Reference prior bayes estimator for bivariate normal covariance matrix with risk comparison

The explicit form of the reference prior bayes estimator due to Yang and Ber-ger (1994) for bivariate normal covariance matrix under entropy loss is given in terms of Legendre polynomials when degrees of freedom is even and in terms of hypergeometric functions in general case. The finite series expression of the density function of the ratio of latent roots of bivariate Wishart matrix is obtained and the exact risk is compared with those of James-Stein minimax estimator and other orthogonally equivariant estimators. It is found numerically that the reference prior bayes estimator has the smallest risk among the class of equivariant estimators compared, when the ratio of the largest to the smallest population latent roots of covariance matrix lies in the middle of the interval [1, ∞]. It has larger risk than that of James-Stein minimax estimator when the ratio is large. Moreover it has larger risk than that of MLE when, for instance, degrees of freedom is 20 and the ratio lies between 4 and 8.     

Convergence of Sample Eigenvectors of Spiked Population Model

In this paper, for the general non Gaussian spiked population model, where a few fixed eigenvalues of the population covariance matrix are separated from others, we investigate the convergence properties of the eigenvectors of sample covariance matrices corresponding to the spiked population eigenvalues and angle between the population eigenvectors and sample eigenvectors as both the sample size and population size are large.     

Unbiased Estimator for a Covariance Matrix Under Two-Step Monotone Incomplete Sample

In this article, we consider an inference for a covariance matrix under two-step monotone incomplete sample. The maximum likelihood estimator of the mean vector is unbiased but that of the covariance matrix is biased. We derive an unbiased estimator for the covariance matrix using some fundamental properties of the Wishart matrix. The properties of the estimators are investigated and the accuracies are checked by a numerical simulation.     

AMEMIYA'S FORM OF THE WEIGHTED LEAST SQUARES ESTIMATOR

Amemiya's estimator is a weighted least squares estimator of the regression coefficients in a linear model with heteroscedastic errors. It is attractive because the heteroscedasticity is not parametrized and the weights (which depend on the error covariance matrix) are estimated nonparametrically. This paper derives an asymptotic expansion for Amemiya's form of the weighted least squares estimator, and uses it to discuss the effects of estimating the weights, of the number of iterations, and of the choice of the initial estimate. The paper also discusses the special case of normally distributed errors and clarifies the particular consequences of assuming normality.     

Ridge regression estimators in the linear regression models with non-spherical errors

The present paper considers a family of ordinary ridge regression estimators in the linear regression model when the disturbances covariance matrix depends upon a few unknown parameters. An asymptotic expansion for the distribution of the ridge regression estimator is developed and under the quadratic loss function its asymptotic risk is compared with that of the feasible GLS estimator.     

Sensitivity analysis in covariance structure analysis with equality constraints

Influence functions are derived for covariance structure analysis with equality constraints, where the parameters are estimated by minimizing a discrepancy function between the assumed covariance matrix and the sample covariance matrix. As a special case maximum likelihood exploratory factor analysis is studied precisely with a numerical example. Comparison is made with the the results of Tanaka and Odaka (1989), who have proposed a sensitivity analysis procedure in maximum likelihood exploratory factor analysis using the perturbation expansion of a certain function of eigenvalues and eigenvectors of a real symmetric matrix. Also the present paper gives a generalization of Tanaka, Watadani and Moon (1991) to the case with equality constraints.     

Another look at Huber's estimator: A new minimax estimator in regression with stochastically bounded noise

Huber's estimator has had a long lasting impact, particularly on robust statistics. It is well known that under certain conditions, Huber's estimator is asymptotically minimax. A moderate generalization in rederiving Huber's estimator shows that Huber's estimator is not the only choice. We develop an alternative asymptotic minimax estimator and name it regression with stochastically bounded noise (RSBN). Simulations demonstrate that RSBN is slightly better in performance, although it is unclear how to justify such an improvement theoretically. We propose two numerical solutions: an iterative numerical solution, which is extremely easy to implement and is based on the proximal point method; and a solution by applying state-of-the-art nonlinear optimization software packages, e.g., SNOPT. Contribution: the generalization of the variational approach is interesting and should be useful in deriving other asymptotic minimax estimators in other problems.     

The Asymptotic Covariance Matrix of the Least Squares Estimator in the Stochastic Linear Regression Model: The Case of Elliptically Symmetric Distribution

This article considers the unconditional asymptotic covariance matrix of the least squares estimator in the linear regression model with stochastic explanatory variables. The asymptotic covariance matrix of the least squares estimator of regression parameters is evaluated relative to the standard asymptotic covariance matrix when the joint distribution of the dependent and explanatory variables is in the class of elliptically symmetric distributions. An empirical example using financial data is presented. Numerical examples and simulation experiments are given to illustrate the difference of the two asymptotic covariance matrices.     

Regularized covariance matrix estimation under the common principal components model

The common principal components (CPC) model provides a way to model the population covariance matrices of several groups by assuming a common eigenvector structure. When appropriate, this model can provide covariance matrix estimators of which the elements have smaller standard errors than when using either the pooled covariance matrix or the per group unbiased sample covariance matrix estimators. In this article, a regularized CPC estimator under the assumption of a common (or partially common) eigenvector structure in the populations is proposed. After estimation of the common eigenvectors using the Flury–Gautschi (or other) algorithm, the off-diagonal elements of the nearly diagonalized covariance matrices are shrunk towards zero and multiplied with the orthogonal common eigenvector matrix to obtain the regularized CPC covariance matrix estimates. The optimal shrinkage intensity per group can be estimated using cross-validation. The efficiency of these estimators compared to the pooled and unbiased estimators is investigated in a Monte Carlo simulation study, and the regularized CPC estimator is applied to a real dataset to demonstrate the utility of the method.     

A Note on Bayesian Analyses of Capture–Recapture Data with Perfect Recaptures

The present article deals with the problem of misspecifying the disturbance-covariance matrix as scalar, when it is locally non scalar. We consider a family of shrinkage estimators based on OLS estimator and compare its asymptotic properties with the properties of OLS estimator. We proposed a similar family of estimators based on FGLS and compared its asymptotic properties with the shrinkage estimator based on OLS under a Pitman's drift process. The effect of misspecifying the disturbances covariance matrix was analyzed with the help of a numerical simulation.     

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.     

ASYMPTOTIC DISTRIBUTION OF THE LARGEST EIGENVALUE

ABSTRACT

This paper studies the asymptotic distribution of the largest eigenvalue of the sample covariance matrix. The multivariate distribution for the population is assumed to be elliptical with finite kurtosis 3κ. An expression as an expectation is obtained for the distribution function of the largest eigenvalue regardless of the multiplicity, m, of the population's largest eigenvalue. The asymptotic distribution function and density function are evaluated numerically for m = 2,3,4,5. The bootstrap of the average of the m largest eigenvalues is shown to be consistent for any underlying distribution with finite fourth-order cumulants.     

An identity for reflexive g-inverses in the context with BLU estimators

In sampling from finite populations conditional minimax estimators are equivalent to BLU estimators. This has been shown by Gabler (1988). If the covariance matrix is singular the BLU estimator can be represented in two different ways.     

Estimation and Test for Multi-Dimensional Regression Models

This work is concerned with the estimation of multi-dimensional regression and the asymptotic behavior of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this article that if we choose to minimize the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.     

Asymptotic distributions of functions of a sample covariance matrix under the elliptical distribution

This paper is concerned with asymptotic distributions of functions of a sample covariance matrix under the elliptical model. Simple but useful formulae for calculating asymptotic variances and covariances of the functions are derived. Also, an asymptotic expansion formula for the expectation of a function of a sample covariance matrix is derived; it is given up to the second-order term with respect to the inverse of the sample size. Two examples are given: one of calculating the asymptotic variances and covariances of the stepdown multiple correlation coefficients, and the other of obtaining the asymptotic expansion formula for the moments of sample generalized variance.     

An Efficient Estimation for Switching Regression Models: A Monte Carlo Study

This article investigates an efficient estimation method for a class of switching regressions based on the characteristic function (CF). We show that with the exponential weighting function, the CF-based estimator can be achieved from minimizing a closed form distance measure. Due to the availability of the analytical structure of the asymptotic covariance, an iterative estimation procedure is developed involving the minimization of a precision measure of the asymptotic covariance matrix. Numerical examples are illustrated via a set of Monte Carlo experiments examining the implementation, finite sample property and the efficiency of the proposed estimator.     

Fast Covariance Estimation for Innovations Computed from a Spatial Gibbs Point Process

In this paper, we derive an exact formula for the covariance of two innovations computed from a spatial Gibbs point process and suggest a fast method for estimating this covariance. We show how this methodology can be used to estimate the asymptotic covariance matrix of the maximum pseudo‐likelihood estimator of the parameters of a spatial Gibbs point process model. This allows us to construct asymptotic confidence intervals for the parameters. We illustrate the efficiency of our procedure in a simulation study for several classical parametric models. The procedure is implemented in the statistical software R , and it is included in spatstat , which is an R package for analyzing spatial point patterns.     

Near Optimal Prediction from Relevant Components

Abstract. The random x regression model is approached through the group of rotations of the eigenvectors for the x ‐covariance matrix together with scale transformations for each of the corresponding regression coefficients. The partial least squares model can be constructed from the orbits of this group. A generalization of Pitman's Theorem says that the best equivariant estimator under a group is given by the Bayes estimator with the group's invariant measure as the prior. A straightforward application of this theorem turns out to be impossible since the relevant invariant prior leads to a non‐defined posterior. Nevertheless we can devise an approximate scale group with a proper invariant prior leading to a well‐defined posterior distribution with a finite mean. This Bayes estimator is explored using Markov chain Monte Carlo technique. The estimator seems to require heavy computations, but can be argued to have several nice properties. It is also a valid estimator when p>n.     

Estimation of proportion ratio in non-compliance randomized trials with repeated measurements in binary data

It is not uncommon to encounter a randomized clinical trial (RCT) in which each patient is treated with several courses of therapies and his/her response is taken after treatment with each course because of the nature of a treatment design for a disease. On the basis of a simple multiplicative risk model proposed elsewhere for repeated binary measurements, we derive the maximum likelihood estimator (MLE) for the proportion ratio (PR) of responses between two treatments in closed form without the need of modeling the complicated relationship between patient’s compliance and patient’s response. We further derive the asymptotic variance of the MLE and propose an asymptotic interval estimator for the PR using the logarithmic transformation. We also consider two other asymptotic interval estimators. One is derived from the principle of Fieller’s Theorem and the other is derived by using the randomization-based approach suggested elsewhere. To evaluate and compare the finite-sample performance of these interval estimators, we apply the Monte Carlo simulation. We find that the interval estimator using the logarithmic transformation of the MLE consistently outperforms the other two estimators with respect to efficiency. This gain in efficiency can be substantial especially when there are patients not complying with their assigned treatments. Finally, we employ the data regarding the trial of using macrophage colony stimulating factor (M-CSF) over three courses of intensive chemotherapies to reduce febrile neutropenia incidence for acute myeloid leukemia patients to illustrate the use of these estimators.     

CONVERGENCE AND PREDICTION OF PRINCIPAL COMPONENT SCORES IN HIGH-DIMENSIONAL SETTINGS

A number of settings arise in which it is of interest to predict Principal Component (PC) scores for new observations using data from an initial sample. In this paper, we demonstrate that naive approaches to PC score prediction can be substantially biased towards 0 in the analysis of large matrices. This phenomenon is largely related to known inconsistency results for sample eigenvalues and eigenvectors as both dimensions of the matrix increase. For the spiked eigenvalue model for random matrices, we expand the generality of these results, and propose bias-adjusted PC score prediction. In addition, we compute the asymptotic correlation coefficient between PC scores from sample and population eigenvectors. Simulation and real data examples from the genetics literature show the improved bias and numerical properties of our estimators.