Classification of a screened data into one of two normal populations perturbed by a screening scheme

In normal classification analysis, there may be cases where the population distributions are perturbed by a screening scheme. This paper considers a new classification method for screened data that is obtained from the perturbed normal distributions. Properties of each population distribution is considered and the best region for classifying the screened data is obtained. These developments yield yet another optimal rule for the classification. The rule is studied from several aspects such as a linear approximation, error rates, and estimation of the rule using the EM algorithm. Relationships among these aspects as well as investigation of the rule’s performance are also considered. The screened classification ideas are illustrated in detail using numerical examples.     

A new test for sphericity of the covariance matrix for high dimensional data

In this paper we propose a new test procedure for sphericity of the covariance matrix when the dimensionality, p, exceeds that of the sample size, N=n+1. Under the assumptions that (A) as p→∞ for i=1,…,16 and (B) p/n→c<∞ known as the concentration, a new statistic is developed utilizing the ratio of the fourth and second arithmetic means of the eigenvalues of the sample covariance matrix. The newly defined test has many desirable general asymptotic properties, such as normality and consistency when (n,p)→∞. Our simulation results show that the new test is comparable to, and in some cases more powerful than, the tests for sphericity in the current literature.     

Combining conditional and unconditional moment restrictions with missing responses

Many statistical models, e.g. regression models, can be viewed as conditional moment restrictions when distributional assumptions on the error term are not assumed. For such models, several estimators that achieve the semiparametric efficiency bound have been proposed. However, in many studies, auxiliary information is available as unconditional moment restrictions. Meanwhile, we also consider the presence of missing responses. We propose the combined empirical likelihood (CEL) estimator to incorporate such auxiliary information to improve the estimation efficiency of the conditional moment restriction models. We show that, when assuming responses are strongly ignorable missing at random, the CEL estimator achieves better efficiency than the previous estimators due to utilization of the auxiliary information. Based on the asymptotic property of the CEL estimator, we also develop Wilks’ type tests and corresponding confidence regions for the model parameter and the mean response. Since kernel smoothing is used, the CEL method may have difficulty for problems with high dimensional covariates. In such situations, we propose an instrumental variable-based empirical likelihood (IVEL) method to handle this problem. The merit of the CEL and IVEL are further illustrated through simulation studies.     

A characterization of multivariate normality through univariate projections

This paper introduces a new characterization of multivariate normality of a random vector based on univariate normality of linear combinations of its components.     

Stochastic comparisons of multivariate mixtures

In this paper we establish multivariate hazard rate, multivariate reverse hazard rate, and multivariate likelihood ratio stochastic orderings among multivariate random mapping (mixture) distributions. The new results streamline and simplify the proofs of some partial results that have recently appeared in the literature. Some applications in reliability theory and risk management are described.     

Difference based estimation for partially linear regression models with measurement errors

This paper is concerned with the estimating problem of the partially linear regression models where the linear covariates are measured with additive errors. A difference based estimation is proposed to estimate the parametric component. We show that the resulting estimator is asymptotically unbiased and achieves the semiparametric efficiency bound if the order of the difference tends to infinity. The asymptotic normality of the resulting estimator is established as well. Compared with the corrected profile least squares estimation, the proposed procedure avoids the bandwidth selection. In addition, the difference based estimation of the error variance is also considered. For the nonparametric component, the local polynomial technique is implemented. The finite sample properties of the developed methodology is investigated through simulation studies. An example of application is also illustrated.     

The multiple hybrid bootstrap — Resampling multivariate linear processes

The paper reconsiders the autoregressive aided periodogram bootstrap (AAPB) which has been suggested in Kreiss and Paparoditis (2003) [18]. Their idea was to combine a time domain parametric and a frequency domain nonparametric bootstrap to mimic not only a part but as much as possible the complete covariance structure of the underlying time series. We extend the AAPB in two directions. Our procedure explicitly leads to bootstrap observations in the time domain and it is applicable to multivariate linear processes, but agrees exactly with the AAPB in the univariate case, when applied to functionals of the periodogram. The asymptotic theory developed shows validity of the multiple hybrid bootstrap procedure for the sample mean, kernel spectral density estimates and, with less generality, for autocovariances.     

Empirical Bayes predictive densities for high-dimensional normal models

This paper addresses the problem of estimating the density of a future outcome from a multivariate normal model. We propose a class of empirical Bayes predictive densities and evaluate their performances under the Kullback–Leibler (KL) divergence. We show that these empirical Bayes predictive densities dominate the Bayesian predictive density under the uniform prior and thus are minimax under some general conditions. We also establish the asymptotic optimality of these empirical Bayes predictive densities in infinite-dimensional parameter spaces through an oracle inequality.     

Concordance measures for multivariate non-continuous random vectors

A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendall’s tau, Spearman’s rho and Spearman’s footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Nešlehová (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well.     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.     

Multivariate logistic regression with incomplete covariate and auxiliary information

In this article, we propose and explore a multivariate logistic regression model for analyzing multiple binary outcomes with incomplete covariate data where auxiliary information is available. The auxiliary data are extraneous to the regression model of interest but predictive of the covariate with missing data. Horton and Laird [N.J. Horton, N.M. Laird, Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information, Biometrics 57 (2001) 34–42] describe how the auxiliary information can be incorporated into a regression model for a single binary outcome with missing covariates, and hence the efficiency of the regression estimators can be improved. We consider extending the method of [9] to the case of a multivariate logistic regression model for multiple correlated outcomes, and with missing covariates and completely observed auxiliary information. We demonstrate that in the case of moderate to strong associations among the multiple outcomes, one can achieve considerable gains in efficiency from estimators in a multivariate model as compared to the marginal estimators of the same parameters.     

Testing model assumptions in functional regression models

In the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L²-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.     

Subsampling tests for variance changes in the presence of autoregressive parameter shifts

In this paper, we consider the problem of testing for variance changes in the linear autoregressive processes including AR(p) processes when there are autoregressive parameter shifts. In performing a test, we employ the conventional residual CUSUM of squares test (RCUSQ) statistic. The RCUSQ test is based on the subsampling method introduced by Jach and Kokoszka (2004) [16] to eliminate the influence caused by autoregressive parameter shifts. It is shown that under regularity conditions, the test statistic behaves asymptotically the function of a standard Brownian bridge. We establish the asymptotic validity of this method and assess its performance both theoretically and numerically.     

Bayesian inference for dependent elliptical measurement error models

In this article we provide a Bayesian analysis for dependent elliptical measurement error models considering nondifferential and differential errors. In both cases we compute posterior distributions for structural parameters by using squared radial prior distributions for the precision parameters. The main result is that the posterior distribution of location parameters, for specific priors, is invariant with respect to changes in the generator function, in agreement with previous results obtained in the literature under different assumptions. Finally, although the results obtained are valid for any elliptical distribution for the error term, we illustrate those results by using the student-t distribution and a real data set.     

On the limiting spectral distribution of the covariance matrices of time-lagged processes

We consider two continuous-time Gaussian processes, one being partially correlated to a time-lagged version of the other. We first give the limiting spectral distribution for the covariance matrices of the increments of the processes when the span between two observations tends to zero. Then, we derive the limiting distribution of the eigenvalues of the sample covariance matrices. This result is obtained when the number of paths of the processes is asymptotically proportional to the number of observations for each single path. As an application, we use the second moment of this distribution together with auxiliary volatility and correlation estimates to construct an adaptive estimator of the time lag between the two processes. Finally, we provide an asymptotic theory for our estimation procedure.     

On a model selection problem from high-dimensional sample covariance matrices

Modern random matrix theory indicates that when the population size p is not negligible with respect to the sample size n, the sample covariance matrices demonstrate significant deviations from the population covariance matrices. In order to recover the characteristics of the population covariance matrices from the observed sample covariance matrices, several recent solutions are proposed when the order of the underlying population spectral distribution is known. In this paper, we deal with the underlying order selection problem and propose a solution based on the cross-validation principle. We prove the consistency of the proposed procedure.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

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.     

Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints

We study a spline-based likelihood method for the partly linear model with monotonicity constraints. We use monotone B-splines to approximate the monotone nonparametric function and apply the generalized Rosen algorithm to compute the estimators jointly. We show that the spline estimator of the nonparametric component achieves the possible optimal rate of convergence under the smooth assumption and that the estimator of the regression parameter is asymptotically normal and efficient. Moreover, a spline-based semiparametric likelihood ratio test is established to make inference of the regression parameter. Also an observed profile information method to consistently estimate the standard error of the spline estimator of the regression parameter is proposed. A simulation study is conducted to evaluate the finite sample performance of the proposed method. The method is illustrated by an air pollution study.