Nonnull distributions of some statistics associated with testing for the equality of two covariance matrices

The nonnull distribution of some statistics, used for testing Σ₁ = Σ₂ are obtained as mixtures of incomplete beta functions as well as mixtures of incomplete gamma functions. The introduction of the convergence factors and certain recurrence relations are useful in the computation of the power of the tests as well as computation of exact percentage points for tests of significance.     

Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations

The authors investigated the asymptotic joint distributions of certain functions of the eigenvalues of the sample covariance matrix, correlation matrix, and canonical correlation matrix in nonnull situations when the population eigenvalues have multiplicities. These results are derived without assuming that the underlying distribution is multivariate normal. In obtaining these expressions, Edgeworth type expansions were used.     

On the asymptotic joint distributions of certain functions of the eigenvalues of four random matrices

In this paper, the authors obtained asymptotic expressions for the joint distributions of certain functions of the eigenvalues of the Wishart matrix, correlation matrix, MANOVA matrix and canonical correlation matrix when the population roots have multiplicity.     

Asymptotic expansions for the distributions of functions of a correlation matrix

This paper deals with asymptotic expansions for the non-null distributions of certain test statistics concerning a correlation matrix in a multivariate normal distribution. For this purpose an asymptotic expansion is given for the distribution of a function of the sample correlation matrix. As special cases of the resulting expansion, asymptotic expansions for the distributions of the sample correlation coefficient, Fisher's z-transformation and arcsine transformation are also given.     

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 distributions of two “synthetic data” estimators for censored single-index models

The censored single-index model provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored and the link function is unknown. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure time models for survival analysis. This paper proposes two methods for estimation of single-index models with randomly censored samples. We first transform the censored data into synthetic data or pseudo-responses unbiasedly, then obtain estimates of the index coefficients by the rOPG or rMAVE procedures of Xia (2006) [1]. Finally, we estimate the unknown nonparametric link function using techniques for univariate censored nonparametric regression. The estimators for the index coefficients are shown to be root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodologies.     

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.     

Asymptotic nonnull distributions for tests for reality of a covariance matrix

In this paper asymptotic nonnull distributions are derived for two statistics used in testing for the reality of the covariance matrix in a complex Gaussian distribution.     

Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality

Asymptotic expansions of the distributions of typical estimators in canonical correlation analysis under nonnormality are obtained. The expansions include the Edgeworth expansions up to order O(1/n) for the parameter estimators standardized by the population standard errors, and the corresponding expansion by Hall's method with variable transformation. The expansions for the Studentized estimators are also given using the Cornish-Fisher expansion and Hall's method. The parameter estimators are dealt with in the context of estimation for the covariance structure in canonical correlation analysis. The distributions of the associated statistics (the structure of the canonical variables, the scaled log likelihood ratio and Rozeboom's between-set correlation) are also expanded. The robustness of the normal-theory asymptotic variances of the sample canonical correlations and associated statistics are shown when a latent variable model holds. Simulations are performed to see the accuracy of the asymptotic results in finite samples.     

Multivariate extreme value distributions for stationary Gaussian sequences

Under weak regularity conditions of the covariance sequence, it is shown that the joint limiting distribution of the maxima on each coordinate of a stationary Gaussian multivariate sequence is that of independent random variables with marginal Gumbel distributions.     

A link-free method for testing the significance of predictors

One important step in regression analysis is to identify significant predictors from a pool of candidates so that a parsimonious model can be obtained using these significant predictors only. However, most of the existing methods assume linear relationships between response and predictors, which may be inappropriate in some applications. In this article, we discuss a link-free method that avoids specifying how the response depends on the predictors. Therefore, this method has no problem of model misspecification, and it is suitable for selecting significant predictors at the preliminary stage of data analysis. A test statistic is suggested and its asymptotic distribution is derived. Examples are used to demonstrate the proposed method.     

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.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions

Let X be a p-dimensional random vector with density f(6X?θ6) where θ is an unknown location vector. For p ≥ 3, conditions on f are given for which there exist minimax estimators θ?(X) satisfying 6Xt6 · 6θ?(X) ? X6 ≤ C, where C is a known constant depending on f. (The positive part estimator is among them.) The loss function is a nondecreasing concave function of 6θ?? θ6². If θ is assumed likely to lie in a ball in $\text{R}$^p, then minimax estimators are given which shrink from the observation X outside the ball in the direction of P(X) the closest point on the surface of the ball. The amount of shrinkage depends on the distance of X from the ball.     

An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory

The article is devoted to the nonparametric estimation of the quadratic covariation of non-synchronously observed Itô processes in an additive microstructure noise model. In a high-frequency setting, we aim at establishing an asymptotic distribution theory for a generalized multiscale estimator including a feasible central limit theorem with optimal convergence rate on convenient regularity assumptions. The inevitably remaining impact of asynchronous deterministic sampling schemes and noise corruption on the asymptotic distribution is precisely elucidated. A case study for various important examples, several generalizations of the model and an algorithm for the implementation warrant the utility of the estimation method in applications.     

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n^?2, for the ₀F₀ function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.     

Joint distributions of numbers of occurrences of a discrete pattern and weak convergence of an empirical process for the pattern

For a {0, 1}-pattern of finite length, an empirical process is introduced in order to describe the number of overlapping occurrences of the pattern at each level t[0,1] in a sequence of the corresponding indicators of i.i.d. [0, 1]-valued observations of length n. A method for obtaining the exact finite-dimensional distributions of the empirical process is given. The weak convergence of the process to a Gaussian process in D[0,1] as n tends to infinity is also established. The limiting process depends on the given pattern. The exact covariance function is compared with the asymptotic covariance function in a numerical example.     

The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend

It is shown that the likelihood ratio of an autoregressive time series of finite order with a regression trend is asymptotically normal. This result is used to derive the power of a test for positive correlation of the residuals under local autoregressive alternatives. The test is based on the Durbin-Watson statistics.     

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.     

On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling

We discuss the local asymptotic behavior of the likelihood function associated with all the four characterizing parameters (α,β,δ,μ) of the Meixner Lévy process under high-frequency sampling scheme. We derive the optimal rate of convergence for each parameter and the Fisher information matrix in a closed form. The skewness parameter β exhibits a slower rate alone, relative to the other three parameters free of sampling rate. An unusual aspect is that the Fisher information matrix is constantly singular for full joint estimation of the four parameters. This is a particular phenomenon in the regular high-frequency sampling setting and is of essentially different nature from low-frequency sampling. As soon as either α or δ is fixed, the Fisher information matrix becomes diagonal, implying that the corresponding maximum likelihood estimators are asymptotically orthogonal.