Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.     

Simultaneous confidexce intervals in an extended crow iii curve model

This paper deals with estimation problems under an extended growth curve model with two hierarchical within-individuals design matrices. The model in cludes the one whose mean structure consists of polynomial growth curves with two different degrees. First we propose certain simple estimators of the mean and covariance parameters which are closely related to the MEE's. Some basic properties of the estimators are given. Simultaneous confidence intervals are constructed, based on the estimators, for each and both of two growth curves. We give asymptotic approximations for the corresponding critical points. A numerical example is also given.     

The effects on the distributions of sample canonical correlations under nonnormality

In this paper, we study the effects of nonnormality on the distributions of sample canonical correlations when the population canonical correlations are simple. In order to achieve the purpose, we derive asymptotic expansion formulas for the distributions of a function of the canonical correlations as well as the individual canonical correlations under nonnormal populations. We particularly discuss the distribution of sample canonical correlations under the class of elliptical population. These expansions are given by using a perturbation method. Simulation results are also given.     

Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size

This paper deals with the distributions of test statistics for the number of useful discriminant functions and the characteristic roots in canonical discriminant analysis. These asymptotic distributions have been extensively studied when the number p   of variables is fixed, the number q+1$q+1$ of groups is fixed, and the sample size N tends to infinity. However, these approximations become increasingly inaccurate as the value of p increases for a fixed value of N. On the other hand, we encounter to analyze high-dimensional data such that p is large compared to n. The purpose of the present paper is to derive asymptotic distributions of these statistics in a high-dimensional framework such that q   is fixed, p→∞$p\to \infty$, m=n-p+q→∞$m=n-p+q\to \infty$, and p/n→c∈(0,1)$p/n\to c\in (0,1)$, where n=N-q-1$n=N-q-1$. Numerical simulation revealed that our new asymptotic approximations are more accurate than the classical asymptotic approximations in a considerably wide range of (n,p,q)$(n,p,q)$.     

Sample size calculations for single-arm survival studies using transformations of the Kaplan–Meier estimator

In single-arm clinical trials with survival outcomes, the Kaplan–Meier estimator and its confidence interval are widely used to assess survival probability and median survival time. Since the asymptotic normality of the Kaplan–Meier estimator is a common result, the sample size calculation methods have not been studied in depth. An existing sample size calculation method is founded on the asymptotic normality of the Kaplan–Meier estimator using the log transformation. However, the small sample properties of the log transformed estimator are quite poor in small sample sizes (which are typical situations in single-arm trials), and the existing method uses an inappropriate standard normal approximation to calculate sample sizes. These issues can seriously influence the accuracy of results. In this paper, we propose alternative methods to determine sample sizes based on a valid standard normal approximation with several transformations that may give an accurate normal approximation even with small sample sizes. In numerical evaluations via simulations, some of the proposed methods provided more accurate results, and the empirical power of the proposed method with the arcsine square-root transformation tended to be closer to a prescribed power than the other transformations. These results were supported when methods were applied to data from three clinical trials.     

Rank-dominant strategy and sincere voting

Theory and Decision - This study considers a voting rule wherein each player sincerely votes when he/she has no information about the preferences of the other players. We introduce the concept of...     

A Variable Selection Criterion for Two Sets of Principal Component Scores in Principal Canonical Correlation Analysis

Canonical correlation analysis (CCA) is often used to analyze the correlation between two random vectors. However, sometimes interpretation of CCA results may be hard. In an attempt to address these difficulties, principal canonical correlation analysis (PCCA) was proposed. PCCA is CCA between two sets of principal component (PC) scores. We consider the problem of selecting useful PC scores in CCA. A variable selection criterion for one set of PC scores has been proposed by Ogura (2010), here, we propose a variable selection criterion for two sets of PC scores in PCCA. Furthermore, we demonstrate the effectiveness of this criterion.     

Partial differential equations for hypergeometric functions 3F2 of matrix argument

Many multivariate non-null distributions and moment formulas can be expressed in terms of hypergeometric functions _pF_q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial differential equations for the functions of ₂F₁ of one argument matrix and two argument matrices, respectively. Such differential equations have been used to obtain asymptotic expansions of the functions (Muirhead [7], [8], [9], Sugiura [10]). The purpose of this paper is to derive partial differential equations for the functions ₃F₂ (a₁ a₂, a₃; b₁, b₂, R) and ₃F₂ (a₁, a₂, a₃; b₁, b₂; R, S). Differential equations for ₂F₂ are also obtained.