Asymptotic expansions for distributions of latent roots in multivariate analysis

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S₁(S₁ + S₂)^?1, where S₁ is $\text{W}{}_{\mathrm{<mtext>m</mtext>}}\text{(n}{}_1\text{, Σ, Ω)}$ and S₂ is W_m(n₂, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n^?1S, where S in W_m(n, Σ), for large n, and S₁S₂^?1, where S₁ is W_m(n₁, Σ) and S₂ is W_m(n₂, Σ), for large n₁ + n₂.     

Asymptotic expansions of the distributions of the latent roots in MANOVA and the canonical correlations

Asymptotic expansions are given for the density function of the normalized latent roots of S₁S₂^?1 for large n under the assumption of $\text{Ω = O(n)}$, where S₁ and S₂ are independent noncentral and central Wishart matrices having the $\text{W}{}_{\mathrm{p}}\text{(b, Σ; Ω)}$ and W_p(n, Σ) distributions, respectively. The expansions are obtained by using a perturbation method. Asymptotic expansions are also obtained for the density function of the normalized canonical correlations when some of the population canonical correlations are zero.     

Asymptotic joint distribution of the largest roots of several multivariate F matrices I. Quasi-independent case

An asymptotic expansion of the joint distribution of k largest characteristic roots C_M⁽ⁱ⁾(S_iS₀^?1), i = 1,…, k, is given, where S'_is and S₀ are independent Wishart matrices with common covariance matrix Σ. The modified second-approximation procedure to the upper percentage points of the maximum of C_M⁽ⁱ⁾(S_iS₀^?1), i = 1,…, k, is also considered. The evaluation of the expansion is based on the idea for studentization due to Welch and James with the use of differential operators and of the perturbation procedure.     

Asymptotic expansions of the distributions of the latent roots and the latent vector of the Wishart and multivariate F matrices

Asymptotic expansions of the joint distributions of the latent roots of the Wishart matrix and multivariate F matrix are obtained for large degrees of freedom when the population latent roots have arbitrary multiplicity. Asymptotic expansions of the distributions of the latent vectors of the above matrices are also derived when the corresponding population root is simple. The effect of normalizations of the vector is examined.     

Asymptotic expansions for the distributions of latent roots ofS h S e −1 and of certain test statistics in MANOVA

S _e andS _n are independent central and noncentral Wishart matrices having Wishart distributionsW _p (n _e , Σ) andW _p (n _h , Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS _h S _e ⁻¹ and of certain test statistics in MANOVA under the assumption thatn=n _e +n _h becomes large with a fixed ration _e ∶n _h =e∶h(e+h=1,e>0,h>0) andΩ=O(n).     

On some distribution problems in Manova and discriminant analysis

Asymptotic expansions, valid for large error degrees of freedom, are given for the multivariate noncentral F distribution and for the distribution of latent roots in MANOVA and discriminant analysis. The asymptotic results are expressed in terms of elementary functions which are easy to compute and the results of some numerical work are included. The Bartlett test of the null hypothesis that some of the noncentrality parameters in discriminant analysis are zero is also briefly discussed.     

The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Summary The joint density function of the latent roots ofS ₁ S ₂ ⁻¹ under violations is obtained whereS ₁ has a complex non-central Wishart distributionW _c (p,n ₁,Σ ₁,Ω) andS ₂, an independent complex central Wishart,W _c (p,n ₂,Σ ₂, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.     

Asymptotic distributions of the latent roots with multiple population roots in multiple discriminant analysis

Asymptotic expansions are derived for the confluent hypergeometric function₁ F ₁(a; c; R, S) with two argument matrices, which appears in the joint density function of the latent roots in multiple discriminant analysis, whenR is “large” and each of the latent roots ofR assumes the general multiplicity. Laplace's method and a partial differential equation method are utilized in the derivation.     

Nonconcave penalized inverse regression in single-index models with high dimensional predictors

In this paper we aim to estimate the direction in general single-index models and to select important variables simultaneously when a diverging number of predictors are involved in regressions. Towards this end, we propose the nonconcave penalized inverse regression method. Specifically, the resulting estimation with the SCAD penalty enjoys an oracle property in semi-parametric models even when the dimension, p_n, of predictors goes to infinity. Under regularity conditions we also achieve the asymptotic normality when the dimension of predictor vector goes to infinity at the rate of p_n=o(n^1/3) where n is sample size, which enables us to construct confidence interval/region for the estimated index. The asymptotic results are augmented by simulations, and illustrated by analysis of an air pollution dataset.     

Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of ξ in semiparametric models of the form Y=G(ξ^TX,ε) where G(⋅) is an unknown link function, ε is an independent error, and ξ is a p_n×1 vector. To recover the direction of ξ, we first propose an inverse regression approach regardless of the link function G(⋅); to construct a data-driven confidence region for the direction of ξ, we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G(⋅) or its derivative. When p_n remains fixed, the empirical likelihood ratio without bias correlation can be asymptotically standard chi-square. Moreover, the asymptotic normality of the empirical likelihood ratio holds true even when the dimension p_n follows the rate of p_n=o(n^1/4) where n is the sample size. Simulation studies are carried out to assess the performance of our proposal, and a real data set is analyzed for further illustration.     

Asymptotic expansions for the distributions of some functions of the latent roots of matrices in three situations

In this paper we derive asymptotic expansions for the distributions of some functions of the latent roots of the matrices in three situations in multivariate normal theory, i.e., (i) principal component analysis, (ii) MANOVA model and (iii) canonical correlation analysis. These expansions are obtained by using a perturbation method. Confidence intervals for the functions of the corresponding population roots are also obtained.     

Improved approximations to distributions of the largest and the smallest latent roots of a wishart matrix

Summary Normalizing transformations of the largest and the smallest latent roots of a sample covariance matrix in a normal sample are obtained, when the corresponding population roots are simple. Using our results, confidence intervals for population roots may easily be constructed. Some numerical comparisons of the resulting approximations are made in a bivariate case, based on exact values of the probability integral of latent roots.     

On the distributions of the latent roots and traces of certain random matrices

It is shown that differential equations given by the author may be used recursively to construct certain multivariate null distributions in reduced form. These include the distributions of individual latent roots of B = S₁(S₁ + S₂)⁻¹, and distributions of Tr B and Tr S₁S₂⁻¹, for small numbers of variates.     

High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between and with q≤p, based on a sample of size of N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m=n−p→∞ and c=p/n→c₀∈[0,1), assuming that and have a joint (q+p)-variate normal distribution. An extended Fisher’s z-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,q, and n and the population canonical correlations.     

An asymptotic expansion of the distribution of Rao's U-statistic under a general condition

In this paper we consider the problem of testing the hypothesis about the sub-mean vector. For this propose, the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n^-1. The same problem in the k-sample case is also investigated. We find that the asymptotic distribution of generalized U-statistic in the k-sample case is identical to that of the generalized Hotelling's T² distribution up to n^-1. A simulation experiment is carried out and its results are presented. It shows that the asymptotic distributions have significant improvement when comparing with the limiting distributions both in the small sample case and the large sample case. It also demonstrates the equivalence of two testing statistics mentioned above.     

A multidimensional Newton-Raphson method and its applications to the existence of asymptotic Fn-estimators and their stochastic expansions

Given a suitable function F_n we define a class of estimators called asymptotic F_n-estimators (i.e., estimators which approximate the solution of F_n(θ) = 0). It is proved that this class is nonvoid if appropriate regularity conditions are fulfilled and if one has at hand a suitable initial estimator. Furthermore, it is shown that F_n-estimators admit a stochastic expansion (which enables to give results on asymptotic expansions for the distribution of these estimators).     

Lower bounds for the distributions of certain multivariate test statistics

A lower (upper) bound is given for the distribution of each d_j, j = k + 1, …, p (j = 1, …, s), the jth latent root of AB^?1, where A and B are independent noncentral and central Wishart matrices having $\text{W}{}_{\mathrm{p}}\text{(q, Σ; Ω)}$ with rank $\text{(Ω) ≤ k = p ? s}$ and W_p(n, Σ), respectively. Similar bound are also given for the distributions of noncentral means and canonical correlations. The results are applied to obtain lower bounds for the null distributions of some multivariate test statistics in Tintner's model, MANOVA and canonical analysis.     

Asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis when the dimension is large

This paper examines asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis based on a sample of size N=n+1 on two sets of variables, i.e.,  and . These problems are related to dimension reduction. The asymptotic approximations of the statistics have been studied extensively when dimensions p₁ and p₂ are fixed and the sample size N tends to infinity. However, the approximations worsen as p₁ and p₂ increase. This paper derives asymptotic expansions of the test statistics when both the sample size and dimension are large, assuming that and have a joint (p₁+p₂)-variate normal distribution. Numerical simulations revealed that this approximation is more accurate than the classical approximation as the dimension increases.     

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β₀ in the linear errors-in-variables model: Y_i=x_i^τβ₀+ε_i,X_i=x_i+u_i,(1?i?n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β₀. It is shown that the coverage error of the confidence region is of order n⁻¹, and Bartlett corrections can reduce the coverage errors to n⁻². An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.