Reduced-rank estimation of the difference between two covariance matrices

We consider m×m$m\times m$ covariance matrices, _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$, which satisfy _Σ2-_Σ1=Δ${\Sigma }_2-{\Sigma }_1=\Delta$, where Δ$\Delta$ has a specified rank. Maximum likelihood estimators of _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$ are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)$\text{rank}(\Delta )$ is known. The likelihood ratio statistic for a test about the value of rank(Δ)$\text{rank}(\Delta )$ is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.     

Likelihood ratio test for one-sided hypothesis of covariance matrices of two normal populations

The likelihood ratio test is derived for a one-sided hypothesis about the covariance matrices from two multivariate normal populations. In the case of equal sample sizes, the limiting distribution of -21og ?_n is given, where ?_n denotes the likelihood ratio criterion. When dimension p=2, for some alternatives, the power of -21og ?_n of size 0.05 is compared with those of several well-known test statistics using Monte Carlo Methods.     

Quadratic perturbation expansions of certain functions of eigenvalues and eigenvectors and their application to sensitivity analysis in multivariate methods

Tanaka (1988) lias derived the influence functions, which are equivalent to the perturbation expansions up to linear terms, of two functions of eigenvalues and eigenvectors of a real symmetric matrix, and applied them to principal component analysis. The present paper deals with the perturbation expansions up to quadratic terms of the same functions and discusses their application to sensitivity analysis in multivariate methods, in particular, principal component analysis and principal factor analysis. Numerical examples are given to show how the approximation improves with the quadratic terms.     

Combination tests for the equality of the covariance matrices of two dependent bivariate normals

To test the equality of the covariance matrices of two dependent bivariate normals, we derive five combination tests using the Simes method. We compare the performance of these tests using simulation to each other and to the competing tests. In particular, simulations show that one of the combination tests has the best performance in terms of controlling the type I error rate even for small samples with similar power compared to other tests. We also apply the recommended test to real data from a crossover bioavailability study.     

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.     

Performance evaluation of likelihood-ratio tests for assessing similarity of the covariance matrices of two multivariate normal populations

Abstract

In analyzing two multivariate normal data sets, the assumption about equality of covariance matrices is usually used as a default for doing subsequence inferences. If this equality doesn’t hold, later inferences will be more complex and usually approximate. If one detects some identical components between two decomposed non equal covariance matrices and uses this extra information, one expects that subsequence inferences can be more accurately performed. For this purpose, in this article we consider some statistical tests about the equality of components of decomposed covariance matrices of two multivariate normal populations. Our emphasis is on the spectral decomposition of these matrices. Hypotheses about the equalities of sizes, shapes, and set of directions as components of these two covariance matrices are tested by the likelihood ratio test (LRT). Some simulation studies are carried out to investigate the accuracy and power of the LRT. Finally, analyses of two real data sets are illustrated.     

On Sensitivity of Inverse Response Plot Estimation and the Benefits of a Robust Estimation Approach

Abstract. Inverse response plots are a useful tool in determining a response transformation function for response linearization in regression. Under some mild conditions it is possible to seek such transformations by plotting ordinary least squares fits versus the responses. A common approach is then to use nonlinear least squares to estimate a transformation by modelling the fits on the transformed response where the transformation function depends on an unknown parameter to be estimated. We provide insight into this approach by considering sensitivity of the estimation via the influence function. For example, estimation is insensitive to the method chosen to estimate the fits in the initial step. Additionally, the inverse response plot does not provide direct information on how well the transformation parameter is being estimated and poor inverse response plots may still result in good estimates. We also introduce a simple robustified process that can vastly improve estimation.     

A robust procedure for testing the equality of mean vectors of two bivariate populations with unequal covariance matrices

A robust test is developed for testing equality of the mean vectors of two bivariate (multivariate) populations when the variance-covariance matrices are not necessarily equal. The test is an extension of the univariate robust test given by Tiku and Singh (1981).     

Two-stage procedures for the difference of two multinormal means with covariance matrices different only by unknown scalar multipliers

We consider the problem of constructing a fixed-size confidence region for the difference of means of two multivariate normal populations It is assumed that the variance-covariance matrices of two populations are different only by unknown scalar multipliers Two-stage procedures are presented to derive such a confidence region We also discuss the asymptotic efficiency of the procedure.     

A simulation comparison of estimators for a regression coefficient under differential non-response

We consider the estimation of a regression coefficient in a linear regression when observations are missing due to nonresponse. Response is assumed to be determined by a nonobservable variable which is linearly related to an observable variable. The values of the observable variable are assumed to be available for the whole sample but the variable is not includsd in the regression relationship of interest . Several alternative estimators have been proposed for this situation under various simplifying assumptions. A sampling theory approach provides three alternative estimatrs by considering the observatins as obtained from a sub-sample, selected on the basis of the fully observable variable , as formulated by Nathan and Holt (1980). Under an econometric approach, Heckman (1979) proposed a two-stage (probit and OLS) estimator which is consistent under specificconditions. A simulation comparison of the four estimators and the ordinary least squares estimator , under multivariate normality of all the variables involved, indicates that the econometric approach estimator is not robust to departures from the conditions underlying its derivation, while two of the other estimators exhibit a similar degree of stable performance over a wide range of conditions. Simulations for a non-normal distribution show that gains in performance can be obtained if observations on the independent variable are available for the whole population.     

The relationship between two parameterizations of linear patterned mean and covariance models

Andrade and Helms (1984) study problems involving estimation and testing of linearly patterned mean and covariance matrices. They parameterize their models under the null hypothesis by using linear constraints on the alternative hypothesis parameterization. In this paper, we show that the nested models that Andrade and Helms consider can be transformed into the nested models considered by Anderson (1969, 1970, 1973) and Szatrowski (1979, 1980, 1981, 1983, 1985).     

Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian covariance matrices

Power studies of tests of equality of covariance matrices of two p-variate complex normal populations σ₁ = σ₂ against two-sided alternatives have been made based on the following five criteria: (1) Roy's largest root, (2) Hotelling's trace, (4) Wilks' criterion and (5) Roy's largest and smallest roots. Some theorems on transformations and Jacobians in the two-sample complex Gaussian case have been proved in order to obtain a general theorem for establishing the local unbiasedness conditions connecting the two critical values for tests (1)–(5). Extensive unbiased power tabulations have been made for p=2, for various values of n₁, n₂, λ₁ and λ₂ where n₁ is the df of the SP matrix from the ith sample and λ₁ is the ith latent root of σ₁σ^-1₂ (i=1, 2). Equal tail areas approach has also been used further to compute powers of tests (1)–(4) for p=2 for studying the bias and facilitating comparisons with powers in the unbiased case. The inferences have been found similar to those in the real case. (Chu and Pillai, Ann. Inst. Statist. Math. 31.     

Unbiased and upper critical values of mean trace of multivariate beta for testing difference of two covariance matrices or several mean vectors

For the invariant unbiased level-α test of equality of two co-variance matrices, the quantities b and B satisfying the equations P(b≤T≤B) = 1-α, E(T|b≤T≤B) = E(T), where T is the mean trace of a multivariate beta, are required. Five and one per cent values of B are tabulated for m = 2,3(2)11,16; b can be obtained from B. Upper five and one per cent values of T are also included, as these are required for the locally most powerful invariant test of nullity of any source of difference in several mean vectors and the locally most powerful invariant one-sided test of equality of two covariance matrices. Lower critical values may be obtained from upper critical values.     

Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality

A test for homogeneity of g ? 2 covariance matrices is presented when the dimension, p, may exceed the sample size, n_i, i = 1, …, g, and the populations may not be normal. Under some mild assumptions on covariance matrices, the asymptotic distribution of the test is shown to be normal when n_i, p → ∞. Under the null hypothesis, the test is extended for common covariance matrix to be of a specified structure, including sphericity. Theory of U-statistics is employed in constructing the tests and deriving their limits. Simulations are used to show the accuracy of tests.     

Asymptotic distribution of hotelling's trace for two unequal covariance matrices and robustness study of test of equality of mean vectors

Asymptotic expansions for the percentiles and c.d.f., up to terms of order $\text{1}\text{n}{}^2$ of the statistic $\text{T =m}\text{T}\text{r}\text{S}{}_1\text{S}{}^{-1}{}_2$, where mS₁ and nS₂ independently distributed W(m, p, Σ₁) and W(n, p, Σ₂) respectively, are obtained using methods similar to those of Ito [4], Chattopadhyay and Pillai [2]. These expansions hold when $\text{Σ}{}_1\text{Σ}{}^{-1}{}_2\text{=}\text{I}\text{+}\text{F}$ and$\text{|}\text{Ch}{}_{\mathrm{i}}\text{(}\text{F}\text{)| < 1}$. Tables of powers of T for p = 3 and p = 4 for m = 4 and various values of n are given and comparison made with the exact powers for p = 3. These powers are useful for the study of (i) the test of equality of covariance matrices in two p-variate normal populations and (ii) robustness of test of equality of mean vectors of l normal populations against the violation of the assumption of equality of covariance matrices.     

A monte carlo study on two methods of calculating the mle's covariance matrix in a seemingly unrelated nonlinear regression.

Econometric techniques to estimate output supply systems, factor demand systems and consumer demand systems have often required estimating a nonlinear system of equations that have an additive error structure when written in reduced form. To calculate the ML estimate's covariance matrix of this nonlinear system one can either invert the Hessian of the concentrated log likelihood function, or invert the matrix calculated by pre-multiplying and post multiplying the inverted MLE of the disturbance covariance matrix by the Jacobian of the reduced form model. Malinvaud has shown that the latter of these methods is the actual limiting distribution's covariance matrix, while Barnett has shown that the former is only an approximation.

In this paper, we use a Monte Carlo simulation study to determine how these two covariance matrices differ with respect to the nonlinearity of the model, the number of observations in the dataet, and the residual process. We find that the covariance matrix calculated from the Hessian of the concentrated likelihood function produces Wald statistics that are distributed above those calculated with the other covariance matrix. This difference becomes insignificant as the sample size increases to one-hundred or more observations, suggesting that the asymptotics of the two covariance matrices are quickly reached.     

A comparison of two varying coefficient meta-analysis methods for an average risk difference

Two interval estimation methods for a general linear function of binomial proportions have been proposed. One method [Zou GY, Huang W, Zhang X. A note on confidence interval estimation for a linear function of binomial proportions. Comput Statist Data Anal. 2009;53:1080–1085] combines Wilson interval estimates of individual proportions, and the other method [Price RM, Bonett DG. An improved confidence interval for a linear function of binomial proportions. Comput Statist Data Anal. 2004;45:449–456] uses an adjusted Wald interval. Both methods are appropriate in varying coefficient meta-analysis models where the risk differences are allowed to vary across studies. The two methods were compared in a simulation study under realistic meta-analysis conditions and the adjusted Wald method was found to have the best performance characteristics.     

A power comparison of three tests for design effects in a random effects covariance model

A preliminary testing procedure for design ettecta in a ran-dom effects covariance model is Compared with the usual procedure to see if the power of the latter can be improved. A procedure which ignores the random covariate effects is included for comparison and for study of misspecification effects. Methodology is based on Roebruck's (1982) results for regular linear models.     

A comparison of bootstrap method and Edgeworth expansion in approximating the distribution of sample variance — one sample and two sample cases

The performance of the bootstrap method and the Edgeworth expansion in approximating the distribution of sample variance are compared when the data are from a non-normal population. Both approximations are very good. so long as the parent population is close to normal.     

A monte carlo study on two methods of calculating the mle's covariance matrix in a seemingly unrelated nonlinear regression. *

Econometric techniques to estimate output supply systems, factor demand systems and consumer demand systems have often required estimating a nonlinear system of equations that have an additive error structure when written in reduced form. To calculate the ML estimate's covariance matrix of this nonlinear system one can either invert the Hessian of the concentrated log likelihood function, or invert the matrix calculated by pre-multiplying and post multiplying the inverted MLE of the disturbance covariance matrix by the Jacobian of the reduced form model. Malinvaud has shown that the latter of these methods is the actual limiting distribution's covariance matrix, while Barnett has shown that the former is only an approximation.

In this paper, we use a Monte Carlo simulation study to determine how these two covariance matrices differ with respect to the nonlinearity of the model, the number of observations in the dataet, and the residual process. We find that the covariance matrix calculated from the Hessian of the concentrated likelihood function produces Wald statistics that are distributed above those calculated with the other covariance matrix. This difference becomes insignificant as the sample size increases to one-hundred or more observations, suggesting that the asymptotics of the two covariance matrices are quickly reached.