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.     

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 invariant and locally minimax test of independence

Let X₁ X₂ … X_N be independent normal p-vectors with common mean vector $$ = ($$) and common nonsingular covariance matrix $$ = Diag ($sG_i) [(1–p) I + pE] Diag ($sG_i), $sG_i> 0, i = 1… p, 1>p>=1/p–1. Write r_ij = sample correlation between the i th and the j th variable i j = 1,… p. It has been proved that for testing the hypothesis H₀ : p = 0 against the alternative H₁ : p>0 where $$ and $sG₁,…, $sG_p are unknown, the test which rejects H₀ for large value of $$ r_ij is locally best invariant for every $aL: 0 > $aL > 1 and locally minimax as p $$ 0 in the sense of Giri and Kiefer, 1964, for every $aL: 0 > $aL $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

Testing for and against a set of linear inequality constraints in a multinomial setting

There are numerous situations in categorical data analysis where one wishes to test hypotheses involving a set of linear inequality constraints placed upon the cell probabilities. For example, it may be of interest to test for symmetry in k × k contingency tables against one-sided alternatives. In this case, the null hypothesis imposes a set of linear equalities on the cell probabilities (namely p_ij = P_ji ×i > j), whereas the alternative specifies directional inequalities. Another important application (Robertson, Wright, and Dykstra 1988) is testing for or against stochastic ordering between the marginals of a k × k contingency table when the variables are ordinal and independence holds. Here we extend existing likelihood-ratio results to cover more general situations. To be specific, we consider testing H_t,0 against H₁ - H₀ and H₁ against H₂ - H ₁ when H₀:k × i=1 p_ix_ji = 0, j = 1,…, s, H₁:k × i=1 p_ix_ji × 0, j = 1,…, s, and does not impose any restrictions on p. The x_ji's are known constants, and s × k - 1. We show that the asymptotic distributions of the likelihood-ratio tests are of chi-bar-square type, and provide expressions for the weighting values.     

Locally minimax tests for multiple correlations

Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X ₁, X ₍₂₎', X '₍₃₎)', where X₁ is one-dimension, X₍₂₎ is p₂- dimensional, and so 1 + p₁ + p₂ = p. Let ρ₁ and ρ be the multiple correlation coefficients of X₁ with X₍₂₎ and with ( X '₍₂₎, X '₍₃₎)', respectively. Write ρ2/2 = ρ² - ρ2/1. We shall cosider the following two problems     

Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality

A test for homogeneity of g ? 2 covariance matrices is presented when the dimension, p, may exceed the sample size, n_i, i = 1, …, g, and the populations may not be normal. Under some mild assumptions on covariance matrices, the asymptotic distribution of the test is shown to be normal when n_i, p → ∞. Under the null hypothesis, the test is extended for common covariance matrix to be of a specified structure, including sphericity. Theory of U-statistics is employed in constructing the tests and deriving their limits. Simulations are used to show the accuracy of tests.     

Asymptotic Distribution for Symmetric Spacing Statistics

A new procedure for testing the H ₀: μ₁ = ··· = μ _k against the alternative H _u :μ₁ ≥ ··· ≥μ _r  ≤ ··· ≤ μ _k with at least one strict inequality, where μ _i is the location parameter of the ith two-parameter exponential distribution, i = 1,…, k, is proposed. Exact critical constants are computed using a recursive integration algorithm. Tables containing these critical constants are provided to facilitate the implementation of the proposed test procedure. Simultaneous confidence intervals for certain contrasts of the location parameters are derived by inverting the proposed test statistic. In comparison to existing tests, it is shown, by a simulation study, that the new test statistic is more powerful in detecting U-shaped alternatives when the samples are derived from exponential distributions. As an extension, the use of the critical constants for comparing Pareto distribution parameters is discussed.     

On the distribution of hotelling's t2 and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T² and the null distribution of multiple correlation R² are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ₁ and μ₂ respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R² is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T² and R² tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.     

ON TESTING AGAINST RESTRICTED ALTERNATIVES FOR THE VARIANCES OF GAUSSIAN MODELS

Let π₁,…,π_p be p independent normal populations with means μ₁…, μ_p and variances σ²₁,…, σ²_p respectively. Let X(n_i) be a simple random sample of size n_i from π_i, i = 1,…,p. Given the simple random samples X(n₁),…, X(n_p) from π₁,…,π_p respectively, a test has been proposed for testing the homogeneity of variances H₀: σ²₁=…σ²_p, against the restricted alternative, H₁: σ²₁≥…≥σ²_p, with at least one strict inequality. Some properties of the test are discussed and critical values are tabulated.     

Optimal design results for 2n factorial paired comparison experiments

Four general classes of partially balanced designs for 2ⁿ factorials, corresponding to four different forms of a general null hypothesis H on factorial effects, are presented. For the typical design in each class, the simplified form of the non-centrality parameter λ² of the asymptotic chi-square distribution of the likelihood ratio statistic for testing the corresponding form of H₀ is derived under defined local alternatives. Optimal designs d₁ maximizing λ² in the i-th class and minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix, i =1,…,4, are determined.     

On comparing two poisson intensity functions

Given realizations of two possion processes with unknown intensities A(·) and F(·) observed over the interval (t₁,t₂), we suppose that it is desired to distinution between H₀ Ξ(·)/λ(·) is constant on (t₁,t₂) versus H₊:Ξ(·)/λ(·) increases on (t₁,t₂). We propose a decision rule which uses the percentage points of the Mann-Whitney U-distribution. We show that the decision rule is unbiased and that the set of alternatives in H₊ can be weakly ordered, specifically: if Ξ(·)/λ(·), β(·)/λ(·) and Ξ(·)/β(·) are increasing on (t₁, t₂) then P{H₀ is rejected |Ξ(·)}≧P{H₀ is rejected|B(·)}≧P{H₀ is rejected|H₀}.     

Identifiability of distributions under competing risks and complementary risks model

Let X₁,X₂,…,X_p be p random variables with cdf's F₁(x),F₂(x),…,F_p(x)respectively. Let U = min(X₁,X₂,…,X_p) and V = max(X₁,X₂,…,X_p).In this paper we study the problem of uniquely determining and estimating the marginal distributions F₁,F₂,…,F_p given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X₁'s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X₁'s are dependent is considered next. Finally the problem of estimation is considered.     

An exact decomposition for a class of exponential dispersion models: Application to sequential testing

We define the Wishart distribution on the cone of positive definite matrices and an exponential distribution on the Lorentz cone as exponential dispersion models. We show that these two distributions possess a property of exact decomposition, and we use this property to solve the following problem: given q samples (y_il,… y_iNj), i = l,…,q, from a N(μ_i,Σ_i,) distribution, test H:Σ₁ = Σ₂ = … = σ_q. Using the exact decomposition property, the classical test statistic for H, involving q parameters p_i = (N_i, - l)/2, i = 1,…,q, is replaced by a sequence of q - l test statistics for the sequence of tests H_i,:σ₁ =σ₂ = … =σ_i given that H_i-1 is true, i = 2,…,q. Each one of these test statistics involves two parameters only, p._i-1 = p₁ + … + p_i-1 and p_i. We also use the exact decomposition property to test equality of the “direction parameters” for q sample points from the exponential distribution on the Lorentz cone. We give a table of critical values for the distribution on the three-dimensional Lorentz cone. Tables of critical values in higher dimensions can easily be computed following the same method as in dimension three.     

Maximum likelihood estimation of the parameters of the bivariate binomial distribution

We consider the problem of maximum likelihood estimation of the parameters of the bxvariate binomial distribution, In the statistical literature, this problem is solved when the observed sample is available in the form of a 2x2 contingency table, that is, with all four cell fre quencies given,, The present paper provides a solution for this problem when only the marginal totals of the 2x2 table are observed, which is the natural set-up in a bivariate sampling situation.. Thus, based on a sample [(X_i,Y_i:), i = 1, …, k] from a bivariate binomial population, we derive maximum likelihood (ML) estimators for the two marginal parameters p₁,p₂: and the covariance parameter p₁₁: It. turns out that the ML estimators for P₁: and P₂: are expressed explicitly in terms of the sample values, whereas the ML estimator for p₁₁: can only be obtained numerically by iterative methods Two nu merical illustrations are also presented     

Dependence Structure of Spacings of Order Statistics

Let X_1:n ≤ X_2:n ≤···≤ X_n:n denote the order statistics of a sample of n independent random variables X₁, X₂,…, X_n, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (X_k1:n, X_k2:n ? X_k1:n,…, X_kr:n ? X_kr?1:n) is MTP₂ [S-MRR₂] whenever 1 ≤ k₁ < k₂ <···< k_r ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.     

A one-sided test for testino homogeneity of scale parameters against simple ordered alternative

In an earlier paper the authors (1997) extended the results of Hayter (1990) to the two parameter exponential probability model. This paper addressee the extention to the scale parameter case under location-scale probability model. Consider k (k≧3) treatments or competing firms such that an observation from with treatment or firm follows a distribution with cumulative distribution function (cdf) F_i(x)=F[(x-μ_i)/Q_i], where F(·) is any absolutely continuous cdf, i=1,…,k. We propose a test to test the null hypothesis H₀:θ₁=…=θ_k against the simple ordered alternative H₁:θ₁≦…≦θ_k, with at least one strict inequality, using the data X_i,j, i=1,…k; j=1,…,n₁. Two methods to compute the critical points of the proposed test have been demonstrated by talking k two parameter exponential distributions. The test procedure also allows us to construct simultaneous one sided confidence intervals (SOCIs) for the ordered pairwise ratios θ_j/θ_i, 1≦i<j≦k. Statistical simulation revealed that: 9i) actual sizes of the critical points are almost conservative and (ii) power of the proposed test relative to some existing tests is higher.     

Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables

Let X ₁,X ₂,…,X _n be independent exponential random variables such that X _i has hazard rate λ for i = 1,…,p and X _j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D _i:n (λ, λ*) = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…,n, where X _0:n ≡ 0. It is shown that the spacings (D _1,n ,D _2,n ,…,D _n:n ) are MTP₂, strengthening one result of Khaledi and Kochar (2000), and that (D _1:n (λ₂, λ*),…,D _n:n (λ₂, λ*)) ≤ _lr (D _1:n (λ₁, λ*),…,D _n:n (λ₁, λ*)) for λ₁ ≤ λ* ≤ λ₂, where ≤ _lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ₁ < λ₂.     

Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X₁₁,X₂₁),(X₁₂,X₂₂),… (X_1n,X_2n) having a bivariate elliptically contoured density of the form where θ₁ θ₂ are location parameters and Δ = ((λ_ik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X₁ and X₂ is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.