Explicit estimators of parameters in the Growth Curve model with linearly structured covariance matrices

Estimation of parameters in the classical Growth Curve model, when the covariance matrix has some specific linear structure, is considered. In our examples maximum likelihood estimators cannot be obtained explicitly and must rely on optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linear structured covariance matrix.     

Estimation of multivariate normal covariance and precision matrices in a star-shape model with missing data

In this paper, we study the problem of estimating the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in a star-shape model with missing data. By considering a type of Cholesky decomposition of the precision matrix Ω=Ψ^′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we get the MLEs of the covariance matrix and precision matrix and prove that both of them are biased. Based on the MLEs, unbiased estimators of the covariance matrix and precision matrix are obtained. A special group G, which is a subgroup of the group consisting all lower triangular matrices, is introduced. By choosing the left invariant Haar measure on G as a prior, we obtain the closed forms of the best equivariant estimates of Ω under any of the Stein loss, the entropy loss, and the symmetric loss. Consequently, the MLE of the precision matrix (covariance matrix) is inadmissible under any of the above three loss functions. Some simulation results are given for illustration.     

Some properties of the lattice points and their application to differential algebra *

In this paper we prove some charateristic conditions about the dimension of an autoreduced subset E of N^m . As an application to differential algebra we find a counter-example to a conjecture about an upper bound for the order of a system of algebraic differential equations ([8], p. 199).     

Nonparametric estimation of the anisotropic probability density of mixed variables

The problem of nonparametric estimation of the joint probability density of a vector of continuous and ordinal/nominal categorical random variables with bounded support is considered. There are numerous publications devoted to the cases of either continuous or categorical variables, and the curse of dimensionality and strong regularity assumptions are the two familiar issues in the literature. Mixed variables occur in practically all applications of the statistical science and, nonetheless, the literature devoted to the joint density estimation is practically next to none. This paper develops the theory of estimation of the density of mixed variables which is on par with results known for simpler settings. Specifically, a data-driven estimator is developed that adapts to unknown anisotropic smoothness of the joint density and, whenever the density depends on a smaller number of variables, performs a dimension reduction that implies the corresponding optimal rate of the mean integrated squared error (MISE) convergence. The results hold without traditional, in the density estimation literature, minimal regularity assumptions like differentiability or continuity of the density. The procedure of estimation is based on mimicking an oracle-estimator that knows the underlying density, and the main theoretical result is the oracle inequality which relates the MISEs of the estimator and the oracle-estimator. The proof is based on a new exponential inequality for Sobolev statistics which is of interest on its own merits.     

Robust dimension reduction based on canonical correlation

The canonical correlation (CANCOR) method for dimension reduction in a regression setting is based on the classical estimates of the first and second moments of the data, and therefore sensitive to outliers. In this paper, we study a weighted canonical correlation (WCANCOR) method, which captures a subspace of the central dimension reduction subspace, as well as its asymptotic properties. In the proposed WCANCOR method, each observation is weighted based on its Mahalanobis distance to the location of the predictor distribution. Robust estimates of the location and scatter, such as the minimum covariance determinant (MCD) estimator of Rousseeuw [P.J. Rousseeuw, Multivariate estimation with high breakdown point, Mathematical Statistics and Applications B (1985) 283-297], can be used to compute the Mahalanobis distance. To determine the number of significant dimensions in WCANCOR, a weighted permutation test is considered. A comparison of SIR, CANCOR and WCANCOR is also made through simulation studies to show the robustness of WCANCOR to outlying observations. As an example, the Boston housing data is analyzed using the proposed WCANCOR method.     

LOCAL INFLUENCE ASSESSMENT BASED ON LOCAL DIVERGENCE IN STOCHASTIC REGRESSION MODEL

Abstract. This paper describes the local influence assessment for parameter inferenceof a statistlcual model by using curvatures assoclated with Iota| divergence under ageneric perturbatlon scheme. The results are applied to examine the local influence instochastlc regresslon model under two perturbation schemes. An economic examp|e isanalyzed to ~llustrate results here.     

Jet spaces of varieties over differential and difference fields

We prove some new structural results on finite-dimensional differential algebraic varieties and difference algebraic varieties in characteristic zero, using elementary methods involving jet spaces. Some partial results and problems are given in the positive characteristic cases. The impact of these methods and results on proofs of the Mordell-Lang conjecture for function fields will also be discussed.     

MOMENT ESTIMATION FOR MULTIVARIATE EXTREME VALUE DISTRIBUTION

Moment estimation for multivariate extreme value distribution is described in this paper. Asymptotic covariance matrix of the estimators is given. The relative efficiencies of moment estimators as compared with the maximum likelihood and the stepwise estimators are computed. We show that when there is strong dependence between the variates, the generalized variance of moment estimators is much lower than the stepwise estimators. It becomes more obvious when the dimension increases.     

Sample covariance shrinkage for high dimensional dependent data

For high dimensional data sets the sample covariance matrix is usually unbiased but noisy if the sample is not large enough. Shrinking the sample covariance towards a constrained, low dimensional estimator can be used to mitigate the sample variability. By doing so, we introduce bias, but reduce variance. In this paper, we give details on feasible optimal shrinkage allowing for time series dependent observations.     

Robust factor analysis

Our aim is to construct a factor analysis method that can resist the effect of outliers. For this we start with a highly robust initial covariance estimator, after which the factors can be obtained from maximum likelihood or from principal factor analysis (PFA). We find that PFA based on the minimum covariance determinant scatter matrix works well. We also derive the influence function of the PFA method based on either the classical scatter matrix or a robust matrix. These results are applied to the construction of a new type of empirical influence function (EIF), which is very effective for detecting influential data. To facilitate the interpretation, we compute a cutoff value for this EIF. Our findings are illustrated with several real data examples.     

Sensitivity of GLS estimators in random effects models

This paper studies the sensitivity of random effects estimators in the one-way error component regression model. Maddala and Mount (1973) [6] give simulation evidence that in random effects models the properties of the feasible GLS estimator are not affected by the choice of the first-step estimator used for the covariance matrix. Taylor (1980) [8] gives a theoretical example of this effect. This paper provides a reason for this in terms of sensitivity. The properties of are transferred via an uncorrelated (and independent under normality) link, called sensitivity. The sensitivity statistic counteracts the improvement in . A Monte Carlo experiment illustrates the theoretical findings.     

Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure

In this paper we study the properties of a kurtosis matrix and propose its eigenvectors as interesting directions to reveal the possible cluster structure of a data set. Under a mixture of elliptical distributions with proportional scatter matrix, it is shown that a subset of the eigenvectors of the fourth-order moment matrix corresponds to Fisher’s linear discriminant subspace. The eigenvectors of the estimated kurtosis matrix are consistent estimators of this subspace and its calculation is easy to implement and computationally efficient, which is particularly favourable when the ratio n/p is large.     

Kernel estimation of density level sets

Let f be a multivariate density and f_n be a kernel estimate of f drawn from the n-sample X₁,…,X_n of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set {f?t} and its plug-in estimator {f_n?t}. As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.     

Estimation of high-dimensional linear factor models with grouped variables

We introduce a generalization of the approximate factor model that divides the observable variables into groups, allows for arbitrarily strong cross-correlation between the disturbance terms of variables that belong to the same group, and for weak correlation between the disturbances of variables that belong to different groups. We call this model the Grouped Variable Approximate Factor Model. We establish identification, propose an estimation approach based on instrumental variable conditions that hold in the limit, and prove consistency in a dual limit framework. Monte Carlo simulations are used to investigate the performance of the estimator, and the techniques are applied to an analysis of industrial output in the US.     

On nonparametric classification with missing covariates

General procedures are proposed for nonparametric classification in the presence of missing covariates. Both kernel-based imputation as well as Horvitz-Thompson-type inverse weighting approaches are employed to handle the presence of missing covariates. In the case of imputation, it is a certain regression function which is being imputed (and not the missing values). Using the theory of empirical processes, the performance of the resulting classifiers is assessed by obtaining exponential bounds on the deviations of their conditional errors from that of the Bayes classifier. These bounds, in conjunction with the Borel-Cantelli lemma, immediately provide various strong consistency results.     

An asymptotic approximation for EPMC in linear discriminant analysis based on two-step monotone missing samples

In this paper, we consider the expected probabilities of misclassification (EPMC) in the linear discriminant function (LDF) based on two-step monotone missing samples and derive an asymptotic approximation for the EPMC with an explicit form for the considered LDF. For this purpose, we also provide some results of the expectations for the inverted Wishart matrices in this paper. Finally, we conduct the Monte Carlo simulation for evaluating our result.     

Multivariate Behrens-Fisher problem with missing data

Inference about the difference between two normal mean vectors when the covariance matrices are unknown and arbitrary is considered. Assuming that the incomplete data are of monotone pattern, a pivotal quantity, similar to the Hotelling T² statistic, is proposed. A satisfactory moment approximation to the distribution of the pivotal quantity is derived. Hypothesis testing and confidence estimation based on the approximate distribution are outlined. The accuracy of the approximation is investigated using Monte Carlo simulation. Monte Carlo studies indicate that the approximate method is very satisfactory even for moderately small samples. The proposed methods are illustrated using an example.