A multivariate ultrastructural errors-in-variables model with equation error

This paper deals with asymptotic results on a multivariate ultrastructural errors-in-variables regression model with equation errors. Sufficient conditions for attaining consistent estimators for model parameters are presented. Asymptotic distributions for the line regression estimators are derived. Applications to the elliptical class of distributions with two error assumptions are presented. The model generalizes previous results aimed at univariate scenarios.     

Estimation in multivariate errors-in-variables models

This paper reviews and extends some of the known results in the estimation in “errors-in-variables” models, treating the structural and the functional cases on a unified basis. The generalized least-squares method proposed by some previous authors is extended to the case where the error covariance matrix contains an unknown vector parameter. This alleviates the difficulty of multiple roots arising from defining estimators as roots to a set of unbiased estimating equations. An alternative method is also considered for cases with both known and unknown error covariance matrix. The relationship between this method and the usual maximum-likelihood and generalized least-squares approaches is also investigated, and it is shown that in a special case they do not necessarily give identical results in finite samples. Finally, asymptotic results are presented.     

Generalized least squares estimation of the functional multivariate linear errors-in-variables model

Estimators of the parameters of the functional multivariate linear errors-in-variables model are obtained by the application of generalized least squares to the sample matrix of mean squares and products. The generalized least squares estimators are shown to be consistent and asymptotically multivariate normal. Relationships between generalized least squares estimation of the functional model and of the structural model are demonstrated. It is shown that estimators constructed under the assumption of normal x are appropriate for fixed x.     

Discrete-time reduced-order estimator design for bilinear stochastic systems with error covariance assignment

This paper is concerned with the design of reduced-order stateestimators for bilinear stochastic discrete-time systems subjectedto estimation error covariance assignment. The purpose of theproblem addressed is to design the reduced-order state estimators for the bilinear stochastic discrete-time systems such thatthe steady-state estimation error covariances achieve the prespecified values. A simple, effective matrix inequality approach is developed to solve this problem. Specifically, (1) the parameterisationof estimation error covariances that certain bilinear errordynamic processes may possess is presented, (2) the characterisationof all reduced-order state estimators that assign such errorcovariances is explicitly derived, and (3) the solvabilityof the assignability conditions is discussed. Furthermore,an illustrative example is used to demonstrate the effectivenessof the proposed design procedure.     

Consistent nonparametric estimation of error distributions in linear model

For the linear modely _i =x _i +e _i ,i=1, 2 ···, let the error sequence {e _i } _i=1 be iid r.v.'s, with unknown densityf(x). In this paper, a nonparametric estimation method based on the residuals is proposed for estimatingf(x) and the consistency of the estimators is obtained.The project supported by National Natural Science Foundation of China Crant 18971061.     

Permutation tests using least distance estimator in the multivariate regression model

This paper proposes two permutation tests based on the least distance estimator in a multivariate regression model. One is a type of t test statistic using the bootstrap method, and the other is a type of F test statistic using the sum of distances between observed and predicted values under the full and reduced models. We conducted a simulation study to compare the power of the proposed permutation tests with that of the parametric tests based on the least squares estimator for three types of hypotheses in several error distributions. The results indicate that the power of the proposed permutation tests is greater than that of the parametric tests when the error distribution is skewed like the Wishart distribution, has a heavy tail like the Cauchy distribution, or has outliers.     

Consistent estimation and testing in heteroscedastic polynomial errors-in-variables models

The paper concentrates on consistent estimation and testing in functional polynomial measurement errors models with known heterogeneous variances. We rest on the corrected score methodology which allows the derivation of consistent and asymptotically normal estimators for line parameters and also consistent estimators for the asymptotic covariance matrix. Hence, Wald and score type statistics can be proposed for testing the hypothesis of a reduced linear relationship, for example, with asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of small scale simulation studies are reported to illustrate the agreement between theoretical and empirical distributions of the test statistics studied. An application to a real data set is also presented.     

GAMMA－MINIMAX ESTIMATORS FOR THE MEAN OF A MULTIVARIATE NORMAL DISTRIBUTION WITH PARTIALLY UNKNOWN COVARIANCE MATRIX

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The covariance structure and likelihood function for multivariate dyadic data

This paper extends the results in Li and Loken [A unified theory of statistical analysis and inference for variance component models for dyadic data, Statist. Sinica 12 (2002) 519-535] on the statistical analysis of measurements taken on dyads to the situations in which more than one attribute are measured on each dyad. Starting from the covariance structure for the univariate case obtained in Li and Loken (2002), the covariance structure for the multivariate case is derived based on the group symmetry induced by the assumed exchangeability in the units. Our primary objective is to document the Gaussian likelihood and the sufficient statistics for multivariate dyadic data in closed form, so that they can be referenced by researchers as they analyze those data. The derivation carried out can also serve as an example of multivariate extension of univariate models based on exchangeability.     

Asymptotic bounds for the expected L error of a multivariate kernel density estimator

The kernel estimator of a multivariate probability density function is studied. An asymptotic upper bound for the expected L¹ error of the estimator is derived. An asymptotic lower bound result and a formula for the exact asymptotic error are also given. The goodness of the smoothing parameter value derived by minimizing an explicit upper bound is examined in numerical simulations that consist of two different experiments. First, the L¹ error is estimated using numerical integration and, second, the effect of the choice of the smoothing parameter in discrimination tasks is studied.     

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.     

Estimation of the multivariate normal precision and covariance matrices in a star-shape model

In this paper, we introduce the star-shape models, where the precision matrix Ω (the inverse of the covariance matrix) is structured by the special conditional independence. We want to estimate the precision matrix under entropy loss and symmetric loss. We show that the maximal likelihood estimator (MLE) of the precision matrix is biased. Based on the MLE, an unbiased estimate is obtained. We consider a type of Cholesky decomposition of Ω, in the sense that Ω=Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements. A special group , which is a subgroup of the group consisting all lower triangular matrices, is introduced. General forms of equivariant estimates of the covariance matrix and precision matrix are obtained. The invariant Haar measures on , the reference prior, and the Jeffreys prior of Ψ are also discussed. We also introduce a class of priors of Ψ, which includes all the priors described above. The posterior properties are discussed and the closed forms of Bayesian estimators are derived under either the entropy loss or the symmetric loss. We also show that the best equivariant estimators with respect to is the special case of Bayesian estimators. Consequently, the MLE of the precision matrix is inadmissible under either entropy or symmetric loss. The closed form of risks of equivariant estimators are obtained. Some numerical results are given for illustration. The project is supported by the National Science Foundation grants DMS-9972598, SES-0095919, and SES-0351523, and a grant from Federal Aid in Wildlife Restoration Project W-13-R through Missouri Department of Conservation.     

Estimation in the nonlinear errors-in-variables model

In the nonlinear structural errors-in-variables model we propose a consistent estimator of the unknown parameter, using a modified least squares criterion. Its rate of convergence strongly related to the regularity of the regression function, is generally slower than the parametric rate of convergence n^−1/2. Nevertheless, it is of order (log n)r/√n, r > 0, for some analytic regression functions.     

Minimax and admissible minimax estimators of the mean of a multivariate normal distribution for unknown covariance matrix

Let X be a p-variate (p ≥ 3) vector normally distributed with mean μ and covariance Σ, and let A be a p × p random matrix distributed independent of X, according to the Wishart distribution W(n, Σ). For estimating μ, we consider estimators of the form δ = δ(X, A). We obtain families of Bayes, minimax and admissible minimax estimators with respect to the quadratic loss function (δ ? μ)′ Σ^?1(δ ? μ) where Σ is unknown. This paper extends previous results of the author [1], given for the case in which the covariance matrix of the distribution is of the form σ²I, where σ is known.     

Robustness of Stein-type estimators under a non-scalar error covariance structure

The Stein-rule (SR) and positive-part Stein-rule (PSR) estimators are two popular shrinkage techniques used in linear regression, yet very little is known about the robustness of these estimators to the disturbances’ deviation from the white noise assumption. Recent studies have shown that the OLS estimator is quite robust, but whether this is so for the SR and PSR estimators is less clear as these estimators also depend on the F statistic which is highly susceptible to covariance misspecification. This study attempts to evaluate the effects of misspecifying the disturbances as white noise on the SR and PSR estimators by a sensitivity analysis. Sensitivity statistics of the SR and PSR estimators are derived and their properties are analyzed. We find that the sensitivity statistics of these estimators exhibit very similar properties and both estimators are extremely robust to MA(1) disturbances and reasonably robust to AR(1) disturbances except for the cases of severe autocorrelation. The results are useful in light of the rising interest of the SR and PSR techniques in the applied literature.     

Admissibilities of linear estimator in a class of linear models with a multivariate t error variable

This paper discusses admissibilities of estimators in a class of linear models,which include the following common models:the univariate and multivariate linear models,the growth curve model,the extended growth curve model,the seemingly unrelated regression equations,the variance components model,and so on.It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms,called as Model II,are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function.It is also proved under Model II that the usual estimators of β are admissible for p 2 with a quadratic loss function,and are admissible for any p with a matrix loss function,where p is the dimension of β.     

Characterization of the multivariate Gauss-Markoff model with singular covariance matrix and missing values

The aim of this paper is to characterize the Multivariate Gauss-Markoff model (MGM) as in (2.1) with singular covariance matrix and missing values. MGMDP2 model and completed MGMDP2Q model are obtained by three transformations D, P and Q (cf. (3.21)) of MGM. The unified theory of estimation (Rao, 1973) which is of interest with respect to MGM has been used.The characterization is reached by estimation of parameters: scalar ² and linear combination as in (4.8), (4.6), (4.7) as well as by the model of the form (5.1) (cf. Th. 5.1). Moreover, testing linear hypothesis in the available model MGMDP2 by test function F as in (6.3) and (6.4) is considered.It is known (Oktaba 1992) that ten quantities in models MGMDP2, and MGMDP2Q are identical (invariant). They permit to say that formulas for estimation and testing in both models are identical (Oktaba et al., 1988, Baksalary and Kala, 1981, Drygas, 1983).An algorithm and the UMGMBO program for calculations concerning estimation and testing in MGM have been presented by Oktaba and Osypiuk (1993).     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.