Exact robustness studies of the test of independence based on four multivariate criteria and their distribution problems under violations

Summary Exact robustness studies against non-normality have been carried out for test of independence based on the four multivariate criteria: Hotelling's trace,U ^(p) , Pillai's trace,V ^(p) , Wilks' criterion,W ^(p) , and Roy's largest root,L _(p) . The density functions ofU ^(p) ,W ^(p) andL _(p) have been obtained in the canonical correlation case and further the moments ofU ^(p) and m.g.f. ofV ^(p) have been derived. All of the study is based on Pillai's distribution of the characteristic roots under violations. Numerical results for the power function have been tabulated for the two-roots case. Slight non-normality does not affect the independence test seriously.V ⁽²⁾ is found to be most robust against nonnormality. Yu-Sheng Hsu is now with Georgia State University, Atlanta.     

On the robustness of least squares procedures in regression models

The criterion robustness of the standard likelihood ratio test (LRT) under the multivariate normal regression model and also the inference robustness of the same test under the univariate set up are established for certain nonnormal distributions of errors. Restricting attention to the normal distribution of errors in the context of univariate regression models, conditions on the design matrix are established under which the usual LRT of a linear hypothesis (under homoscedasticity of errors) remains valid if the errors have an intraclass covariance structure. The conditions hold in the case of some standard designs. The relevance of C. R. Rao's (1967 In Proceedings Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 355–372) and G. Zyskind's (1967, Ann. Math. Statist.38 1092–1110) conditions in this context is discussed.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.     

Types of laws of random vectors and best estimators

Optimality of estimators of a vector parameter in terms of the probability of the estimators being contained in suitable region(s) around the parameter point is defined. Conditions under which optimal estimators in the usual senses are also optimal in the above sense are investigated.     

The Spearman footrule and a Markov chain property

An equivalent representation of the Spearman footrule is considered and a characterization in terms of a Markov chain is established. A martingale approach is thereby incorporated in the study of the asymptotic normality of the statistics.     

A characterization of k-normal varieties

Let v be a valuation of terms of type , assigning to each term t of type a value v(t) 0. Let k 1 be a natural number. An identity of type is called k-normal if either s = t or both s and t have value k, and otherwise is called non-k-normal. A variety V of type is said to be k-normal if all its identities are k-normal, and non-k-normal otherwise. In the latter case, there is a unique smallest k-normal variety to contain V , called the k-normalization of V. Inthe case k = 1, for the usual depth valuation of terms, these notions coincide with the well-known concepts of normal identity, normal variety, and normalization of a variety. I. Chajda has characterized the normalization of a variety by means of choice algebras. In this paper we generalize his results to a characterization of the k-normalization of a variety, using k-choice algebras. We also introduce the concept of a k-inflation algebra, and for the case that v is the usual depth valuation of terms, we prove that a variety V is k-normal iff it is closed under the formation of k-inflations, and that the k-normalization of V consists precisely of all homomorphic images of k-inflations of algebras in V .     

The converse theorem for Minkowski's inequality

Let (Ω, Σ, μ) be a measure space and ?, ψ : (0, ∞) → (0, ∞) some bijective functions. Suppose that the functional _?,ψ defined on class of μ-integrable simple functions χ : Ω → [0, ∞), μ({? : χ(?) > 0} > 0, by the formula

     

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2.     

Iterated monoidal categories

We develop a notion of an n-fold monoidal category and show that it corresponds in a precise way to the notion of an n-fold loop space. Specifically, the group completion of the nerve of such a category is an n-fold loop space, and free n-fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher H-space structure of an n-fold loop space. We also show directly that this operad has the same homotopy type as the n-th Smith filtration of the Barratt-Eccles operad and the n-th filtration of Berger's complete graph operad. Moreover, this operad contains an equivalent preoperad which gives rise to Milgram's small model for when n=2 and is very closely related to Milgram's model of 62" alt="View the MathML source" title="View the MathML source" src="http://ars.els-cdn.com/content/image/1-s2.0-S0001870803000653-si2.gif"> for n>2.