Variable selection in finite mixture of median regression models using skew-normal distribution

A regression model with skew-normal errors provides a useful extension for traditional normal regression models when the data involve asymmetric outcomes.Moreover,data that arise from a heterogeneous population can be efficiently analysed by a finite mixture of regression models.These observations motivate us to propose a novel finite mixture of median regression model based on a mixture of the skew-normal distributions to explore asymmetrical data from several subpopulations.With the appropriate choice of the tuning parameters,we establish the theoretical properties of the proposed procedure,including consistency for variable selection method and the oracle property in estimation.A productive nonparametric clustering method is applied to select the number of components,and an efficient EM algorithm for numerical computations is developed.Simulation studies and a real data set are used to illustrate the performance of the proposed methodologies.

A LARGE SAMPLE ESTIMATE IN MEDIAN LINEAR REGRESSION MODEL Ⅰ: NONTRUNCATED CASE

This paper uses a grouping-adjusting procedure to the data from a median linear regression model, and estimtes the regression coefficients by the method of weighted least squares. This method simplifies computation and in the meantime, preserves the same asymptotic normal distribution for the estimator, as in the ordinary minimum L_1-norm estimates.

LOCAL MEDIAN ESTIMATION OF VARIANCE FUNCTION

This paper considers local median estimation in fixed design regression problems. The proposed method is employed to estimate the median function and the variance function of a heteroscedastic regression model. Strong convergence rates of the proposed estimators are obtained. Simulation results are given to show the performance of the proposed methods.

RELATIVE STABILITY FOR LOCAL MEDIAN ESTIMATE

Consider the nonparametric median regression model Y-ni = g(x(ni)) + epsilon(ni), 1 less than or equal to i less than or equal to n, where Y-ni's are the observations at the fixed design points x(ni) is an element of [0, 1], is an element of(ni)'s are independent identically distributed random variables with median zero, g(x) is the smooth function of interest, Suppose the local median estimate (g) over tilde(n, h)(x) of g(x) admits the Bahadur's representation. Under some regular conditions, the relative stability of the local median estimate is established in the L-2 sense.

M-Cross-Validation in Local Median Estimation

M-cross-validation criterion is proposed for selecting a smoothing parameter in a nonparametric median regression model in which a uniform weak convergency rate for the M-cross-validated local median estimate, and the upper and lower bounds of the smoothing parameter selected by the proposed criterion are established. The main contribution of this study shows a drastic difference from those encountered in the classical L2-, L1- cross-validation technique, which leads only to the consistency in the sense of the average. Obviously, our results are novel and nontrivial from the point of view of mathematics and statistics, which provides insight and possibility for practitioners substituting maximum deviation for average deviation to evaluate the performance of the data-driven technique.

Rate of Convergence of Normal Approximation for Local Median Estimate

Consider the fixed\|design nonparametric median regression model Y \{ni =g(x \{ni )+ε \{ni ,1≤i≤n, where ε \{ni are iid random variables with median zero. In estimating the regression function g(x), local median estimates \{nh (x) are employed, where h is the number of neighbors of x. Under some regularity conditions, the asymptotic normality and rate of convergence of normal approximation are obtained.

MAXIMUM INFORMATION AND OPTIMUM ESTIMATING FUNCTION

In order to construct estimating functions in some parametric models, this paper introducestwo classes of information matrices. Some necessary and sufficient conditions for the informationmatrices achieving their upper bounds are given. For the problem of estimating the median,some optimum estimating functions based on the information matrices are acquired. Undersome regularity conditions, an approach to carrying out the best basis function is introduced. Innonlinear regression models, an optimum estimating function based on the information matricesis obtained. Some examples are given to illustrate the results. Finally, the concept of optimumestimating function and the methods of constructing optimum estimating function are developedin more general statistical models.