Combining least-squares and quantile regressions

Least-squares and quantile regressions are method of moments techniques that are typically used in isolation. A leading example where efficiency may be gained by combining least-squares and quantile regressions is one where some information on the error quantiles is available but the error distribution cannot be fully specified. This estimation problem may be cast in terms of solving an over-determined estimating equation (EE) system for which the generalized method of moments (GMM) and empirical likelihood (EL) are approaches of recognized importance. The major difficulty with implementing these techniques here is that the EEs associated with the quantiles are non-differentiable. In this paper, we develop a kernel-based smoothing technique for non-smooth EEs, and derive the asymptotic properties of the GMM and maximum smoothed EL (MSEL) estimators based on the smoothed EEs. Via a simulation study, we investigate the finite sample properties of the GMM and MSEL estimators that combine least-squares and quantile moment relationships. Applications to real datasets are also considered.     

Method of Moments Estimation in Linear Regression with Errors in both Variables

Recently, in this journal, there has been revised attention on estimating the parameters of the errors in variables, linear structural model. For example, O’Driscoll and Ramirez (2011) used a geometric approach to give insight into the performance of various slope estimators for the linear structural model as introduced by the present author. This article aims to provide a unified method of moments approach for estimating the parameters in the linear structural model, concentrating attention on estimators using the higher moments, which to date has received only little attention in the literature.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.     

矩约束模型的最优矩条件选取方法

大量经济、金融以及企业管理等领域研究对象的行为特征可以通过矩约束模型来刻画。然而,该模型中参数的估计对矩条件的选取非常敏感。如何选取最优的矩条件,进而得到更准确的参数估计和更精确的统计推断,是实证研究面临的重要问题。本文从估计量均方误差（MSE）最小的角度,研究了一般矩约束模型两步有效广义矩（GMM）估计的最优矩条件选取方法。首先,利用迭代的方法,推导出两步有效GMM估计的高阶MSE,然后通过Nagar分解,求出了两步有效GMM估计量的近似MSE。接着,基于近似MSE表达式,给出了两步有效GMM估计矩条件选取准则的一般理论,即定义了最优的矩条件,提出了两步有效GMM估计的最优矩条件选取准则,并证明了选取准则的渐近有效性。模拟结果表明,本文提出的矩条件选取方法能够很好地改善两步有效GMM估计量的有限样本表现,降低估计量的有效样本偏差。本研究为实证研究中面临的矩条件选择问题提供了理论依据。     

Computational comparison of the general weighted moments estimators of the scale parameter of a Pareto distribution with a known shape parameter with other estimators based on a multiply type II-censored sample

Wu et al. [Computational comparison for weighted moments estimators and BLUE of the scale parameter of a Pareto distribution with known shape parameter under type II multiply censored sample, Appl. Math. Comput. 181 (2006), pp. 1462–1470] proposed the weighted moments estimators (WMEs) of the scale parameter of a Pareto distribution with known shape parameter on a multiply type II-censored sample. They claimed that some WMEs are better than the best linear unbiased estimator (BLUE) based on the exact mean-squared error (MSE). In this paper, the general WME (GWME) is proposed and the computational comparison of the proposed estimator with the WMEs and BLUE is done on the basis of the exact MSE for given sample sizes and different censoring schemes. As a result, the GWME is performing better than the best estimator among 12 WMEs and BLUE for all cases. Therefore, GWME is recommended for use. At last, one example is given to demonstrate the proposed GWME.     

A method for estimating parameters and quantiles of the three-parameter inverse Gaussian distribution based on statistics invariant to unknown location

The inverse Gaussian (IG) distribution, also known as the Wald distribution, is a long-tailed positively skewed distribution and a well-known lifetime distribution. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter IG distribution, which is based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared with other prominent methods in terms of bias and variance. Finally, we present two illustrative examples.     

Linear estimation for the extended exponential power distribution

Tiao and Lund [The use of OLUMV estimators in inference robustness studies of the location parameter of a class of symmetric distributions. J Amer Statist Assoc. 1970;65(329):370–386] tabulated the coefficients of the best linear unbiased estimators (BLUEs) of location and scale for a particular family of symmetric distributions. This family was a reparameterization of the extended exponential power distribution (EEPD) with the shape parameter restricted to be greater than or equal to one. In this work, we consider the BLU estimation of the location and scale parameters of the EEPD when the shape parameter is one-third and one-half. We obtain closed-form expressions for the single and product moments of the order statistics when the shape parameter is in general in the form of a reciprocal of an integer. These expressions are then used to determine the BLUEs and the corresponding variances for complete samples of size 20 and less. We consider some other linear estimators of the location and scale parameters and then compare them with the BLUEs. Finally, we present a numerical example to illustrate the developed results.     

Estimating the probability of obtaining nonfeasible parameter estimates of the generalized pareto distribution

In this paper we consider the problem of estimating the parameters of the generalized Pareto distribution. Both the method of moments and probability-weighted moments do not guarantee that their respective estimates will be consistent with the observed data. We present simple programs to predict the probability of obtaining such nonfeasible estimates. Our estimation techniques are based on results from intensive simulations and the successful modelling of the lower tail of the distribution of the upper bound of the support. More simulations are performed to validate the new procedure.     

GMM in linear regression for longitudinal data with multiple covariates measured with error

Griliches and Hausman 5 Griliches, Z. and Hausman, J. A. 1986. Errors in variables in panel data. J. Econometrics, 32: 93–118. [Crossref], [Web of Science ®] [Google Scholar] and Wansbeek 11 Wansbeek, T. J. 2001. GMM estimation in panel data models with measurement error. J. Econometrics, 104: 259–268. [Crossref], [Web of Science ®] [Google Scholar] proposed using the generalized method of moments (GMM) to obtain consistent estimators in linear regression models for longitudinal data with measurement error in one covariate, without requiring additional validation or replicate data. For usefulness of this methodology, we must extend it to the more realistic situation where more than one covariate are measured with error. Such an extension is not straightforward, since measurement errors across different covariates may be correlated. By a careful construction of the measurement error correlation structure, we are able to extend Wansbeek's GMM and show that the extended Griliches and Hausman's GMM is equivalent to the extended Wansbeek's GMM. For illustration, we apply the extended GMM to data from two medical studies, and compare it with the naive method and the method assuming only one covariate having measurement error.