An application of approximate finite sample results to parameter estimation in a linear errors-in-variables model

In the simple errors-in-variables model the least squares estimator of the slope coefficient is known to be biased towards zero for finite sample size as well as asymptotically. In this paper we suggest a new corrected least squares estimator, where the bias correction is based on approximating the finite sample bias by a lower bound. This estimator is computationally very simple. It is compared with previously proposed corrected least squares estimators, where the correction aims at removing the asymptotic bias or the exact finite sample bias. For each type of corrected least squares estimators we consider the theoretical form, which depends on an unknown parameter, as well as various feasible forms. An analytical comparison of the theoretical estimators is complemented by a Monte Carlo study evaluating the performance of the feasible estimators. The new estimator proposed in this paper proves to be superior with respect to the mean squared error.     

Sequential estimation of normal mean under asymmetric loss function with a shrinkage stopping rule

The problem of estimating a normal mean with unknown variance is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, a sequential stopping rule and two sequential estimators of the mean are proposed and second-order asymptotic expansions of their risk functions are derived. It is demonstrated that the sample mean becomes asymptotically inadmissible, being dominated by a shrinkage-type estimator. Also a shrinkage factor is incorporated in the stopping rule and similar inadmissibility results are established. Received September 1997     

Estimation strategies for the regression coefficient parameter matrix in multivariate multiple regression

We consider improved estimation strategies for the parameter matrix in multivariate multiple regression under a general and natural linear constraint. In the context of two competing models where one model includes all predictors and the other restricts variable coefficients to a candidate linear subspace based on prior information, there is a need of combining two estimation techniques in an optimal way. In this scenario, we suggest some shrinkage estimators for the targeted parameter matrix. Also, we examine the relative performances of the suggested estimators in the direction of the subspace and candidate subspace restricted type estimators. We develop a large sample theory for the estimators including derivation of asymptotic bias and asymptotic distributional risk of the suggested estimators. Furthermore, we conduct Monte Carlo simulation studies to appraise the relative performance of the suggested estimators with the classical estimators. The methods are also applied on a real data set for illustrative purposes.     

Kantorovich inequalities and efficiency comparisons for several classes of estimators in linear models

New matrix, determinant and trace versions of the Kantorovich inequality (KI) involving two positive definite matrices are presented. Some of these are used to study the efficiencies of minimum-distance (MD) estimators, generalized method-of-moments (GMM) estimators and several estimators specific to longitudinal or panel-data analysis. They are also used to give upper bounds for the determinant and trace of the asymptotic variance matrix of a weighted least-squares (WLS) estimator in the generalized linear model.     

Estimators for the common principal components model based on reweighting: influence functions and Monte Carlo study

The common principal components model for several groups of multivariate observations is a useful parsimonious model for the scatter structure which assumes equal principal axes but different variances along those axes for each group. Due to the lack of resistance of the classical maximum likelihood estimators for the parameters of this model, several robust estimators have been proposed in the literature: plug-in estimators and projection-pursuit (PP) type estimators. In this paper, we show that it is possible to improve the low efficiency of the projection-pursuit estimators by applying a reweighting step. More precisely, we consider plug-in estimators obtained by plugging a reweighted estimator of the scatter matrices into the maximum likelihood equations defining the principal axes. The weights considered penalize observations with large values of the influence measures defined by Boente et al. (2002). The new estimators are studied in terms of theoretical properties (influence functions and asymptotic variances) and are compared with other existing estimators in a simulation study.     

Model averaging based on James–Stein estimators

Existing model averaging methods are generally based on ordinary least squares (OLS) estimators. However, it is well known that the James–Stein (JS) estimator dominates the OLS estimator under quadratic loss, provided that the dimension of coefficient is larger than two. Thus, we focus on model averaging based on JS estimators instead of OLS estimators. We develop a weight choice method and prove its asymptotic optimality. A simulation experiment shows promising results for the proposed model average estimator.     

Estimation of the precision matrix of multivariate Pearson type II model

In this paper, the problem of estimating the precision matrix of a multivariate Pearson type II-model is considered. A new class of estimators is proposed. Moreover, the risk functions of the usual and the proposed estimators are explicitly derived. It is shown that the proposed estimator dominates the MLE and the unbiased estimator, under the quadratic loss function. A simulation study is carried out and confirms these results. Improved estimator of tr (Σ ⁻¹) is also obtained.     

Restricted maximum likelihood estimation under Eisenhart model Ill

For a balanced two-way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non-negativity requirements of the variance components by L ee and K apadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non-negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.     

Hybrid estimators for small diffusion processes based on reduced data

We deal with the Bayes type estimators and the maximum likelihood type estimators of both drift and volatility parameters for small diffusion processes defined by stochastic differential equations with small perturbations from high frequency data. From the viewpoint of numerical analysis, initial Bayes type estimators for both drift and volatility parameters based on reduced data are required, and adaptive maximum likelihood type estimators with the initial Bayes type estimators, which are called hybrid estimators, are proposed. The asymptotic properties of the initial Bayes type estimators based on reduced data are derived and it is shown that the hybrid estimators have asymptotic normality and convergence of moments. Furthermore, a concrete example and simulation results are given.     

On estimating the mean of the selected normal population under the LINEX loss function

Following Parsian and Farsipour (1999), we consider the problem of estimating the mean of the selected normal population, from two normal populations with unknown means and common known variance, under the LINEX loss function. Some admissibility results for a subclass of equivariant estimators are derived and a sufficient condition for the inadmissibility of an arbitrary equivariant estimator is provided. As a consequence, several of the estimators proposed by Parsian and Farsipour (1999) are shown to be inadmissible and better estimators are obtained. Received January 2001/Revised May 2002     

Instrumental variables estimators of nonparametric models with discrete endogenous regressors

This paper discusses estimation of nonparametric models whose regressor vectors consist of a vector of exogenous variables and a univariate discrete endogenous regressor with finite support. Both identification and estimators are derived from a transform of the model that evaluates the nonparametric structural function via indicator functions in the support of the discrete regressor. A two-step estimator is proposed where the first step constitutes nonparametric estimation of the instrument and the second step is a nonparametric version of two-stage least squares. Linear functionals of the model are shown to be asymptotically normal, and a consistent estimator of the asymptotic covariance matrix is described. For the binary endogenous regressor case, it is shown that one functional of the model is a conditional (on covariates) local average treatment effect, that permits both unobservable and observable heterogeneity in treatments. Finite sample properties of the estimators from a Monte Carlo simulation study illustrate the practicability of the proposed estimators.     

Two-stage point estimation with a shrinkage stopping rule

Consider the problem of estimating a mean vector in ap-variate normal distribution under two-stage sequential sampling schemes. The paper proposes a stopping rule motivated by the James-Stein shrinkage estimator, and shows that the stopping rule and the corresponding shrinkage estimator asymptotically dominate the usual two-stage procedure under a sequence of local alternatives forp3. Also the results of Monte Carlo simulation for average sample sizes and risks of estimators are stated.     

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi–Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.     

Least squares, preliminary test and Stein-type estimation in general vector AR(p) models

In the general vector autoregressive process AR ( p ), multivariate least square estimation (LSE)/maximum likelihood estimation (MLE) of a subset of the parameters is considered when the complementary subset is suspected to be redundant. This may be viewed as a special case of linear constraints of autoregressive parameters. We incorporate this nonsample information in the estimation process and propose preliminary test and Stein-type estimators for the target subset of parameters. Under local alternatives their asymptotic properties are investigated and compared with those of unrestricted and restricted LSE. The dominance picture of the estimators is presented.     

Extension of some important identities in shrinkage-pretest strategies

In this paper, we establish three identities which play a crucial role in deriving the asymptotic distributional risk function and the asymptotic distributional bias of a large class of estimators of a matrix parameter. In particular, we generalize the results in Judge and Bock (The statistical implication of pre-test and Stein-rule estimators in econometrics. North Holland, Amsterdam, 1978). The established results are useful in risk analysis of a class of Stein-rule type matrix estimators.     

A Comparative Analysis of Different IV and GMM Estimators of Dynamic Panel Data Models

It is well known that the usual procedures for estimating panel data models are inconsistent in the dynamic setting. A large number of consistent estimators however, have been proposed in the literature. This paper provides a survey of the majority of mainstream estimators, which tend to consist of IV and GMM ones. It also considers a newly proposed extension to the promising Wansbeek–Bekker estimator (Harris & Mátyás, 2000). To provide guidance to the applied researcher working on micro-datasets, the small sample performance of these estimators is evaluated using a set of Monte Carlo experiments.     

An efficient GMM estimator of spatial autoregressive models

In this paper, we consider GMM estimation of the regression and MRSAR models with SAR disturbances. We derive the best GMM estimator within the class of GMM estimators based on linear and quadratic moment conditions. The best GMM estimator has the merit of computational simplicity and asymptotic efficiency. It is asymptotically as efficient as the ML estimator under normality and asymptotically more efficient than the Gaussian QML estimator otherwise. Monte Carlo studies show that, with moderate-sized samples, the best GMM estimator has its biggest advantage when the disturbances are asymmetrically distributed. When the diagonal elements of the spatial weights matrix have enough variation, incorporating kurtosis of the disturbances in the moment functions will also be helpful.     

Estimation of the characteristic roots of the scale matrix

The simultaneous estimation of the characteristic roots of the scale matrix of the multivariatet-model is considered. The improved estimation strategies are developed in the light of a quadratic loss function. It is demonstrated analytically and numerically that the class of proposed estimators outperforms the class of usual estimators in the sense of having smaller risk.     

Sequential estimation of a linear function of normal means under asymmetric loss function

The problem of estimating a linear function of k normal means with unknown variances is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, sequential stopping rules satisfying a general set of assumptions are considered. Two estimators are proposed and second-order asymptotic expansions of their risk functions are derived. It is shown that the usual estimator, namely the linear function of the sample means, is asymptotically inadmissible, being dominated by a shrinkage-type estimator. An example illustrates the use of different multistage sampling schemes and provides asymptotic expansions of the risk functions. Received: August 1999     

Spatial local M-estimation under association

We investigate the asymptotic behavior of a robust version of local linear regression estimators with variable bandwidth for spatial associated processes. The weak consistency of the proposed estimators is given under appropriate conditions. Furthermore, we establish the asymptotic normality of the estimators, from which expressions for the asymptotic bias and variance can be derived.