Song Xi Chen

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ∞) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n−4/5x−1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.

Beta kernel estimators for density functions

Kernel estimators using non-negative kernels are considered to estimate probability density functions with compact supports. The kernels are chosen from a family of beta densities. The beta kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The proposed beta kernel estimators have two features. One is that the different amount of smoothing is allocated by naturally varying kernel shape without explicitly changing the value of the smoothing bandwidth. Another feature is that the support of the beta kernels can match the support of the density function; this leads to larger effective sample sizes used in the density estimation and can produce density estimates that have smaller finite-sample variance than some other estimators.

Tests for High-Dimensional Covariance Matrices

We propose tests for sphericity and identity of high-dimensional covariance matrices. The tests are nonparametric without assuming a specific parametric distribution for the data. They can accommodate situations where the data dimension is much larger than the sample size, namely the “large p, small n” situations. We demonstrate by both theoretical and empirical studies that the tests have good properties for a wide range of dimensions and sample sizes. We applied the proposed test on a microarray dataset on Yorkshire Gilts and tested for the covariance structure for the expression levels for sets of genes.

On the accuracy of empirical likelihood confidence regions for linear regression model

The coverage errors of the empirical likelihood confidence regions for β in a linear regression model,Yi=xiβ+εi, 1≤i≤n, are of ordern−1. Bartlett corrections may be employed to reduce the order of magnitude of the coverage errors ton−2. For practical implementation of Bartlett correction, an empirical Bartlett correction is given.

Empirical likelihood for estimating equations with missing values

We consider an empirical likelihood inference for parameters defined by general estimating equations when some components of the random observations are subject to missingness. As the nature of the estimating equations is wide-ranging, we propose a nonparametric imputation of the missing values from a kernel estimator of the conditional distribution of the missing variable given the always observable variable. The empirical likelihood is used to construct a profile likelihood for the parameter of interest. We demonstrate that the proposed nonparametric imputation can remove the selection bias in the missingness and the empirical likelihood leads to more efficient parameter estimation. The proposed method is further evaluated by simulation and an empirical study on a genetic dataset on recombinant inbred mice.

An empirical likelihood goodness‐of‐fit test for time series

The testing of a computing model for a stationary time series is a standard task in statistics. When a parametric approach is used to model the time series, the question of goodness-of-fit arises. In this paper, we employ the empirical likelihood for an a-mixing process and formulate a statistic test measures the goodness-of-fit of a parametric model. The technique is based on comparison with kernel smoothing estimators. The goodness of- fit test proposed is based on the asymptotics of the empirical likelihood, which has two attractive features. One is its automatic consideration of the variation associated with the nonparametric fit due to the empirical likelihoods ability to studentise internally. The other one is that the asymptotic distributions of the test statistic are free of unknown parameters which avoids secondary plug-in estimation. We apply the empirical likelihood based test to a discretised diffusion model which has been recently considered in financial market analysis.

Two Sample Tests for High Dimensional Covariance Matrices

We propose two tests for the equality of covariance matrices between two high-dimensional populations. One test is on the whole variance-covariance matrices, and the other is on offdiagonal sub-matrices which define the covariance between two non-overlapping segments of the high-dimensional random vectors. The tests are applicable (i) when the data dimension is much larger than the sample sizes, namely the “large p, small n” situations and (ii) without assuming parametric distributions for the two populations. These two aspects surpass the capability of the conventional likelihood ratio test. The proposed tests can be used to test on covariances associated with gene ontology terms.

Beta-Bernstein Smoothing for Regression Curves with Compact Support

ABSTRACT. The problem of boundary bias is associated with kernel estimation for regression curves with compact support. This paper proposes a simple and uni(r)ed approach for remedying boundary bias in non‐parametric regression, without dividing the compact support into interior and boundary areas and without applying explicitly different smoothing treatments separately. The approach uses the beta family of density functions as kernels. The shapes of the kernels vary according to the position where the curve estimate is made. Theyare symmetric at the middle of the support interval, and become more and more asymmetric nearer the boundary points. The kernels never put any weight outside the data support interval, and thus avoid boundary bias. The method is a generalization of classical Bernstein polynomials, one of the earliest methods of statistical smoothing. The proposed estimator has optimal mean integrated squared error at an order of magnitude n−4/5, equivalent to that of standard kernel estimators when the curve has an unbounded support.

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β0 in the linear errors-in-variables model: Yi = xiτβ0 + ei,Xi, = xi + ui,(1≤i≤n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β0. It is shown that the coverage error of the confidence region is of order n-1, and Bartlett corrections can reduce the coverage errors to n-2. An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.

Tests for high-dimensional regression coefficients with factorial designs

We propose simultaneous tests for coefficients in high-dimensional linear regression models with factorial designs. The proposed tests are designed for the “large p, small n” situations where the conventional F-test is no longer applicable. We derive the asymptotic distribution of the proposed test statistic under the high-dimensional null hypothesis and various scenarios of the alternatives, which allow power evaluations. We also evaluate the power of the F-test for models of moderate dimension. The proposed tests are employed to analyze a microarray data on Yorkshire Gilts to find significant gene ontology terms which are significantly associated with the thyroid hormone after accounting for the designs of the experiment.