On optimum invariant tests of equality of intraclass correlation coefficients

In this paper we address the problem of testing the equality ofk intraclass correlation coefficients based on samples from independentp-variate normal populations, and explore various aspects of optimality through invariance. A UMPIU test is derived fork=2, and LMMPIU test of SenGupta and Vermeire (1986) is indicated fork>2. Several approximately optimum invariant tests are also proposed. The tests are compared with the approximate LR tests and Fisher'sZ-tests derived in Konishi and Gupta (1987, 1989). As expected, the performance of the proposed tests turns out to be quite satisfactory and superior to the LR tests andZ-tests.     

Asymptotic optimality of nonparametric tests for restricted alternatives in some multivariate mixed models

For some mixed models (involving both stochastic and nonstochastic predictors), a general class of permutationally distribution-free rank tests for some restricted alternative problems is considered. The proposed tests are asymptotically optimal in the light of the restricted likelihood ratio tests. For an ordered alternative problem in a two-way analysis of covariance model, the proposed tests are asymptotically optimal.     

Locally most powerful rank tests for testing randomness and symmetry

Let X _i, 1 i N, be N independent random variables (i.r.v.) with distribution functions (d.f.) F _i(x,), 1 i N, respectively, where is a real parameter. Assume furthermore that F _i(·,0) = F(·) for 1 i N. Let R = (R ₁,R _N) and R ⁺,...,R _N ⁺ be the rank vectors of X = (X ₁,X _N) and |X|=(|X ₁|,...,|X _N|), respectively, and let V = (V ₁,V _N) be the sign vector of X. The locally most powerful rank tests (LMPRT) S = S(R) and the locally most powerful signed rank tests (LMPSRT) S = S(R ⁺, V) will be found for testing = 0 against > 0 or < 0 with F being arbitrary and with F symmetric, respectively.     

A note on the uniformly most powerful tests in the presence of nuisance parameters

When a testing problem has nuisance parameters, the uniformly most powerful (UMP) tests do not generally exist. Exceptional examples were given by Dubey (1962, Skand. Aktuarietidskr., 45, 25–38; 1963, Skand. Aktuarietidskr., 46, 1–24) and Takeuchi (1968, Ann. Math. Statist., 40, 1838–1839) for the exponential distributions. What is essential for proving the existence of UMP tests lies in a special relationship between null hypothesis and the alternative. Assuming a similar relationship between them, a similar kind of result can be shown under more general situation.     

Locally minimax tests in symmetrical distributions

In this paper we give an extension of the theory of local minimax property of Giri and Kiefer (1964, Ann. Math. Statist., 35, 21–35) to the family of elliptically symmetric distributions which contains the multivariate normal distribution as a member.This work was partially supported by the Canadian N.S.E.R.C. grant     

On the local minimaxity of a test of independence in incomplete samples

Summary LetX=(X ₁,X ₂) be ap-dimensional normal variable. Suppose that we haven independent observations onX andm, independent observations onX _i ,i=1,2, respectively. For testing the independence betweenX ₁ andX ₂, Eaton and Kariya (1983),Ann. Statist.,11, 654–665) obtains a locally most powerful test among the class of tests invariant under the group of affine transformations. In this paper, this LMPI test is shown to be locally minimax within the class of all tests of the same level.     

Locally most powerful sequential tests for stochastic processes

For a continuous time stochastic process with distribution P_? depending on a one-dimensional parameter ? the problem of sequentially testing ? = 0 against ? > 0 is treated. We assume that the process of likelihood ratios has a certain representation which allows to obtain identities of the Wald type for stopping times. These identities are then used to derive a result on locally most powerful tests for which a problem of optimal stopping is solved.     

Modified likelihood ratio test for homogeneity in normal mixtures with two samples

This paper investigates the modified likelihood ratio test（LRT） for homogeneity in normal mixtures of two samples with mixing proportions unknown. It is proved that the limit distribution of the modified likelihood ratio test is X^2（1）.     

线性模型下F-检验最优的误差协方差结构

文献中回归参数线性假设的F-检验统计量主要包括基于广义最小二乘估计F- 统计量F(θ),基于最小二乘估计的F-统计量FLSE以及Wu C．F．J．等于1988年提出的调整的F-统计量FA(θ)．其中后两者因形式简单而常常被广泛采用．本文主要研究了FA(θ)和FLSE的最优性,并分别获得了FA(θ)=F(θ)和ELSE=F(θ)的充要条件．最后,我们将所得的结果应用到医药领域的两类重要模型．     

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

Summary Local 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.     

Modified likelihood ratio test for homogeneity in bivariate normal mixtures with presence of a structural parameter

This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ22 null limiting distribution.     

Asymptotically efficient rank MANOVA tests for restricted alternatives in randomized block designs

In a randomized block design MANOVA model, for intrablock as well as aligned rank tests for homogeneity of treatment effects against some restricted alternatives, asymptotic optimality is studied by reference to the corresponding restrieted likelihood ratio tests. Tests based on aligned ranks are better than intra-block rank tests when the error distributions are homogeneous aeross the blocks.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0987.     

Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

One-sided tests for independence of seemingly unrelated regression equations

In a seemingly unrelated regression model with p(2) equations, this paper considers the problem of testing independence of equations against a one-sided alternative hypothesis. The power functions of invariant tests are evaluated and the locally most mean powerful invariant test is obtained.     

Second-Order Efficient Test for Inhomogeneous Poisson Processes

We observe a realization X ⁽ⁿ⁾ of a Poisson process on the set with intensity function depending on the unknown real parameter . Based on X ⁽ⁿ⁾ we test simple null hypothesis against one sided alternative for given . We improve the level of the well-known locally asymptotically uniformly most powerful (LAUMP) test by using the Edgeworth type expansion for stochastic integral. We show that the improved test is second-order efficient under certain regularity conditions.      

A result on hypothesis testing for a multivariate normal distribution when some observations are missing

Let U_i = (X_i, Y_i), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μ_x, μ_y) and covariance matrix

. Let X_i, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H₀ : μ = 0 vs. H₁ : μ ≠ 0. We prove that Hotelling's T² test, which uses (X_i, Y_i), i = 1, 2,…, n (and discards X_i, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T²-test cannot be beaten. A similar result is also proved for the problem of testing μ_x ? μ_y = 0. A Bayes test and other competitors which are similar tests are discussed.     

Tests for the marginal probabilities in the two-way contingency table under restricted alternatives

Testing hypotheses on the marginal probabilities of a two-way contingency table is discussed. Three statistics are considered for testing the hypothesis of specified probabilities in the margins against alternatives with certain kind of order restriction. The properties of these statistics are discussed and their asymptotic behaviors are compared in depth. An appliction which motivated the consideration of the original testing problem is illustrated with a practical data.     

Local likelihood ratio tests in the normal mixture model

The test for homogeneity in the mixture normal model is difficult to study due to the breakdown of the regularity conditions under standard theory. The asymptotic optimality of the likelihood ratio test is questionable and its distributional properties are also difficult to evaluate. In this paper, we consider and compare several tests based on the local likelihood ratio which are shown to be quite competitive compared with the generalized likelihood ratio test.     

Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function

Let X ₁,...,X _n be independent observations on a random variable X. This paper considers a class of omnibus procedures for testing the hypothesis that the unknown distribution of X belongs to the family of Cauchy laws. The test statistics are weighted integrals of the squared modulus of the difference between the empirical characteristic function of the suitably standardized data and the characteristic function of the standard Cauchy distribution. A large-scale simulation study shows that the new tests compare favorably with the classical goodness-of-fit tests for the Cauchy distribution, based on the empirical distribution function. For small sample sizes and short-tailed alternatives, the uniformly most powerful invariant test of Cauchy versus normal beats all other tests under discussion.     

A locally most powerful invariant test for the equality of means associated with covariate discriminant analysis

In this paper, the authors propose a locally most powerful invariant test for the equality of means in the presence of covariate variables. Also the null and nonnull distributions associated with the above test are developed. This problem arises in covariate discriminant analysis and has been treated by various authors, notably Cochran and Bliss 1948, Ann. Math. Statist.19, 151–176 and Rao 1949, Sankhyã9 343–366; 1966. The test derived here locally dominates in power the tests proposed so far. It is also shown that the Cochran-Bliss test is uniformly most powerful in the class of conditional invariant tests.