Multivariate dependence of spacings of generalized order statistics

Multivariate dependence of spacings of generalized order statistics is studied. It is shown that spacings of generalized order statistics from DFR (IFR) distributions have the CIS (CDS) property. By restricting the choice of the model parameters and strengthening the assumptions on the underlying distribution, stronger dependence relations are established. For instance, if the model parameters are decreasingly ordered and the underlying distribution has a log-convex decreasing (log-concave) hazard rate, then the spacings satisfy the MTP₂ (S- MRR₂) property. Some consequences of the results are given. In particular, conditions for non-negativity of the best linear unbiased estimator of the scale parameter in a location-scale family are obtained. By applying a result for dual generalized order statistics, we show that in the particular situation of usual order statistics the assumptions can be weakened.     

Multivariate analysis of variance with fewer observations than the dimension

In this article, we consider the problem of testing a linear hypothesis in a multivariate linear regression model which includes the case of testing the equality of mean vectors of several multivariate normal populations with common covariance matrix Σ, the so-called multivariate analysis of variance or MANOVA problem. However, we have fewer observations than the dimension of the random vectors. Two tests are proposed and their asymptotic distributions under the hypothesis as well as under the alternatives are given under some mild conditions. A theoretical comparison of these powers is made.     

Semiparametric analysis based on weighted estimating equations for transformation models with missing covariates

Missing covariate data are very common in regression analysis. In this paper, the weighted estimating equation method (Qi et al., 2005) [25] is used to extend the so-called unified estimation procedure (Chen et al., 2002) [4] for linear transformation models to the case of missing covariates. The non-missingness probability is estimated nonparametrically by the kernel smoothing technique. Under missing at random, the proposed estimators are shown to be consistent and asymptotically normal, with the asymptotic variance estimated consistently by the usual plug-in method. Moreover, the proposed estimators are more efficient than the weighted estimators with the inverse of true non-missingness probability as weight. Finite sample performance of the estimators is examined via simulation and a real dataset is analyzed to illustrate the proposed methods.     

Methods for improvement in estimation of a normal mean matrix

This paper is concerned with the problem of estimating a matrix of means in multivariate normal distributions with an unknown covariance matrix under invariant quadratic loss. It is first shown that the modified Efron-Morris estimator is characterized as a certain empirical Bayes estimator. This estimator modifies the crude Efron-Morris estimator by adding a scalar shrinkage term. It is next shown that the idea of this modification provides a general method for improvement of estimators, which results in the further improvement on several minimax estimators. As a new method for improvement, an adaptive combination of the modified Stein and the James-Stein estimators is also proposed and is shown to be minimax. Through Monte Carlo studies of the risk behaviors, it is numerically shown that the proposed, combined estimator inherits the nice risk properties of both individual estimators and thus it has a very favorable risk behavior in a small sample case. Finally, the application to a two-way layout MANOVA model with interactions is discussed.     

Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data

In this article, the Stein-Haff identity is established for a singular Wishart distribution with a positive definite mean matrix but with the dimension larger than the degrees of freedom. This identity is then used to obtain estimators of the precision matrix improving on the estimator based on the Moore-Penrose inverse of the Wishart matrix under the Efron-Morris loss function and its variants. Ridge-type empirical Bayes estimators of the precision matrix are also given and their dominance properties over the usual one are shown using this identity. Finally, these precision estimators are used in a quadratic discriminant rule, and it is shown through simulation that discriminant methods based on the ridge-type empirical Bayes estimators provide higher correct classification rates.     

Estimation of means of multivariate normal populations with order restriction

Multivariate isotonic regression theory plays a key role in the field of statistical inference under order restriction for vector valued parameters. Two cases of estimating multivariate normal means under order restricted set are considered. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but are restricted by partial order. This paper shows that when covariance matrices are known, the estimator given by this paper always dominates unrestricted maximum likelihood estimator uniformly, and when covariance matrices are unknown, the plug-in estimator dominates unrestricted maximum likelihood estimator under the order restricted set of covariance matrices. The isotonic regression estimators in this paper are the generalizations of plug-in estimators in unitary case.     

On weighting of bivariate margins in pairwise likelihood

Composite and pairwise likelihood methods have recently been increasingly used. For clustered data with varying cluster sizes, we study asymptotic relative efficiencies for various weighted pairwise likelihoods, with weight being a function of cluster size. For longitudinal data, we also study weighted pairwise likelihoods with weights that can depend on lag. Good choice of weights are needed to avoid the undesirable behavior of estimators with low efficiency. Some analytic results are obtained using the multivariate normal distribution. For clustered data, a practically good choice of weight is obtained after study of relative efficiencies for an exchangeable multivariate normal model; they are different from weights that had previously been suggested. For longitudinal data, there are advantages to only include bivariate margins of adjacent or nearly adjacent pairs in the weighted pairwise likelihood.     

Statistical inference using a weighted difference-based series approach for partially linear regression models

Partially linear regression models with fixed effects are useful tools for making econometric analyses and normalizing microarray data. Baltagi and Li (2002) [7] proposed a computation friendly difference-based series estimation (DSE) for them. We show that the DSE is not asymptotically efficient in most cases and further propose a weighted difference-based series estimation (WDSE). The weights in it do not involve any unknown parameters. The asymptotic properties of the resulting estimators are established for both balanced and unbalanced cases, and it is shown that they achieve a semiparametric efficient boundary. Additionally, we propose a variable selection procedure for identifying significant covariates in the parametric part of the semiparametric fixed-effects regression model. The method is based on a combination of the nonconcave penalization (Fan and Li, 2001 [13]) and weighted difference-based series estimation techniques. The resulting estimators have the oracle property; that is, they can correctly identify the true model as if the true model (the subset of variables with nonvanishing coefficients) were known in advance. Simulation studies are conducted and an application is given to demonstrate the finite sample performance of the proposed procedures.     

Bias correction of cross-validation criterion based on Kullback-Leibler information under a general condition

This paper deals with the bias correction of the cross-validation (CV) criterion to estimate the predictive Kullback-Leibler information. A bias-corrected CV criterion is proposed by replacing the ordinary maximum likelihood estimator with the maximizer of the adjusted log-likelihood function. The adjustment is just slight and simple, but the improvement of the bias is remarkable. The bias of the ordinary CV criterion is O(n^-1), but that of the bias-corrected CV criterion is O(n^-2). We verify that our criterion has smaller bias than the AIC, TIC, EIC and the ordinary CV criterion by numerical experiments.     

Analysis of NMAR missing data without specifying missing-data mechanisms in a linear latent variate model

It is natural to assume that a missing-data mechanism depends on latent variables in the analysis of incomplete data in latent variate modeling because latent variables are error-free and represent key notions investigated by applied researchers. Unfortunately, the missing-data mechanism is then not missing at random (NMAR). In this article, a new estimation method is proposed, which leads to consistent and asymptotically normal estimators for all parameters in a linear latent variate model, where the missing mechanism depends on the latent variables and no concrete functional form for the missing-data mechanism is used in estimation. The method to be proposed is a type of multi-sample analysis with or without mean structures, and hence, it is easy to implement. Complete-case analysis is shown to produce consistent estimators for some important parameters in the model.     

Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data

Wong and Yu [Generalized MLE of a joint distribution function with multivariate interval-censored data, J. Multivariate Anal. 69 (1999) 155-166] discussed generalized maximum likelihood estimation of the joint distribution function of a multivariate random vector whose coordinates are subject to interval censoring. They established uniform consistency of the generalized MLE (GMLE) of the distribution function under the assumption that the random vector is independent of the censoring vector and that both of the vector distributions are discrete. We relax these assumptions and establish consistency results of the GMLE under a multivariate mixed case interval censorship model. van der Vaart and Wellner [Preservation theorems for Glivenko-Cantelli and uniform Glivenko-Cantelli class, in: E. Gine, D.M. Mason, J.A. Wellner (Eds.), High Dimensional Probability, vol. II, Birkhäuser, Boston, 2000, pp. 115-133] and Yu [Consistency of the generalized MLE with multivariate mixed case interval-censored data, Ph.D Dissertation, Binghamton University, 2000] independently proved strong consistency of the GMLE in the L₁(μ)-topology, where μ is a measure derived from the joint distribution of the censoring variables. We establish strong consistency of the GMLE in the topologies of weak convergence and pointwise convergence, and eventually uniform convergence under appropriate distributional assumptions and regularity conditions.     

Nonparametric estimation of competing risks models with covariates

In competing risks model, several failure times arise potentially. The smallest failure time and its index only are observed. Without specific assumptions, the joint or even the marginal distribution functions of the underlying failure times are not identifiable (A. Tsiatis, Proc. Natl. Acad. Sci. USA 72 (1975) 20). Nonetheless, if each individual is characterized by a “sufficiently informative” set of covariates, these distributions are identifiable under some conditions of regularity (J.J. Heckman and B. Honoré, Biometrika 76 (1989) 325). In this paper, nonparametric kernel estimators of the joint distribution function of failure times conditional on the covariates are proposed. Their weak and strong consistency are discussed.     

Smooth depth contours characterize the underlying distribution

The Tukey depth is an innovative concept in multivariate data analysis. It can be utilized to extend the univariate order concept and advantages to a multivariate setting. While it is still an open question as to whether the depth contours uniquely determine the underlying distribution, some positive answers have been provided. We extend these results to distributions with smooth depth contours, with elliptically symmetric distributions as special cases. The key ingredient of our proofs is the well-known Cramér-Wold theorem.     

Multivariate copulas, quasi-copulas and lattices

We investigate some properties of the partially ordered sets of multivariate copulas and quasi-copulas. Whereas the set of bivariate quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind-MacNeille completion of the set of bivariate copulas, we show that this is not the case in higher dimensions.     

Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors

We consider a panel data semiparametric partially linear regression model with an unknown parameter vector for the linear parametric component, an unknown nonparametric function for the nonlinear component, and a one-way error component structure which allows unequal error variances (referred to as heteroscedasticity). We develop procedures to detect heteroscedasticity and one-way error component structure, and propose a weighted semiparametric least squares estimator (WSLSE) of the parametric component in the presence of heteroscedasticity and/or one-way error component structure. This WSLSE is asymptotically more efficient than the usual semiparametric least squares estimator considered in the literature. The asymptotic properties of the WSLSE are derived. The nonparametric component of the model is estimated by the local polynomial method. Some simulations are conducted to demonstrate the finite sample performances of the proposed testing and estimation procedures. An example of application on a set of panel data of medical expenditures in Australia is also illustrated.     

A profile-type smoothed score function for a varying coefficient partially linear model

The varying coefficient partially linear model is considered in this paper. When the plug-in estimators of coefficient functions are used, the resulting smoothing score function becomes biased due to the slow convergence rate of nonparametric estimations. To reduce the bias of the resulting smoothing score function, a profile-type smoothed score function is proposed to draw inferences on the parameters of interest without using the quasi-likelihood framework, the least favorable curve, a higher order kernel or under-smoothing. The resulting profile-type statistic is still asymptotically Chi-squared under some regularity conditions. The results are then used to construct confidence regions for the parameters of interest. A simulation study is carried out to assess the performance of the proposed method and to compare it with the profile least-squares method. A real dataset is analyzed for illustration.     

A two-stage approach to semilinear in-slide models

The semilinear in-slide models (SLIMs) have been shown to be effective methods for normalizing microarray data [J. Fan, P. Tam, G. Vande Woude, Y. Ren, Normalization and analysis of cDNA micro-arrays using within-array replications applied to neuroblastoma cell response to a cytokine, Proceedings of the National Academy of Science (2004) 1135-1140]. Using a backfitting method, [J. Fan, H. Peng, T. Huang, Semilinear high-dimensional model for normalization of microarray data: a theoretical analysis and partial consistency, Journal of American Statistical Association, 471, (2005) 781-798] proposed a profile least squares (PLS) estimation for the parametric and nonparametric components. The general asymptotic properties for their estimator is not developed. In this paper, we consider a new approach, two-stage estimation, which enables us to establish the asymptotic normalities for both of the parametric and nonparametric component estimators. We further propose a plug-in bandwidth selector using the asymptotic normality of the nonparametric component estimator. The proposed method allow for the modeling of the aggregated SLIMs case where we can explicitly show that taking the aggregated information into account can improve both of the parametric and nonparametric component estimator by the proposed two-stage approach. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedures.     

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.     

Statistical inference of minimum BD estimators and classifiers for varying-dimensional models

Stochastic modeling for large-scale datasets usually involves a varying-dimensional model space. This paper investigates the asymptotic properties, when the number of parameters grows with the available sample size, of the minimum- estimators and classifiers under a broad and important class of Bregman divergence (), which encompasses nearly all of the commonly used loss functions in the regression analysis, classification procedures and machine learning literature. Unlike the maximum likelihood estimators which require the joint likelihood of observations, the minimum-BD estimators are useful for a range of models where the joint likelihood is unavailable or incomplete. Statistical inference tools developed for the class of large dimensional minimum- estimators and related classifiers are evaluated via simulation studies, and are illustrated by analysis of a real dataset.