Asymptotic optimality of estimators of a linear functional eeiation if the ratio of the error variances is known 2

For the problem of estimating a linear functional relation when the ratio of the error variances is known a general class of estimators is introduced. They include as special cases the instrumental variable and replication cases and some others. Conditions are given for consistency, asymptotic normality and asymptotic optimality within this class based on the variance of the limit distribution. Fisheb's lower bound for asymptotic variances is established, and under normality the asymptotically optimal estimators are shown to be best asymptotically normal. For an inhomogeneous linear relation only estimators which are invariant with respect to a translation of the origin are considered, and asymptotically optimal invariant and, under normality, best asymptotically normal invariant estimators are obtained. Several special cases are discussed.     

Characterization of distributions in invariant models

The theorem of Kagan et al. (1967), (Sankhya Ser. A. 27, 405) on the characterization of the normal law is extended to the characterization of a broad class of distribution shapes, also in the linear regression model, and the stability of this characterization is considered. The results enable, among others, to construct an asymptotically efficient estimator from a subclass of equivariant asymptotically linear estimators of θ.     

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.     

Two-Stage Bounded-lnfluence Estimators for Simultaneous-Equations Models

This article presents a class of estimators for linear structural models that are robust to heavytailed disturbance distributions, gross errors in either the endogenous or exogenous variables, and certain other model failures. The class of estimators modifies ordinary two-stage least squares by replacing each least squares regression by a bounded-influence regression. Conditions under which the estimators are qualitatively robust, consistent, and asymptotically normal are established, and an empirical example is presented.     

Empirical likelihood-based inference in nonlinear regression models with missing responses at random

This paper investigates the estimations of regression parameters and response mean in nonlinear regression models in the presence of missing response variables that are missing with missingness probabilities depending on covariates. We propose four empirical likelihood (EL)-based estimators for the regression parameters and the response mean. The resulting estimators are shown to be consistent and asymptotically normal under some general assumptions. To construct the confidence regions for the regression parameters as well as the response mean, we develop four EL ratio statistics, which are proven to have the χ² distribution asymptotically. Simulation studies and an artificial data set are used to illustrate the proposed methodologies. Empirical results show that the EL method behaves better than the normal approximation method and that the coverage probabilities and average lengths depend on the selection probability function.     

M-estimation in regression models for censored data

In this paper, we study M-estimators of regression parameters in semiparametric linear models for censored data. A class of consistent and asymptotically normal M-estimators is constructed. A resampling method is developed for the estimation of the asymptotic covariance matrix of the estimators.     

EMPIRICAL BAYES ESTIMATION WITH NON-IDENTICAL COMPONENTS. CONTINUOUS CASE.

In this paper a variant of the standard empirical Bayes estimation problem is considered where the component problems in the sequence are not identical in that the conditional distribution of the observations may vary with the component problems. Let {(Θ_n, X_n)} be a sequence of independent random vectors where Θ_n? G and, given Θ_n=Θ_n, X_n -P_Θ,m(n) with {m(n)} a sequence of positive integers where m(n)≤m? < ∞ for all n. With P_Θ,m in a continuous exponential family of distributions, asymptotically optimal empirical Bayes procedures are exhibited which depend on uniform approximations of certain functions on compact sets by polynomials in e^Θ. Such approximations have been explicitly calculated in the case of normal and gamma families. In the case of normal families, asymptotically optimal linear empirical Bayes estimators in the class of all linear estimators are derived and shown to have rate O(n^-1/2).     

Empirical likelihood for heteroscedastic partially linear errors-in-variables model with α-mixing errors

In this paper, we apply the empirical likelihood method to heteroscedastic partially linear errors-in-variables model. For the cases of known and unknown error variances, the two different empirical log-likelihood ratios for the parameter of interest are constructed. If the error variances are known, the empirical log-likelihood ratio is proved to be asymptotic chi-square distribution under the assumption that the errors are given by a sequence of stationary α-mixing random variables. Furthermore, if the error variances are unknown, we show that the proposed statistic is asymptotically standard chi-square distribution when the errors are independent. Simulations are carried out to assess the performance of the proposed method.     

Empirical bayes estimation of binomial parameter with symmetric priors

This paper deals with the problem of estimating the binomial parameter via the nonparametric empirical Bayes approach. This estimation problem has the feature that estimators which are asymptotically optimal in the usual empirical Bayes sense do not exist (Robbins (1958, 1964)), However, as pointed out by Liang (1934) and Gupta and Liang (1988), it is possible to construct asymptotically optimal empirical Bayes estimators if the unknown prior is symmetric about the point 1/2, In this paper, assuming symmetric priors a monotone empirical Bayes estimator is constructed by using the isotonic regression method. This estimator is asymptotically optimal in the usual empirical Bayes sense. The corresponding rate of convergence is investigated and shown to be of order n^-1, where n is the number of past observations at hand.     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Function-based hypothesis testing in censored two-sample location-scale models

Function-based hypothesis testing in two-sample location-scale models has been addressed for uncensored data using the empirical characteristic function. A test of adequacy in censored two-sample location-scale models is lacking, however. A plug-in empirical likelihood approach is used to introduce a test statistic, which, asymptotically, is not distribution free. Hence for practical situations bootstrap is necessary for performing the test. A multiplier bootstrap and a model appropriate resampling procedure are given to approximate critical values from the null asymptotic distribution. Although minimum distance estimators of the location and scale are deployed for the plug-in, any consistent estimators can be used. Numerical studies are carried out that validate the proposed testing method, and real example illustrations are given.

     

Partially linear single-index model with missing responses at random

This paper considers semiparametric partially linear single-index model with missing responses at random. Imputation approach is developed to estimate the regression coefficients, single-index coefficients and the nonparametric function, respectively. The imputation estimators for the regression coefficients and single-index coefficients are obtained by a stepwise approach. These estimators are shown to be asymptotically normal, and the estimator for the nonparametric function is proved to be asymptotically normal at any fixed point. The bandwidth problem is also considered in this paper, a delete-one cross validation method is used to select the optimal bandwidth. A simulation study is conducted to evaluate the proposed methods.     

A note on james-stein and bayes empiricl bayes estimators

In an empirical Bayes decision problem, a simple class of estimators is constructed that dominate the James-Stein

estimator, A prior distribution A is placed on a restricted (normal) class G of priors to produce a Bayes empirical Bayes estimator, The Bayes empirical Bayes estimator is smooth, admissible, and asymptotically optimal. For certain A rate of convergence to minimum Bayes risk is 0(n^-1)uniformly on G. The results of a Monte Carlo study are presented to demonstrate the favorable risk bebhavior of the Bayes estimator In comparison with other competitors including the James-Stein estimator.     

Statistical inference of partially linear panel data regression models with fixed individual and time effects

This article considers a partially linear panel data model with fixed individual and time effects in a setting where both N and T are large. Based on the within transformation and profile likelihood method, we propose an approach to estimating the parametric and non parametric components of the partially linear model. The resultant estimators are shown to be consistent and asymptotically normal. Monte Carlo simulations are also conducted to illustrate the finite-sample performance of the proposed estimators.     

Empirical likelihood dimension reduction inference for partially non-linear error-in-responses models with validation data

ABSTRACT

In this article, partially non linear models when the response variable is measured with error and explanatory variables are measured exactly are considered. Without specifying any error structure equation, a semiparametric dimension reduction technique is employed. Two estimators of unknown parameter in non linear function are obtained and asymptotic normality is proved. In addition, empirical likelihood method for parameter vector is provided. It is shown that the estimated empirical log-likelihood ratio has asymptotic Chi-square distribution. A simulation study indicates that, compared with normal approximation method, empirical likelihood method performs better in terms of coverage probabilities and average length of the confidence intervals.     

Empirical Likelihood for Threshold Autoregressive Models

This article develops empirical likelihood for threshold autoregressive models. We propose general estimating equations based on moment constraint. Under some suitable conditions, we show the empirical likelihood estimators for parameter are asymptotically normally distributed, and the proposed log empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution.     

Empirical Likelihood for Generalized Partially Linear Single-index Models

Empirical-likelihood based inference for the parameters in a generalized partially linear single-index models (GPLSIM) is investigated. Based on the local linear estimators of the nonparametric parts of the GPLSIM, an estimated empirical likelihood-based statistic of the parametric components is proposed. We show that the resulting statistic is asymptotically standard chi-squared distributed, the confidence regions for the parametric components are constructed. Some simulations are conducted to illustrate the proposed method.     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

Local GMM Estimation of Semiparametric Panel Data with Smooth Coefficient Models

In this article, we consider the estimation of semiparametric panel data smooth coefficient models. We propose a class of local generalized method of moments (LGMM) estimators that are simple and easy to implement in practice. We show that the proposed LGMM estimators are consistent and asymptotically normal. Monte Carlo simulations suggest that our proposed estimator performs quite well in finite samples. An empirical application using a large panel of U.K. firms is also presented.     

BAYES EMPIRICAL BAYES APPROACH TO ESTIMATION OF THE FAILURE RATE IN EXPONENTIAL DISTRIBUTION

ABSTRACT

In the empirical Bayes (EB) decision problem consisting of squared error estimation of the failure rate in exponential distribution, a prior Λ is placed on the gamma family of prior distributions to produce Bayes EB estimators which are admissible. A subclass of such estimators is shown to be asymptotically optimal (a.o.). The results of a Monte Carlo study are presented to demonstrate the a.o. property of the Bayes EB estimators.