivariate location with respect to the same statistic based on untransformed data is at least one.     

Three Kolmogorov-Smirnov-type one-sample tests with improved power properties

Some simple point estimators are proposed for the three-parameter Weibull distribution. Both complete and type II censored sampling are considered. The biases and variances of these estimators are studied by Monte Carlo simulation.

Percentage points for the estimator of the shape parameter are also obtained by Monte Carlo simulation, which enables interval estimation and tests of hypotheses to be carried out for the shape parameter.     

Optimal designs for the logit and probit models for binary data

We consider optimal designs for a class of symmetric models for binary data which includes the common probit and logit models. We show that for a large group of optimality criteria which includes the main ones in the literature (e.g. A-, D-, E-, F- and G-optimality) the optimal design for our class of models is a two-point design with support points symmetrically placed about the ED50 but with possibly unequal weighting. We demonstrate how one can further reduce the problem to a one-variable optimization by characterizing various of the common criteria. We also use the results to demonstrate major qualitative differences between the F - and c-optimal designs, two design criteria which have similar motivation.     

Pearson-type goodness-of-fit tests for regression

A procedure for testing the goodness of fit of linear regression models is introduced. For a given partition of the real line into cells, the proposed test is a quadratic form based on the vector of observed minus expected frequencies of the residuals obtained by maximum-likelihood estimation of the regression parameters. The quadratic form is of the same computational difficulty as the traditional Pearson-type tests with uncensored data. A statistic based on only one cell is particularly easy to apply and is used for testing the normality assumption in a real data set from astronomy. A simulation study examines the finite-sample properties of the proposed tests.     

A class of distribution-free tests for two-way layout

In this article a class of distribution-free tests for the hypothesis of no row (treatment) effect in a two-way layout design, with several observations per cell, is proposed. The tests are based on U-statistics, constructed by considering minima of all possible subsamples of same size from each cell.The proposed class of tests is compared with the parametric test, Mack and Skillings test and Yate's test for two-way layout, in terms of Pitman ARE sense. It is seen that for the case of equal number of observations per cell, the proposed tests have better efficiency for exponential and uniform error distributions.     

A Comparative Study of Different Methodologies for Fault Diagnosis in Multivariate Quality Control

Different methodologies for fault diagnosis in multivariate quality control have been proposed in recent years. These methods work in the space of the original measured variables and have performed reasonably well when there is a reduced number of mildly correlated quality and/or process variables with a well-conditioned covariance matrix. These approaches have been introduced by emphasizing their positive or negative virtues, generally on an individual basis, so it is not clear for the practitioner the best method to be used. This article provides a comprehensive study of the performance of diverse methodological approaches when tested on a large number of distinct simulated scenarios. Our primary aim is to highlight key weaknesses and strengths in these methods as well as clarifying their relationships and the requirements for their implementation in practice.     

Generalized cochran-mantel-haenszel test statistics for correlated categorical data

Three new test statistics are introduced for correlated categorical data in stratified R×C tables. They are similar in form to the standard generalized Cochran-Mantel-Haenszel statistics but modified to handle correlated outcomes. Two of these statistics are asymptotically valid in both many-strata (sparse data) and large-strata limiting models. The third one is designed specifically for the many-strata case but is valid even with a small number of strata. This latter statistic is also appropriate when strata are assumed to be random.     

Hayter and Tsui's test with double sampling for the vector mean of multivariate normal population

Abstract

In this paper, we introduce a version of Hayter and Tsui's statistical test with double sampling for the vector mean of a population under multivariate normal assumption. A study showed that this new test was more or as efficient than the well-known Hotelling's T² with double sampling. Some nice features of Hayter and Tsui's test are its simplicity of implementation and its capability of identifying the errant variables when the null hypothesis is rejected. Taking that into consideration, a new control chart called HTDS is also introduced as a tool to monitor multivariate process vector mean when using double sampling.     

Data-driven smooth tests for the extreme value distribution

In this paper we present data-driven smooth tests for the extreme value distribution. These tests are based on a general idea of construction of data-driven smooth tests for composite hypotheses introduced by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. [(1997). Data-driven smooth tests for composite hypotheses. Ann. Statist., 25, 1222–1250] and its modification for location-scale family proposed in Janic-Wróblewska, A. [(2004). Data-driven smooth test for a location-scale family. Statistics, in press]. Results of power simulations show that the newly introduced test performs very well for a wide range of alternatives and is competitive with other commonly used tests for the extreme value distribution.     

Effects of the generalized Box–Cox transformation on Type I error rate and power of Hotelling's T 2

Most multivariate statistical techniques rely on the assumption of multivariate normality. The effects of nonnormality on multivariate tests are assumed to be negligible when variance–covariance matrices and sample sizes are equal. Therefore, in practice, investigators usually do not attempt to assess multivariate normality. In this simulation study, the effects of skewed and leptokurtic multivariate data on the Type I error and power of Hotelling's T ² were examined by manipulating distribution, sample size, and variance–covariance matrix. The empirical Type I error rate and power of Hotelling's T ² were calculated before and after the application of generalized Box–Cox transformation. The findings demonstrated that even when variance–covariance matrices and sample sizes are equal, small to moderate changes in power still can be observed.