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.     

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.     

An asymptotic expansion of Wishart distribution when the population eigenvalues are infinitely dispersed

Takemura and Sheena [A. Takemura, Y. Sheena, Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix, J. Multivariate Anal. 94 (2005) 271–299] derived the asymptotic joint distribution of the eigenvalues and the eigenvectors of a Wishart matrix when the population eigenvalues become infinitely dispersed. They also showed necessary conditions for an estimator of the population covariance matrix to be tail minimax for typical loss functions by calculating the asymptotic risk of the estimator. In this paper, we further examine those distributions and risks by means of an asymptotic expansion. We obtain the asymptotic expansion of the distribution function of relevant elements of the sample eigenvalues and eigenvectors. We also derive the asymptotic expansion of the risk function of a scale and orthogonally equivariant estimator with respect to Stein’s loss. As an application, we prove non-minimaxity of Stein’s and Haff’s estimators, which has been an open problem for a long time.     

Further theoretical results and a comparison between two methods for approximating eigenvalues of perturbed covariance matrices

Covariance matrices, or in general matrices of sums of squares and cross-products, are used as input in many multivariate analyses techniques. The eigenvalues of these matrices play an important role in the statistical analysis of data including estimation and hypotheses testing. It has been recognized that one or few observations can exert an undue influence on the eigenvalues of a covariance matrix. The relationship between the eigenvalues of the covariance matrix computed from all data and the eigenvalues of the perturbed covariance matrix (a covariance matrix computed after a small subset of the observations has been deleted) cannot in general be written in closed-form. Two methods for approximating the eigenvalues of a perturbed covariance matrix have been suggested by Hadi (1988) and Wang and Nyquist (1991) for the case of a perturbation by a single observation. In this paper we improve on these two methods and give some additional theoretical results that may give further insight into the problem. We also compare the two improved approximations in terms of their accuracies.     

On the limiting distributions of the jackknife statistics for eigenvalues of a sample covariance matrix

The limiting distributions of jackknife statistics for eigenvalues of a sample covariance matrix are derived under the nonnormal situations. Also the numerical examples are given under normal and nonnormal populations.     

A Selection Procedure for the Number of Signals in Presence of Colored Noise

Ranking and selection theory is used to estimate the number of signals present in colored noise. The data structure follows the well-known MUSIC (MUltiple SIgnal Classification) model. We deal with the eigenvalues of a covariance matrix, using the MUSIC model and colored noise. The data matrix can be written as the product of two matrices. The first matrix is the sample covariance matrix of the observed vectors. The second matrix is the inverse of the sample covariance matrix of reference vectors. We propose a multi-step selection procedure to construct a confidence interval on the number of signals present in a data set. Properties of this procedure will be stated and proved. Those properties will be used to compute the required parameters (procedure constants). Numerical examples are given to illustrate our theory.     

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.     

Estimation of parameters in a generalized GMANOVA model based on an outer product analogy and least squares

This paper investigates estimation of parameters in a combination of the multivariate linear model and growth curve model, called a generalized GMANOVA model. Making analogy between the outer product of data vectors and covariance yields an approach to directly do least squares to covariance. An outer product least squares estimator of covariance (COPLS estimator) is obtained and its distribution is presented if a normal assumption is imposed on the error matrix. Based on the COPLS estimator, two-stage generalized least squares estimators of the regression coefficients are derived. In addition, asymptotic normalities of these estimators are investigated. Simulation studies have shown that the COPLS estimator and two-stage GLS estimators are alternative competitors with more efficiency in the sense of sample mean, standard deviations and mean of the variance estimates to the existing ML estimator in finite samples. An example of application is also illustrated.     

Large Dynamic Covariance Matrices

Second moments of asset returns are important for risk management and portfolio selection. The problem of estimating second moments can be approached from two angles: time series and the cross-section. In time series, the key is to account for conditional heteroscedasticity; a favored model is Dynamic Conditional Correlation (DCC), derived from the ARCH/GARCH family started by Engle (1982). In the cross-section, the key is to correct in-sample biases of sample covariance matrix eigenvalues; a favored model is nonlinear shrinkage, derived from Random Matrix Theory (RMT). The present article marries these two strands of literature to deliver improved estimation of large dynamic covariance matrices. Supplementary material for this article is available online.     

A selection procedure for estimating the number of signal components

This paper proposes a selection procedure to estimate the multiplicity of the smallest eigenvalue of the covariance matrix. The unknown number of signals present in a radar data can be formulated as the difference between the total number of components in the observed multivariate data vector and the multiplicity of the smallest eigenvalue. In the observed multivariate data, the smallest eigenvalues of the sample covariance matrix may in fact be grouped about some nominal value, as opposed to being identically equal. We propose a selection procedure to estimate the multiplicity of the common smallest eigenvalue, which is significantly smaller than the other eigenvalues. We derive the probability of a correct selection, P(CS), and the least favorable configuration (LFC) for our procedures. Under the LFC, the P(CS) attains its minimum over the preference zone of all eigenvalues. Therefore, a minimum sample size can be determined from the P(CS) under the LFC, P(CS|LFC), in order to implement our new procedure with a guaranteed probability requirement. Numerical examples are presented in order to illustrate our proposed procedure.     

More on local influence in pllincipal components analysis

Local influence on the eigenvalues of sample covariance matrices in

principal components analysis is examined for a reasonable modification of Shi's (1997) perturbation scheme, The modification is suggested for samples from populations with both unknown mean vector and covariance matrix. While Shi's detection indexes (1997) consist of only quadratic terms, the modified perturbation scheme leads to detection indexes constituted by both linear and quadratic terms associated with centralized observations. These linear and quadratic terms reflect local influences on the first two sample moments. Examples are investigated based on the two detection indexes.     

High-dimensional Markowitz portfolio optimization problem: empirical comparison of covariance matrix estimators

We compare the performance of recently developed regularized covariance matrix estimators for Markowitz's portfolio optimization and of the minimum variance portfolio (MVP) problem in particular. We focus on seven estimators that are applied to the MVP problem in the literature; three regularize the eigenvalues of the sample covariance matrix, and the other four assume the sparsity of the true covariance matrix or its inverse. Comparisons are made with two sets of long-term S&P 500 stock return data that represent two extreme scenarios of active and passive management. The results show that the MVPs with sparse covariance estimators have high Sharpe ratios but that the naive diversification (also known as the ‘uniform (on market share) portfolio’) still performs well in terms of wealth growth.     

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.     

Predicting the Daily Covariance Matrix for S&P 100 Stocks Using Intraday Data—But Which Frequency to Use?

This article investigates the merits of high-frequency intraday data when forming mean-variance efficient stock portfolios with daily rebalancing from the individual constituents of the S&P 100 index. We focus on the issue of determining the optimal sampling frequency as judged by the performance of these portfolios. The optimal sampling frequency ranges between 30 and 65 minutes, considerably lower than the popular five-minute frequency, which typically is motivated by the aim of striking a balance between the variance and bias in covariance matrix estimates due to market microstructure effects such as non-synchronous trading and bid-ask bounce. Bias-correction procedures, based on combining low-frequency and high-frequency covariance matrix estimates and on the addition of leads and lags do not substantially affect the optimal sampling frequency or the portfolio performance. Our findings are also robust to the presence of transaction costs and to the portfolio rebalancing frequency.     

Predicting the Daily Covariance Matrix for S&P 100 Stocks Using Intraday Data—But Which Frequency to Use?

This article investigates the merits of high-frequency intraday data when forming mean-variance efficient stock portfolios with daily rebalancing from the individual constituents of the S&P 100 index. We focus on the issue of determining the optimal sampling frequency as judged by the performance of these portfolios. The optimal sampling frequency ranges between 30 and 65 minutes, considerably lower than the popular five-minute frequency, which typically is motivated by the aim of striking a balance between the variance and bias in covariance matrix estimates due to market microstructure effects such as non-synchronous trading and bid-ask bounce. Bias-correction procedures, based on combining low-frequency and high-frequency covariance matrix estimates and on the addition of leads and lags do not substantially affect the optimal sampling frequency or the portfolio performance. Our findings are also robust to the presence of transaction costs and to the portfolio rebalancing frequency.     

The minimum regularized covariance determinant estimator

Statistics and Computing - The minimum covariance determinant (MCD) approach estimates the location and scatter matrix using the subset of given size with lowest sample covariance determinant. Its...     

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.     

Estimation of the optimal portfolio weights by shrinking the mean vector towards a linear subspace

To obtain estimators of mean-variance optimal portfolio weights, Stein-type estimators of the mean vector that shrink a sample mean towards the grand mean have been applied. However, the dominance of these estimators has not been shown under the loss function used in the estimation problem of the mean-variance optimal portfolio weights, which is different than the quadratic function for the case in which the covariance matrix is unknown. We analytically give the conditions for Stein-type estimators that shrink towards the grand mean, or more generally, towards a linear subspace, to improve upon the classical estimators, which are obtained by simply plugging in sample estimates. We also show the dominance when there are linear constraints on portfolio weights.     

On a statistic useful in dimensionality reduction in multivariable linear stochastic system

In this paper we study the sampling properties of a test statistic which has important applications in the area of linear stochastic control systems with multi-inputs and multi-outputs. The statistic is the ratio of a partial sum of the eigenvalues of a sample covariance matrix and its trace. It turns out that using a method due to Sugiura we may derive a useful approximation for its distribution up to and including terms of order l/n, where n denotes the appropriate size. Numerical illustrations using real data are given.