Normalizing and variance stabilizing transformations of multivariate statistics under an elliptical population

Summary Normalizing and variance stabilizing transformations of a sample correlation, multiple correlation and canonical correlation coefficients are obtained under an elliptical population. It is shown that the Fisher'sz-transformation is efficient for these statistics. A normalizing transformation is also studied for a latent root of a sample covariance matrix in an elliptical sample.     

Inferences on interclass and intraclass correlations in multivariate familial data

Inference procedures for interelass and intraclass correlations are given in the multivariate context of familial data for which measurements are taken on more than one characteristic. Unified estimators are proposed based on a certain class of unbiased estimators of covariance matrices. Asymptotic distributions of the proposed estimators are derived under the assumption of multivariate normality. The results can be used to construct approximate confidence intervals and test procedures.Research supported by the Department of Statistics, the Pennsylvania State University and the Air Force Office of Scientific Research under Grant AFSO-88-0030.Institute of Statistical Mathematics     

Normalizing transformations of some statistics of Gaussian ARMA processes

In this paper, the authors investigate Edgeworth type expansions of certain transformations of some statistics of Gaussian ARMA processes. They also investigate transformations which will make the second order part of the Edgeworth expansions vanish. Some numerical studies are made and they show that the above transformations give better approximations than the usual approximation.This work is supported by Contract N00014-K-0292 of the Office of Naval Research and Contract F49620-85-C-0008 of the Air Force Office of Scientific Research. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.The work of this author was done at the Center for Multivariate Analysis. His permanent address is Department of Mathematics, Hiroshima University, Naka-ku, Hiroshima 730, Japan.     

D n -normal semigroups of transformations

Given a subgroup G of the symmetric group S _n on n letters, a semigroup S of transformations of X _n is G-normal if G _S =G, where G _S consists of all permutations h∈S _n such that h ⁻¹ fh∈S for all f∈S. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups. In 1996, I. Levi showed that the alternating group A _n can not serve as the group G _S for any semigroup of total transformations of X _n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A _n -normal semigroups of partial transformations of X _n . Also, in 1994, I. Levi and R.B. McFadden described all S _n -normal semigroups. In this paper, we show that the dihedral group D _n may serve as the group G _S for semigroups of transformations of X _n . We characterize a large class of D _n -normax semigroups and describe certain D _n -normal semigroups.     

Some properties of BLUE in a linear model and canonical correlations associated with linear transformations

Let (x, Xβ, V) be a linear model and let A′ = (A′₁, A′₂) be a p × p nonsingular matrix such that A₂X = 0, Rank A₂ = p − Rank X. We represent the BLUE and its covariance matrix in alternative forms under the conditions that the number of unit canonical correlations between y₁ ( = A₁x) and y₂ ( = A₂x) is zero. For the second problem, let x′ = (x′₁, x′₂) and let a g-inverse V⁻ of V be written as (V⁻)′ = (A′₁, A′₂). We investigate the reations (if any) between the nonzero canonical correlations {1 ₁ … ₁ > 0} due to y₁ ( = A₁x) and y₂ ( = A₂x), and the nonzero canonical correlations {1 λ₁ … λ_v+r > 0} due to x₁ and x₂. We answer some of the questions raised by Latour et al. (1987, in Proceedings, 2nd Int. Tampere Conf. Statist. (T. Pukkila and S. Puntanen, Eds.), Univ. of Tampere, Finland) in the case of the Moore-Penrose inverse V⁺ = (A′₁, A′₂) of V.     

Improvements of goodness-of-fit statistics for sparse multinomials based on normalizing transformations

We consider multinomial goodness-of-fit tests for a specified simple hypothesis under the assumption of sparseness. It is shown that the asymptotic normality of the PearsonX ² statistic (X _k ² ) and the log-likelihood ratio statistic (G _k ² ) assuming sparseness. In this paper, we improve the asymptotic normality ofX _k ² andG _k ² statistics based on two kinds of normalizing transformation. The performance of the transformed statistics is numerically investigated.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Generalized Hodges-Lehmann estimators for the analysis of variance

For X_i, …, X_n a random sample and K(·, ·) a symmetric kernel this paper considers large sample properties of location estimator satisfying , . Asymptotic normality of is obtained and two forms of interval estimators for parameter θ satisfying EK(X₁ − θ, X₂ − θ) = 0, are discussed. Consistent estimation of the variance parameters is obtained which permits the construction of asymptotically distribution free procedures. The p-variate and multigroup extension is accomplished to provide generalized one-way MANOVA. Monte Carlo results are included.     

A Sufficient Condition for a Smash Product as a Transfinite Left Free Normalizing Extension and Its Application

In this paper, a sufficient condition is given under which the smash product A#H is a transfinite left free normalizing extension of an algebra A. Moreover, the result is applied to a skew semigroup ring, a skew group ring and the quantum group U _q (sl(2)) such that some properties are shown. Received April 14, 1997, Accepted August 10, 1998     

Investigation and reduction of variance of a weighted estimate in numerical statistical simulation

The efficiency of the “value” and “partial value” modeling related to the construction of a modeling distribution for an auxiliary random variable by multiplying the initial density by a value function is investigated. The value function usually corresponds to a solution of the adjoint equation. Conditions under which the value modeling of the initial distribution reduces the variance compared to the direct simulation are obtained. It is proved that the variance of the weighted estimate is bounded in the case of the partial value modeling. This proposition provides a basis for a method for determining whether or not the variance of the weighted estimate is bounded. This method uses the majorizing adjoint equation. Using a practically important problem in transport theory as an example, the asymptotic optimization of the distribution of the mean free path is presented. The application of the proposed method of the investigation of the variance boundedness for the analysis of the classical exponential transformation method of simulating the mean free path of a particle in the one-dimensional and the spherical variants is discussed.     

Cubature formulas for a disk that are invariant under groups of transformations of regular polygons into themselves

A method for constructing cubature formulas of different degrees of accuracy that are invariant under the groups of transformations of regular polygons into themselves is proposed. The cubature sum can contain powers of the Laplace operator applied to the integrand at the origin.     

Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators

Suppose thatX _n =(X ₁,...X _n) is a collection ofm-dimensional random vectorsX _i forming a stochastic process with a parameter . Let be the MLE of . We assume that a transformationA( ) of has thek-thorder Edgeworth expansion (k=2,3). IfA extinguishes the terms in the Edgeworth expansion up tok-th-order (k2), then we say thatA is thek-th-order normalizing transformation. In this paper, we elucidate thek-th-order asymptotics of the normalizing transformations. Some conditions forA to be thek-th-order normalizing transformation will be given. Our results are very general, and can be applied to the i.i.d. case, multivariate analysis and time series analysis. Finally, we also study thek-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

Necessary and sufficient conditions for the existence of stabilizing feedback for control systems

We prove that the existence of a non-smooth control Lyapunovfunction is a necessary and sufficient condition for the existenceof an ordinary smooth time-varying feedback that stabilizesan affine time-varying control system. Results concerning thenon-affine case are also provided.     

On equivalencies between design-based and regression-based variance estimators for randomized experiments

This paper demonstrates that the randomization-based “Neyman” and constant-effects estimators for the variance of estimated average treatment effects are equivalent to a variant of the White “heteroskedasticity-robust” estimator and the homoskedastic ordinary least squares (OLS) estimator, respectively.     

High-dimensional asymptotic expansion of LR statistic for testing intraclass correlation structure and its error bound

This paper deals with the null distribution of a likelihood ratio (LR) statistic for testing the intraclass correlation structure. We derive an asymptotic expansion of the null distribution of the LR statistic when the number of variable p and the sample size N approach infinity together, while the ratio p/N is converging on a finite nonzero limit c(0,1). Numerical simulations reveal that our approximation is more accurate than the classical χ²-type and F-type approximations as p increases in value. Furthermore, we derive a computable error bound for its asymptotic expansion.     

Filters and semigroup compactification properties

Stone-Čech compactifications derived from a discrete semigroup S can be considered as the spectrum of the algebra ℬ(S) or as a collection of ultrafilters on S. What is certain and indisputable is the fact that filters play an important role in the study of Stone-Čech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactifications too. In the present paper, first we review the characterizations of semigroup compactifications in terms of filters and then extend some of the results in Papazyan (Semigroup Forum 41:329–338, 1990) concerning the Stone-Čech compactification to a semigroup compactification associated with a Hausdorff semitopological semigroup.     

Asymptotic equivalence of the jackknife and infinitesimal jackknife variance estimators for some smooth statistics

The jackknife variance estimator and the infinitesimal jackknife variance estimator are shown to be asymptotically equivalent if the functional of interest is a smooth function of the mean or a trimmed L-statistic with Hölder continuous weight function.     

On the neighborhood in stabilizing period-T orbits for chaotic m degree polynomial dynamical system

In this paper, a problem of stabilizing a period-T orbit in discrete chaotic m degree polynomial dynamical systems is studied. The aim is to present a new method for determining the neighborhood of a period-T point in which the system remains stable when subjected to a linear feedback control. A theorem on the existence of neighborhood is rigorously proved using idea from functional analysis and polar coordinate transformation. The ways of implementing the obtained theorem in the Hénon map are proposed. The validity of this method is shown by numerical simulation.     

Improved prediction for a multivariate normal distribution with unknown mean and variance

The prediction problem for a multivariate normal distribution is considered where both mean and variance are unknown. When the Kullback–Leibler loss is used, the Bayesian predictive density based on the right invariant prior, which turns out to be a density of a multivariate t-distribution, is the best invariant and minimax predictive density. In this paper, we introduce an improper shrinkage prior and show that the Bayesian predictive density against the shrinkage prior improves upon the best invariant predictive density when the dimension is greater than or equal to three.