Understanding Estimators of Treatment Effects in Regression Discontinuity Designs

In this paper, we propose two new estimators of treatment effects in regression discontinuity designs. These estimators can aid understanding of the existing estimators such as the local polynomial estimator and the partially linear estimator. The first estimator is the partially polynomial estimator which extends the partially linear estimator by further incorporating derivative differences of the conditional mean of the outcome on the two sides of the discontinuity point. This estimator is related to the local polynomial estimator by a relocalization effect. Unlike the partially linear estimator, this estimator can achieve the optimal rate of convergence even under broader regularity conditions. The second estimator is an instrumental variable estimator in the fuzzy design. This estimator will reduce to the local polynomial estimator if higher order endogeneities are neglected. We study the asymptotic properties of these two estimators and conduct simulation studies to confirm the theoretical analysis.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

A simple method for the estimation of rational distributed lag models

In an earlier paper we suggested a method for the identification and estimation of linear transfer function models. The method was claimed to be especially suitable for polynomial transfer function models. In this paper we shall consider the case of rational transfer function models (distributed lag models) in more detail. A simple method for the estimation of the parameters of multiple input rational distributed lag models is suggested. The method is based on simple linear identities that the parameters always fulfill. The asymptotic distribution of the proposed estimator is derived. Two illustrative examples of the use of the new method are given.     

Superiority of the r-k class estimator over some estimators in a misspecified linear model

ABSTRACT

In this article, we discuss the superiority of r-k class estimator over some estimators in a misspecified linear model. We derive the necessary and sufficient conditions for the superiority of the r-k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion in the misspecified linear model.     

Performances of the Positive-Rule Stein-Type Ridge Estimator in a Linear Regression Model with Spherically Symmetric Error Distributions

In this article, the positive-rule Stein-type ridge estimator (PSRE) is introduced for the parameters in a multiple linear regression model with spherically symmetric error distributions when it is suspected that the parameter vector may be restricted to a linear manifold. The bias and quadratic risk functions of the PSRE are derived and compared with some related competing estimators in literatures. Particularly, some sufficient conditions are derived for superiority of the PSRE over the ordinary ridge estimator, the restricted ridge estimator and the preliminary test ridge estimator, respectively. Furthermore, some graphical results are provided to illustrate some of the theoretical results.     

A new stochastic mixed ridge estimator in linear regression model

This paper is concerned with the parameter estimation in linear regression model with additional stochastic linear restrictions. To overcome the multicollinearity problem, a new stochastic mixed ridge estimator is proposed and its efficiency is discussed. Necessary and sufficient conditions for the superiority of the stochastic mixed ridge estimator over the ridge estimator and the mixed estimator in the mean squared error matrix sense are derived for the two cases in which the parametric restrictions are correct and are not correct. Finally, a numerical example is also given to show the theoretical results.     

A note on classical Stein-type estimators in elliptically contoured models

In this paper the conditions under which a broad class of Stein-type estimators dominates the best invariant unbiased estimator of the mean of an elliptically contoured population have been established. The superiority conditions are derived for both known and unknown scale structures. Also an example is given when the general scale matrix is assumed to be known in linear regression.     

Performance of the principal component two-parameter estimator in misspecified linear regression model

In this article, we consider the performance of the principal component two-parameter estimator in situation of multicollinearity for misspecified linear regression model where misspecification is due to omission of some relevant explanatory variables. The conditions of superiority of the principal component two-parameter estimator over some estimators under the Mahalanobis loss function by the average loss criterion are derived. Furthermore, a real data example and a Monte Carlo simulation study are provided to illustrate some of the theoretical results.     

On the weighted mixed Liu-type estimator under unbiased stochastic restrictions

In this article, we introduce the weighted mixed Liu-type estimator (WMLTE) based on the weighted mixed and Liu-type estimator (LTE) in linear regression model. We will also present necessary and sufficient conditions for superiority of the weighted mixed Liu-type estimator over the weighted mixed estimator (WME) and Liu type estimator (LTE) in terms of mean square error matrix (MSEM) criterion. Finally, a numerical example and a Monte Carlo simulation is also given to show the theoretical results.     

A two-parameter estimator in the negative binomial regression model

In this article, a two-parameter estimator is proposed to combat multicollinearity in the negative binomial regression model. The proposed two-parameter estimator is a general estimator which includes the maximum likelihood (ML) estimator, the ridge estimator (RE) and the Liu estimator as special cases. Some properties on the asymptotic mean-squared error (MSE) are derived and necessary and sufficient conditions for the superiority of the two-parameter estimator over the ML estimator and sufficient conditions for the superiority of the two-parameter estimator over the RE and the Liu estimator in the asymptotic mean-squared error (MSE) matrix sense are obtained. Furthermore, several methods and three rules for choosing appropriate shrinkage parameters are proposed. Finally, a Monte Carlo simulation study is given to illustrate some of the theoretical results.     

On the Restricted Liu Estimator in the Gauss–Markov Model

In this paper, we compare two estimators, the RLE (restricted Liu estimator) and the RLSE (restricted least squares estimator) of parameters in linear models under Gauss–Markov models. Using generalized inverse of matrices, we found some equivalency conditions for the superiority of the RLE with respect to the MSE criterion.     

Efficiency of two classes of stochastic restricted almost unbiased type principal component estimators in linear regression model

In this paper, we introduce two new classes of estimators called the stochastic restricted almost unbiased ridge-type principal component estimator (SRAURPCE) and the stochastic restricted almost unbiased Liu-type principal component estimator (SRAURPCE) to overcome the well-known multicollinearity problem in linear regression model. For the two cases when the restrictions are true and not true, necessary and sufficient conditions for the superiority of the proposed estimators are derived and compared, respectively. Furthermore, a Monte Carlo simulation study and a numerical example are given to illustrate the performance of the proposed estimators.     

An alternative stochastic restricted Liu estimator in linear regression

In this paper, we introduce an alternative stochastic restricted Liu estimator for the vector of parameters in a linear regression model when additional stochastic linear restrictions on the parameter vector are assumed to hold. The new estimator is a generalization of the ordinary mixed estimator (OME) (Durbin in J Am Stat Assoc 48:799–808, 1953; Theil and Goldberger in Int Econ Rev 2:65–78, 1961; Theil in J Am Stat Assoc 58:401–414, 1963) and Liu estimator proposed by Liu (Commun Stat Theory Methods 22:393–402, 1993). Necessary and sufficient conditions for the superiority of the new stochastic restricted Liu estimator over the OME, the Liu estimator and the estimator proposed by Hubert and Wijekoon (Stat Pap 47:471–479, 2006) in the mean squared error matrix (MSEM) sense are derived. Furthermore, a numerical example based on the widely analysed dataset on Portland cement (Woods et al. in Ind Eng Chem 24:1207–1241, 1932) and a Monte Carlo evaluation of the estimators are also given to illustrate some of the theoretical results.     

A comparison of mixed and minimax estimators of linear models

In this paper the stochastic properties of two estimators of linear models, mixed and minimax, based on different types of prior information, are compared using quadratic risk as the criterion for superiority. A necessary and sufficient condition for the minimax estimator to be superior to the comparable mixed estimator is derived as well as a simpler necessary but not sufficient condition.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

Central Limit Theorems of Local Polynomial Threshold Estimator for Diffusion Processes with Jumps

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the non‐parametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the σ ‐field generated by diffusion processes is destroyed, so the classical central limit theorem for martingale difference sequences cannot work. In high‐frequency data, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps by Jacod's stable convergence theorem. We believe that our proof procedure for local polynomial threshold estimators provides a new method in this field, especially in the local linear case.     

Minimum distance estimation in linear regression with strong mixing errors

Abstract

Minimum distance estimation on the linear regression model with independent errors is known to yield an efficient and robust estimator. We extend the method to the model with strong mixing errors and obtain an estimator of the vector of the regression parameters. The goal of this article is to demonstrate the proposed estimator still retains efficiency and robustness. To that end, this article investigates asymptotic distributional properties of the proposed estimator and compares it with other estimators. The efficiency and the robustness of the proposed estimator are empirically shown, and its superiority over the other estimators is established.     

Application of Stein-type estimation in combining regression estimates from replicated experiments

This paper considers the application of Stein-type estimation procedure for the coefficients in a linear regression model when data are available from replicated experiment. Two families of estimators characterized by a single scalar are proposed and their large sample asymptotic properties are derived. These are utilized for comparing the performances of the two estimators along with the conventional estimator and conditions for the superiority of one estimator over the other are deduced.     

Local polynomial quasi-likelihood regression with truncated and dependent data

The local polynomial quasi-likelihood estimation has several good statistical properties such as high minimax efficiency and adaptation of edge effects. In this paper, we construct a local quasi-likelihood regression estimator for a left truncated model, and establish the asymptotic normality of the proposed estimator when the observations form a stationary and α-mixing sequence, such that the corresponding result of Fan et al. [Local polynomial kernel regression for generalized linear models and quasilikelihood functions, J. Amer. Statist. Assoc. 90 (1995), pp. 141–150] is extended from the independent and complete data to the dependent and truncated one. Finite sample behaviour of the estimator is investigated via simulations too.     

Performance of the stein-type two-parameter estimator in multiple linear regression model

In this article, we consider the Stein-type approach to the estimation of the regression parameter in a multiple regression model under a multicollinearity situation. The Stein-type two-parameter estimator is proposed when it is suspected that the regression parameter may be restricted to a subspace. The bias and the quadratic risk of the proposed estimator are derived and compared with the two-parameter estimator (TPE), the restricted TPE and the preliminary test TPE. The conditions of superiority of the proposed estimator are obtained. Finally, a real data example is provided to illustrate some of the theoretical results.