Delta Method,Moment Convergence,and Inference

Statistics, as functions of the observations, are usually given by well-behaved functions. This fact is used to obtain limit distributions for statistics whose components are given by asymptotically linear functions. These results are then extended to the moments of distributions, covariance matrices and confidence regions for parameters of interest. These regions may be used to test, through duality, hypothesis on these parameters. A theoretical application is presented.     

Kernel Density-Based Linear Regression Estimate

For linear regression models with non normally distributed errors, the least squares estimate (LSE) will lose some efficiency compared to the maximum likelihood estimate (MLE). In this article, we propose a kernel density-based regression estimate (KDRE) that is adaptive to the unknown error distribution. The key idea is to approximate the likelihood function by using a nonparametric kernel density estimate of the error density based on some initial parameter estimate. The proposed estimate is shown to be asymptotically as efficient as the oracle MLE which assumes the error density were known. In addition, we propose an EM type algorithm to maximize the estimated likelihood function and show that the KDRE can be considered as an iterated weighted least squares estimate, which provides us some insights on the adaptiveness of KDRE to the unknown error distribution. Our Monte Carlo simulation studies show that, while comparable to the traditional LSE for normal errors, the proposed estimation procedure can have substantial efficiency gain for non normal errors. Moreover, the efficiency gain can be achieved even for a small sample size.     

Empirical Likelihood for Single-Index Regression Models under Negatively Associated Errors

In this article, we use bockwise empirical likelihood technique to construct confidence regions for the parameter of the single-index models under negatively associated errors. It is shown that the blockwise empirical likelihood ratio statistic for the parameter of interest is asymptotically χ²-type distributed. The result can be used to obtain confidence regions for the parameter of interest.     

Adaptive Crossover Design for Normal Responses

In a response-adaptive design, we review and update the trial on the basis of outcomes in order to achive a specific goal. In clinical trials our goal is to allocate a larger number of patients to the better treatment. In the present paper, we use a response adaptive design in a two-treatment two-period crossover trial where the treatment responses are continuous. We provide probability measures to choose between the possible treatment combinations AA, AB, BA, or BB. The goal is to use the better treatment combination a larger number of times. We calculate the allocation proportions to the possible treatment combinations and their standard errors. We also derive some asymptotic results and provide solutions on related inferential problems. The proposed procedure is compared with a possible competitor. Finally, we use a data set to illustrate the applicability of our proposed design.     

A Note on Wald's Type Chi-squared Tests of Fit

The Wald's method for constructing chi-squared tests of fit has been formulated more accurately. It is shown that Wald's type statistics will follow the central chi-squared distribution if and only if the limit covariance matrix of standardized frequencies will not depend on unknown parameters. Several examples that illustrate this important fact are presented. In particular, it is shown that the goodness-of-fit statistic developed by Moore and Stubblebine does not follow the chi-squared limit distribution, and, hence, cannot be used for testing multivariate normality.     

Clustering large number of extragalactic spectra of galaxies and quasars through canopies

Abstract

Cluster analysis is the distribution of objects into different groups or more precisely the partitioning of a data set into subsets (clusters) so that the data in subsets share some common trait according to some distance measure. Unlike classification, in clustering one has to first decide the optimum number of clusters and then assign the objects into different clusters. Solution of such problems for a large number of high dimensional data points is quite complicated and most of the existing algorithms will not perform properly. In the present work a new clustering technique applicable to large data set has been used to cluster the spectra of 702248 galaxies and quasars having 1,540 points in wavelength range imposed by the instrument. The proposed technique has successfully discovered five clusters from this 702,248X1,540 data matrix.     

Quasi-maximum exponential likelihood estimation for a non stationary GARCH(1,1) model

ABSTRACT

This article investigates a quasi-maximum exponential likelihood estimator(QMELE) for a non stationary generalized autoregressive conditional heteroscedastic (GARCH(1,1)) model. Asymptotic normality of this estimator is derived under a non stationary condition. A simulation study and a real example are given to evaluate the performance of QMELE for this model.     

Selection of bandwidth for kernel regression

ABSTRACT

The most important factor in kernel regression is a choice of a bandwidth. Considerable attention has been paid to extension the idea of an iterative method known for a kernel density estimate to kernel regression. Data-driven selectors of the bandwidth for kernel regression are considered. The proposed method is based on an optimally balanced relation between the integrated variance and the integrated square bias. This approach leads to an iterative quadratically convergent process. The analysis of statistical properties shows the rationale of the proposed method. In order to see statistical properties of this method the consistency is determined. The utility of the method is illustrated through a simulation study and real data applications.