On f-tests and linear hypothesis in a general linear model

A simple method of setting linear hypotheses testable by F-tests in a general linear model when the covariance matrix has a general form and is completely unknown, is provided. With some additional conditions imposed on the covariance matrix, there exist the UMP invariant tests of certain linear hypotheses. We derive them to compare the powers with those of F-tests obtained under no restrictions on the covariance matrix. The results are illustrated in a multiple regression model with some examples.     

ON UMP INVARIANT F-TEST PROCEDURES IN A GENERAL LINEAR MODEL

A simple method of setting linear hypotheses for a split mean vector testable by F-tests in a general linear model, when the covariance matrix has a general form and is completely unknown, is provided by extending the method discussed in Ukita et al. The critical functions in these F-tests are constructed as UMP invariants, when the covariance matrix has a known structure. Further critical functions in F-tests of linear hypotheses for the other split mean vector in the model are shown to be UMP invariant if the same known structure of the covariance matrix is assumed.     

Selection of Variables in Multivariate Regression Models for Large Dimensions

The Akaike information criterion, AIC, and Mallows’ C _p statistic have been proposed for selecting a smaller number of regressors in the multivariate regression models with fully unknown covariance matrix. All of these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix is much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge-type estimators for the covariance matrix. In this article, we use a ridge-type estimators for the covariance matrix and obtain the modified AIC and modified C _p statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.     

Comparisons of parameter and hypothesis definitions in a general linear model

In the context of the general linear model E(Y)=Xβ possibly subject to restrictions Rβ=r two secondary parameters may be well defined by Θ_i=G_iE(Y)-Θ_oi=C_i β-Θ_oi,i=1,2, and corresponding nonconstant hypotheses, H_i:Θ_i=0. The following possible relations are defined: Θ₁: is dependent upon /equivalent to/identical to Θ₂:H₁is a subhypothesis of/is identical to H₂. Necessary and sufficient conditions, involving straightforward matrix computations, are presented for each relation. Comparisons of secondary parameters and hypotheses are illustrated with an incomplete, unbalanced 3 × 4 factorial design from Searle in which, using a constrained version of Searle's model, parameters and hypotheses in the incomplete, unbalanced design are shown to be indentical to parameters one would define if complete balanced data were available. Techniques for computing simplified definitions are illustrated.     

Test for the mean matrix in a Growth Curve model for high dimensions

We consider the problem of estimating and testing a general linear hypothesis in a general multivariate linear model, the so-called Growth Curve model, when the p × N observation matrix is normally distributed.

The maximum likelihood estimator (MLE) for the mean is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. We modify the MLE to an unweighted estimator and propose new tests which we compare with the previous likelihood ratio test (LRT) based on the weighted estimator, i.e., the MLE. We show that the performance of these new tests based on the unweighted estimator is better than the LRT based on the MLE.     

Estimating and Testing a Structured Covariance Matrix for Three-Level Multivariate Data

This article considers an approach to estimating and testing a new Kronecker product covariance structure for three-level (multiple time points (p), multiple sites (u), and multiple response variables (q)) multivariate data. Testing of such covariance structure is potentially important for high dimensional multi-level multivariate data. The hypothesis testing procedure developed in this article can not only test the hypothesis for three-level multivariate data, but also can test many different hypotheses, such as blocked compound symmetry, for two-level multivariate data as special cases. The tests are implemented with two real data sets.     

Admissible linear estimation in singular linear models

The admissibility results of Rao (1976), proved in the context of a nonsingular covariance matrix, are exteneded to the situation where the covariance matrix is singular. Admi.s s Lb Le linear estimators in the Gauss-Markoff model are characterized and admis-sibility of the best linear unbiased estimator is investigated.     

Admissible linear estimation in singular linear models with respect to a restricted parameter set

The admissibility results of Hoffmann (1977), proved in the context of a nonsingular covariance matrix are extended to the situation where the covariance matrix is singular. Admissible linear estimators in the Gauss-Markoff model are characterised and admissibility of the Best Linear Unbiased Estimator is investigated.     

Robustness of the Multiple Correlation Coefficient When Sampling From a Mixture of Two Multivariate Normal Populations

The density of the multiple correlation coefficient is derived by direct integration when the sample covariance matrix has a linear non-central distribution. Using the density, we deduce the null and non-null distribution of the multiple correlation coefficient when sampling from a mixture of two multivariate normal populations with the same covariance matrix. We also compute actual significance levels of the test of the hypothesis H_o : ρ_1·2…p = 0 versus H_a:ρ_1·2…p > 0, given the mixture model.     

Estimation of parameters in the singular gauss-markoff model

Consider the Gauss-Markoff model (Y, Xβ, σ² V) in the usual notation (Rao, 1973a, p. 294). If V is singular, there exists a matrix N such that N'Y has zero covariance. The minimum variance unbiased estimator of an estimable parametric function p'β is obtained in the wider class of (non-linear) unbiased estimators of the form f(N'Y) + Y'g(N'Y) where f is a scalar and g is a vector function.     

The General Multivariate Gauss—Markov Model of the Incomplete Block Design

The aim of the paper is to generalize testing and estimation for the multivariate standard incomplete block model (Rao & Mitra, 1971a) to the general multivariate Gauss—Markov incomplete block model with singular covariance matrix. The results of this paper can be applied to particular cases of the multivariate Gauss—Markov incomplete block model, including the Zyskind—Martin model.     

Representations of the covariance matrix for robustness in the gauss-markov model

This paper considers some extensions of the results of Rao and Rao and Mitra. They gave a table of general representations of the covariance matrix in terms of the given design matrix, under which various statistical procedures in the least squares theory based on the simple Gauss-Markov model with the spherical covariance matrix are also valid under the general Gauss-Markov model. We shall give extended tables adding some more results relating to robustness, especially in connection with the estimation and testing of hypotheses on linear parametric functions     

Unbiased tests for some one-sided testing problems

Unbiased tests are found for various testing problems. In the first model considered we test homogeneity of k + 1 independent one-parameter exponential family populations vs. the tree-top ordering alternative. The tree-top alternative is appropriate for one-sided comparisons for treatments with a control. In the next set of models normality is assumed. In one such model k independent populations have different unknown means but have an unknown common variance. An independent estimate of the variance exists. We test homogeneity of means against the alternative of no homogeneity. We also consider the alternative of an ordering of the means as well as the tree-top ordering. The final model considered is when we take a random sample from a multivariate normal population with unknown mean vector and an unknown covariance matrix of the intraclass type. We test the hypothesis that the mean vector is the zero vector against the one-sided alternative that each mean is nonnegative (with at least one positive).     

New HEAVY Models for Fat-Tailed Realized Covariances and Returns

ABSTRACT

We develop a new score-driven model for the joint dynamics of fat-tailed realized covariance matrix observations and daily returns. The score dynamics for the unobserved true covariance matrix are robust to outliers and incidental large observations in both types of data by assuming a matrix-F distribution for the realized covariance measures and a multivariate Student's t distribution for the daily returns. The filter for the unknown covariance matrix has a computationally efficient matrix formulation, which proves beneficial for estimation and simulation purposes. We formulate parameter restrictions for stationarity and positive definiteness. Our simulation study shows that the new model is able to deal with high-dimensional settings (50 or more) and captures unobserved volatility dynamics even if the model is misspecified. We provide an empirical application to daily equity returns and realized covariance matrices up to 30 dimensions. The model statistically and economically outperforms competing multivariate volatility models out-of-sample. Supplementary materials for this article are available online.     

Comparisons of estimators of the number of true null hypotheses and adaptive FDR procedures in multiplicity testing

Many exploratory studies such as microarray experiments require the simultaneous comparison of hundreds or thousands of genes. It is common to see that most genes in many microarray experiments are not expected to be differentially expressed. Under such a setting, a procedure that is designed to control the false discovery rate (FDR) is aimed at identifying as many potential differentially expressed genes as possible. The usual FDR controlling procedure is constructed based on the number of hypotheses. However, it can become very conservative when some of the alternative hypotheses are expected to be true. The power of a controlling procedure can be improved if the number of true null hypotheses (m ₀) instead of the number of hypotheses is incorporated in the procedure [Y. Benjamini and Y. Hochberg, On the adaptive control of the false discovery rate in multiple testing with independent statistics, J. Edu. Behav. Statist. 25(2000), pp. 60–83]. Nevertheless, m ₀ is unknown, and has to be estimated. The objective of this article is to evaluate some existing estimators of m ₀ and discuss the feasibility of these estimators in incorporating into FDR controlling procedures under various experimental settings. The results of simulations can help the investigator to choose an appropriate procedure to meet the requirement of the study.     

A control chart based on sample ranges for monitoring the covariance matrix of the multivariate processes

For the univariate case, the R chart and the S ² chart are the most common charts used for monitoring the process dispersion. With the usual sample size of 4 and 5, the R chart is slightly inferior to the S ² chart in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the chart based on the standardized sample ranges, we call the RMAX chart, is substantially inferior in terms of efficiency in detecting shifts in the covariance matrix than the VMAX chart, which is based on the standardized sample variances. The user's familiarity with sample ranges is a point in favor of the RMAX chart. An example is presented to illustrate the application of the proposed chart.     

Missing values: sparse inverse covariance estimation and?an?extension to sparse regression

We propose an ℓ ₁-regularized likelihood method for estimating the inverse covariance matrix in the high-dimensional multivariate normal model in presence of missing data. Our method is based on the assumption that the data are missing at random (MAR) which entails also the completely missing at random case. The implementation of the method is non-trivial as the observed negative log-likelihood generally is a complicated and non-convex function. We propose an efficient EM algorithm for optimization with provable numerical convergence properties. Furthermore, we extend the methodology to handle missing values in a sparse regression context. We demonstrate both methods on simulated and real data.     

Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.     

Optimal designs for asymmetrical factorial paired comparison experiments

Three forms of a general null hypothesis Ho on the factorial parameters of a general asymmetrical factorial paired comparison experiment are considered. A class of partially balanced designscorresponding to each form of H₀ is constructed and the A,D and ioptimal design, minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix of related maximumlikelihoodestimators, in that class is determined. Moreover, the optimal design in each class maximizes the noncentrality parameter λ² of the asymptotic noncentral chi-square distribution of the likelihood ratiostatistic -2 log λ for testing H_o under defined local alternatives. These results apply directly to symmetrical factorial paired comparison experiments as special casesExamples are given forillustrating applications of the developed results     

Estimation in a truncated bivariate poisson distribution

Moment estimators for parameters in a truncated bivariate Poisson distribution are derived in Hamdan (1972) for the special case of λ₁ = λ₂, Where λ₁, λ₂ are the marginal means. Here we derive the maximum likelihood estimators for this special case. The information matrix is also obtained which provides asymptotic covariance matrix of the maximum likelihood estimators. The asymptotic covariance matrix of moment estimators is also derived. The asymptotic efficiency of moment estimators is computed and found to be very low.