Restricted estimation in multivariate measurement error regression model

We study a multivariate ultrastructural measurement error (MUME) model with more than one response variable. This model is a synthesis of multivariate functional and structural models. Three consistent estimators of regression coefficients, satisfying the exact linear restrictions have been proposed. Their asymptotic distributions are derived under the assumption of a non-normal measurement error and random error components. A simulation study is carried out to investigate the small sample properties of the estimators. The effect of departure from normality of the measurement errors on the estimators is assessed.     

On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which are notoriously heavy-tailed. Let π denote the intractable posterior density that results when this regression model is combined with the standard non-informative prior on the unknown regression coefficients and scale matrix of the errors. Roughly speaking, the posterior is proper if and only if n≥d+k, where n is the sample size, d is the dimension of the response, and k is number of covariates. We provide a method of making exact draws from π in the special case where n=d+k, and we study Markov chain Monte Carlo (MCMC) algorithms that can be used to explore π when n>d+k. In particular, we show how the Haar PX-DA technology studied in Hobert and Marchev (2008) [11] can be used to improve upon Liu’s (1996) [7] data augmentation (DA) algorithm. Indeed, the new algorithm that we introduce is theoretically superior to the DA algorithm, yet equivalent to DA in terms of computational complexity. Moreover, we analyze the convergence rates of these MCMC algorithms in the important special case where the regression errors have a Student’s t distribution. We prove that, under conditions on n, d, k, and the degrees of freedom of the t distribution, both algorithms converge at a geometric rate. These convergence rate results are important from a practical standpoint because geometric ergodicity guarantees the existence of central limit theorems which are essential for the calculation of valid asymptotic standard errors for MCMC based estimates.     

Restricted expected multivariate least squares

A new approach of estimating parameters in multivariate models is introduced. A fitting function will be used. The idea is to estimate parameters so that the fitting function equals or will be close to its expected value. The function will be decomposed into two parts. From one part, which will be independent of the mean parameters, the dispersion matrix is estimated. This estimator is inserted in the second part which then yields the estimators of the mean parameters. The Growth Curve model, extended Growth Curve model and a multivariate variance components model will illustrate the approach.     

Case deletion diagnostics in multilevel models

This paper studies case deletion diagnostics for multilevel models. Using subset deletion, diagnostic measures for identifying influential units at any level are developed for both fixed and random parameters. Two approximate update formulae are derived. The first formula uses one-step approximation, while the second formula also includes the impact of estimating the random parameter. Two examples are used to illustrate the methodology developed.     

Operator trigonometry of multivariate finance

We inquire into an operator-trigonometric analysis of certain multi-asset financial pricing models. Our goal is to provide a new geometric point of view for the understanding and analysis of such financial instruments. Among those instruments which we examine are quantos for currency hedging, spread options for multi-asset pricing, portfolio rebalancing under stochastic interest rates, Black-Scholes volatility models, and risk measures.     

Minimax estimation for singular linear multivariate models with mixed uncertainty

The problem of minimax estimation is examined for the linear multivariate statistically indeterminate observation model with mixed uncertainty. The a priori information on the distributions of model parameters is formulated in terms of second-order moment characteristics. It is shown that in the regular case the minimax estimate is defined explicitly via the solution of the dual optimization problem. For singular models, the method of dual optimization is developed by means of using the Tikhonov regularization techniques. Several particular cases which are widely used in practice are also considered.     

Asymptotic robustness of standard errors in multilevel structural equation models

Data in social and behavioral sciences are often hierarchically organized. Multilevel statistical methodology was developed to analyze such data. Most of the procedures for analyzing multilevel data are derived from maximum likelihood based on the normal distribution assumption. Standard errors for parameter estimates in these procedures are obtained from the corresponding information matrix. Because practical data typically contain heterogeneous marginal skewnesses and kurtoses, this paper studies how nonnormally distributed data affect the standard errors of parameter estimates in a two-level structural equation model. Specifically, we study how skewness and kurtosis in one level affect standard errors of parameter estimates within its level and outside its level. We also show that, parallel to asymptotic robustness theory in conventional factor analysis, conditions exist for asymptotic robustness of standard errors in a multilevel factor analysis model.     

Rank estimation in reduced-rank regression

Reduced rank regression assumes that the coefficient matrix in a multivariate regression model is not of full rank. The unknown rank is traditionally estimated under the assumption of normal responses. We derive an asymptotic test for the rank that only requires the response vector have finite second moments. The test is extended to the nonconstant covariance case. Linear combinations of the components of the predictor vector that are estimated to be significant for modelling the responses are obtained.     

Orthant tail dependence of multivariate extreme value distributions

The orthant tail dependence describes the relative deviation of upper- (or lower-) orthant tail probabilities of a random vector from similar orthant tail probabilities of a subset of its components, and can be used in the study of dependence among extreme values. Using the conditional approach, this paper examines the extremal dependence properties of multivariate extreme value distributions and their scale mixtures, and derives the explicit expressions of orthant tail dependence parameters for these distributions. Properties of the tail dependence parameters, including their relations with other extremal dependence measures used in the literature, are discussed. Various examples involving multivariate exponential, multivariate logistic distributions and copulas of Archimedean type are presented to illustrate the results.     

A family of estimators for multivariate kurtosis in a nonnormal linear regression model

In this paper, we propose a new estimator for a kurtosis in a multivariate nonnormal linear regression model. Usually, an estimator is constructed from an arithmetic mean of the second power of the squared sample Mahalanobis distances between observations and their estimated values. The estimator gives an underestimation and has a large bias, even if the sample size is not small. We replace this squared distance with a transformed squared norm of the Studentized residual using a monotonic increasing function. Our proposed estimator is defined by an arithmetic mean of the second power of these squared transformed squared norms with a correction term and a tuning parameter. The correction term adjusts our estimator to an unbiased estimator under normality, and the tuning parameter controls the sizes of the squared norms of the residuals. The family of our estimators includes estimators based on ordinary least squares and predicted residuals. We verify that the bias of our new estimator is smaller than usual by constructing numerical experiments.     

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of non-zero subdeterminants.     

Tail dependence for multivariate t -copulas and its monotonicity

The tail dependence indexes of a multivariate distribution describe the amount of dependence in the upper right tail or lower left tail of the distribution and can be used to analyse the dependence among extremal random events. This paper examines the tail dependence of multivariate t-distributions whose copulas are not explicitly accessible. The tractable formulas of tail dependence indexes of a multivariate t-distribution are derived in terms of the joint moments of its underlying multivariate normal distribution, and the monotonicity properties of these indexes with respect to the distribution parameters are established. Simulation results are presented to illustrate the results.     

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 adjustment technique in a parameter restricted linear model

We consider an approach yielding a minimax estimator in the linear regression model with a priori information on the parameter vector, e.g., ellipsoidal restrictions. This estimator is computed directly from the loss function and can be motivated by the general Pitman nearness criterion. It turns out that this approach coincides with the projection estimator which is obtained by projecting an initial arbitrary estimate on the subset defined by the restrictions.     

Minimax least squares and quasiminimax estimation in linear models with missing values

We consider the problem of estimating the parameter vector in the linear model when observations on the independent variables are partially missing or incorrect. New estimators are developed, which systematically combine prior information with the incomplete data. We compare these methods with the alternative strategy of deleting incomplete observations.Support by Deutsche Forschungsgemeinschaft, Grant No. 284/1-2 is gratefully acknowledged.     

Linear plus quadratic (LPQ) quasiminimax estimation in the linear regression model

If the errors in the linear regression model are assumed to be independent with nonvanishing third and finite fourth moments, then it is possible to improve all linear estimators by so-called linear plus quadratic (LPQ) estimators. These consist of linear and quadratic terms in the endogeneous variable and depend on the unknown moments of the errors which, in general, have to be estimated from the data. In this paper, we will use LPQ estimators for quasiminimax estimation and some related problems.Support by Deutsche Forschungsgemeinschaft Grant No. Tr 253/1-2 is gratefully acknowledged.     

Measures of influence for the functional linear model with scalar response

This paper studies how to identify influential observations in the functional linear model in which the predictor is functional and the response is scalar. Measurement of the effects of a single observation on estimation and prediction when the model is estimated by the principal components method is undertaken. For that, three statistics are introduced for measuring the influence of each observation on estimation and prediction of the functional linear model with scalar response that are generalizations of the measures proposed for the standard regression model by [D.R. Cook, Detection of influential observations in linear regression, Technometrics 19 (1977) 15-18; D. Peña, A new statistic for influence in linear regression, Technometrics 47 (2005) 1-12] respectively. A smoothed bootstrap method is proposed to estimate the quantiles of the influence measures, which allows us to point out which observations have the larger influence on estimation and prediction. The behavior of the three statistics and the quantile estimation bootstrap based method is analyzed via a simulation study. Finally, the practical use of the proposed statistics is illustrated by the analysis of a real data example, which show that the proposed measures are useful for detecting heterogeneity in the functional linear model with scalar response.     

Nonnegative quadratic estimation of the mean squared errors of minimax estimators in the linear regression model

In this paper, the problem of nonnegative quadratic estimation of the mean squared errors of minimax estimators of in the linear regression modelE(y)=X, VAR(y) = ² is discussed. An explicit formula for the admissible nonnegative minimum biased estimator is given. Some applications to one-way classification model are also considered.     

Probabilistic nonadaptive group testing in the presence of errors and DNA library screening

We use the subset containment relation to construct a probabilistic nonadaptive group testing design and decoding algorithm that, in the presence of testing errors, identifies many positives in a population. We give a lower bound for the expected portion of positives identified as a function of an upper bound on the number of testing errors.The algorithms contained herein are part of The State University of New York Research Foundation invention C1230-125, Probabilistic and Combinatorial Nonadaptive and Two-Stage Group Testing and DNA Library Screening by A. Macula and K. Anne.     

Asymptotic Bayesian structure learning using graph supports for Gaussian graphical models

The theory of Gaussian graphical models is a powerful tool for independence analysis between continuous variables. In this framework, various methods have been conceived to infer independence relations from data samples. However, most of them result in stepwise, deterministic, descent algorithms that are inadequate for solving this issue. More recent developments have focused on stochastic procedures, yet they all base their research on strong a priori knowledge and are unable to perform model selection among the set of all possible models. Moreover, convergence of the corresponding algorithms is slow, precluding applications on a large scale. In this paper, we propose a novel Bayesian strategy to deal with structure learning. Relating graphs to their supports, we convert the problem of model selection into that of parameter estimation. Use of non-informative priors and asymptotic results yield a posterior probability for independence graph supports in closed form. Gibbs sampling is then applied to approximate the full joint posterior density. We finally give three examples of structure learning, one from synthetic data, and the two others from real data.