Depth weighted scatter estimators

General depth weighted scatter estimators are introduced and investigated. For general depth functions, we find out that these affine equivariant scatter estimators are Fisher consistent and unbiased for a wide range of multivariate distributions, and show that the sample scatter estimators are strong and \sqrtn-consistent and asymptotically normal, and the influence functions of the estimators exist and are bounded in general. We then concentrate on a specific case of the general depth weighted scatter estimators, the projection depth weighted scatter estimators, which include as a special case the well-known Stahel-Donoho scatter estimator whose limiting distribution has long been open until this paper. Large sample behavior, including consistency and asymptotic normality, and efficiency and finite sample behavior, including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators, are thoroughly investigated. The influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined. Unlike typical high-breakdown competitors, the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a bounded-influence function and a moderate maximum bias curve. Comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and, consequently, represent very favorable choices of affine equivariant multivariate scatter estimators.Comment: Published at http://dx.doi.org/10.1214/009053604000000922 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Empirical likelihood for median regression model with designed censoring variables

AbstractWe propose a new and simple estimating equation for the parameters in median regression models with designed censoring variables, and then apply the empirical log likelihood ratio statistic to construct confidence region for the parameters. The empirical log likelihood ratio statistic is shown to have a standard chi-square distribution, which makes this method easy to implement. At the same time, another empirical log likelihood ratio statistic is proposed based on an existing estimating equation and the limiting distribution of the empirical likelihood ratio statistic is shown to be a sum of weighted chi-square distributions. We compare the performance of the empirical likelihood confidence region based on the new estimating equation, with that based on the existing estimating equation and a normal approximation method by simulation studies

On the Second Order Properties of Empirical Likelihood with Moment Restrictions1

This paper considers the second order properties of empirical likelihood for a parameter defined by moment restrictions, which is the framework operated upon by the Generalized Method of Moments. It is shown that the empirical likelihood defined for this general framework still admits the delicate second order property of Bartlett correction, which represents a substantial extension of all the established cases of Bartlett correction for the empirical likelihood. An empirical Bartlett correction is proposed, which is shown to work effectively in improving the coverage accuracy of confidence regions for the parameter

Empirical Likelihood Confidence Region for Parameters in Semi-linear Errors-in-Variables Models

This paper proposes a constrained empirical likelihood confidence region for a parameter in the semi-linear errors-in-variables model. The confidence region is constructed by combining the score function corresponding to the squared orthogonal distance with a constraint on the parameter, and it overcomes that the solution of limiting mean estimation equations is not unique. It is shown that the empirical log likelihood ratio at the true parameter converges to the standard chi-square distribution. Simulations show that the proposed confidence region has coverage probability which is closer to the nominal level, as well as narrower than those of normal approximation of generalized least squares estimator in most cases. A real data example is given

On Parameter Estimation for Semi-linear Errors-in-Variables Models

AbstractThis paper studies a semi-linear errors-in-variables model of the formYi=x′iβ+g(Ti)+ei,Xi=xi+ui(1⩽i⩽n). The estimators of parametersβ,σ2and of the smooth functiongare derived by using the nearest neighbor-generalized least square method. Under some weak conditions, it is shown that the estimators of unknown vectorβand the unknown parameterσ2are strongly consistent and asymptotically normal. The estimator ofgalso achieves an optimal rate of convergence

The abstract of doctoral dissertation ‘Some research on hypothesis testing and nonparametric variable screening problems for high dimensional data’

In this thesis, we construct test statistic for association test and independence test in high dimension, respectively, and study the corresponding theoretical properties under some regularity conditions. Meanwhile, we propose a nonparametric variable screening procedure for sparse additive model with multivariate response in untra-high dimension and established some screening properties

On the Second Order Properties of Empirical Likelihood with Moment Restrictions1

This paper considers the second order properties of empirical likelihood for a parameter defined by moment restrictions, which is the framework operated upon by the Generalized Method of Moments. It is shown that the empirical likelihood defined for this general framework still admits the delicate second order property of Bartlett correction, which represents a substantial extension of all the established cases of Bartlett correction for the empirical likelihood. An empirical Bartlett correction is proposed, which is shown to work effectively in improving the coverage accuracy of confidence regions for the parameter.This preprint was published as Song Xi Chen and Hengjian Cui, "On the second-order properties of empirical likelihood with moment restrictions", Journal of Econometrics (2007): 492-516, doi: 10.1016/j.jeconom.2006.10.006</p

Empirical likelihood for median regression model with designed censoring variables

We propose a new and simple estimating equation for the parameters in median regression models with designed censoring variables, and then apply the empirical log likelihood ratio statistic to construct confidence region for the parameters. The empirical log likelihood ratio statistic is shown to have a standard chi-square distribution, which makes this method easy to implement. At the same time, another empirical log likelihood ratio statistic is proposed based on an existing estimating equation and the limiting distribution of the empirical likelihood ratio statistic is shown to be a sum of weighted chi-square distributions. We compare the performance of the empirical likelihood confidence region based on the new estimating equation, with that based on the existing estimating equation and a normal approximation method by simulation studies.Empirical likelihood Designed censoring Fixed censoring Median regression model Confidence region