Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

Minimum disparity estimation based on combined disparities: Asymptotic results

The maximum likelihood estimator (MLE) is asymptotically efficient for most parametric models under standard regularity conditions, but it has very poor robustness properties. On the other hand some of the minimum disparity estimators like the minimum Hellinger distance estimator (MHDE) have strong robustness features but their small sample efficiency at the model turns out to be very poor compared to the MLE. Methods based on the minimization of some combined disparities can substantially improve their small sample performances without affecting their robustness properties (Park et al., 1995). All studies involving the combined disparity have so far been empirical, and there are no results on the asymptotic properties of these estimators. In view of the usefulness of these procedures this is a major gap in theory, which we try to fill through the present work. Some illustrations of the performance of the estimators and the corresponding tests are also provided.     

A note on the efficiency of composite quantile regression

Composite quantile regression (CQR) is motivated by the desire to have an estimator for linear regression models that avoids the breakdown of the least-squares estimator when the error variance is infinite, while having high relative efficiency even when the least-squares estimator is fully efficient. Here, we study two weighting schemes to further improve the efficiency of CQR, motivated by Jiang et al. [Oracle model selection for nonlinear models based on weighted composite quantile regression. Statist Sin. 2012;22:1479–1506]. In theory the two weighting schemes are asymptotically equivalent to each other and always result in more efficient estimators compared with CQR. Although the first weighting scheme is hard to implement, it sheds light on in what situations the improvement is expected to be large. A main contribution is to theoretically and empirically identify that standard CQR has good performance compared with weighted CQR only when the error density is logistic or close to logistic in shape, which was not noted in the literature.     

On the weighted least-squares,the ordinary least-squares and the best linear unbiased estimators under a restricted growth curve model

Necessary and sufficient conditions are given for a restricted growth curve model to be consistent. The general expressions of the weighted least-squares estimators (WLSEs), the ordinary least-squares estimators (OLSEs) and the best linear unbiased estimator (BLUE) under this model are also derived. Moreover, some algebraic and statistical properties of these estimators are presented by rank method.     

Kernel density estimation using weighted data ∗

In this paper we compare the kernel density estimators proposed by Bhattacharyya et al. (1988) and Jones (1991) for length biased data, showing the asymptotic normality of the estimators. A method to construct a new estimator is proposed. Moreover, we extend these results to weighted data and we study an estimator for the weight function.     

Two-stage Huber estimation

In this paper we propose a new robust estimator in the context of two-stage estimation methods directed towards the correction of endogeneity problems in linear models. Our estimator is a combination of Huber estimators for each of the two stages, with scale corrections implemented using preliminary median absolute deviation estimators. In this way we obtain a two-stage estimation procedure that is an interesting compromise between concerns of simplicity of calculation, robustness and efficiency. This method compares well with other possible estimators such as two-stage least-squares (2SLS) and two-stage least-absolute-deviations (2SLAD), asymptotically and in finite samples. It is notably interesting to deal with contamination affecting more heavily the distribution tails than a few outliers and not losing as much efficiency as other popular estimators in that case, e.g. under normality. An additional originality resides in the fact that we deal with random regressors and asymmetric errors, which is not often the case in the literature on robust estimators.     

Jackknife inference for heteroscedastic linear regression models

Inference on the regression parameters in a heteroscedastic linear regression model with replication is considered, using either the ordinary least-squares (OLS) or the weighted least-squares (WLS) estimator. A delete-group jackknife method is shown to produce consistent variance estimators irrespective of within-group correlations, unlike the delete-one jackknife variance estimators or those based on the customary δ-method assuming within-group independence. Finite-sample properties of the delete-group variance estimators and associated confidence intervals are also studied through simulation.     

Using generalized doubly robust estimator to estimate average treatment effects of multiple treatments in observational studies

The generalized doubly robust estimator is proposed for estimating the average treatment effect (ATE) of multiple treatments based on the generalized propensity score (GPS). In medical researches where observational studies are conducted, estimations of ATEs are usually biased since the covariate distributions could be unbalanced among treatments. To overcome this problem, Imbens [The role of the propensity score in estimating dose-response functions, Biometrika 87 (2000), pp. 706–710] and Feng et al. [Generalized propensity score for estimating the average treatment effect of multiple treatments, Stat. Med. (2011), in press. Available at: http://onlinelibrary.wiley.com/doi/10.1002/sim.4168/abstract] proposed weighted estimators that are extensions of a ratio estimator based on GPS to estimate ATEs with multiple treatments. However, the ratio estimator always produces a larger empirical sample variance than the doubly robust estimator, which estimates an ATE between two treatments based on the estimated propensity score (PS). We conduct a simulation study to compare the performance of our proposed estimator with Imbens’ and Feng et al.’s estimators, and simulation results show that our proposed estimator outperforms their estimators in terms of bias, empirical sample variance and mean-squared error of the estimated ATEs.     

A bias-reduced estimator for the mean of a heavy-tailed distribution with an infinite second moment

We use bias-reduced estimators of high quantiles of heavy-tailed distributions, to introduce a new estimator for the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked in a simulation study, by four of the most popular goodness-of-fit tests. The accuracy of the resulting confidence intervals is evaluated as well. We also investigate the finite sample behavior and compare our estimator with some versions of Peng's estimator of the mean (namely those based on Hill, t-Hill and Huisman et al. extreme value index estimators). Moreover, we discuss the robustness of the tail index estimators used in this paper. Finally, our estimation procedure is applied to the well-known Danish fire insurance claims data set, to provide confidence bounds for the means of weekly and monthly maximum losses over a period of 10 years.     

Estimation of average treatment effects based on parametric propensity score model

In this paper, the estimation of average treatment effects is examined given that the propensity score is of a parametric form with some unknown parameters. Under the assumption that the treatment is ignorable given some observed characteristics, the MLEs for those unknown parameters in the probability assignment model have been achieved firstly and then three estimators have been defined by the inverse probability weighted, regression and imputation methods, respectively. All the estimators are shown asymptotically normal and more importantly, the substantial efficiency gains of the first two estimates have been obtained theoretically compared with the existing estimators in Hahn (1998) and Hirano et al. (2003), i.e., the inverse weighted probability estimator and the regression estimator have smaller asymptotic variances. Our simulation analysis verifies the theoretical results in terms of biases, SEs and MSEs.     

A robust Liu regression estimator

The least-squares regression estimator can be very sensitive in the presence of multicollinearity and outliers in the data. We introduce a new robust estimator based on the MM estimator. By considering weights, also the resulting MM-Liu estimator is highly robust, but also the estimation of the biasing parameter is robustified. Also for high-dimensional data, a robust Liu-type estimator is introduced, based on the Partial Robust M-estimator. Simulation experiments and a real dataset show the advantages over the standard estimators and other robustness proposals.     

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.     

Stability under contamination of robust regression estimators based on differences of residuals

A reasonable approach to robust regression estimation is minimizing a robust scale estimator of the pairwise differences of residuals. We introduce a large class of estimators based on this strategy extending ideas of Yohai and Zamar (Am. Statist. (1993) 1824–1842) and Croux et al. (J. Am. Statist. Assoc. (1994) 1271–1281). The asymptotic robustness properties of the estimators in this class are addressed using the maxbias curve. We provide a general principle to compute this curve and present explicit formulae for several particular cases including generalized versions of S-, R- and τ-estimators. Finally, the most stable estimator in the class, that is, the estimator with the minimum maxbias curve, is shown to be the set of coefficients that minimizes an appropriate quantile of the distribution of the absolute pairwise differences of residuals.     

Generalized preliminary test stochastic restricted estimator in the linear regression model

ABSTRACT

In this paper, we propose three generalized estimators, namely, generalized unrestricted estimator (GURE), generalized stochastic restricted estimator (GSRE), and generalized preliminary test stochastic restricted estimator (GPTSRE). The GURE can be used to represent the ridge estimator, almost unbiased ridge estimator (AURE), Liu estimator, and almost unbiased Liu estimator. When stochastic restrictions are available in addition to the sample information, the GSRE can be used to represent stochastic mixed ridge estimator, stochastic restricted Liu estimator, stochastic restricted almost unbiased ridge estimator, and stochastic restricted almost unbiased Liu estimator. The GPTSRE can be used to represent the preliminary test estimators based on mixed estimator. Using the GPTSRE, the properties of three other preliminary test estimators, namely preliminary test stochastic mixed ridge estimator, preliminary test stochastic restricted almost unbiased Liu estimator, and preliminary test stochastic restricted almost unbiased ridge estimator can also be discussed. The mean square error matrix criterion is used to obtain the superiority conditions to compare the estimators based on GPTSRE with some biased estimators for the two cases for which the stochastic restrictions are correct, and are not correct. Finally, a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings of the proposed estimators.     

Parameter estimation for generalized random coefficient autoregressive processes

A generalized random coefficient autoregressive (GRCA) process is introduced in which the random coefficients are permitted to be correlated with the error process. The ordinary random coefficient autoregressive process, the Markovian bilinear model and its generalization, and the random coefficient exponential autoregressive process, among others, are seen to be special cases of the GRCA process. Conditional least squares, and weighted least-squares estimators of the mean of the random coefficient vector are derived and their limit distributions are studied. Estimators of the variance-covariance parameters are also discussed. A simulation study is presented which shows that the weighted least-squares estimator dominates the unweighted least-squares estimator.     

A shift parameter estimation based on smoothed Kolmogorov–Smirnov

A new procedure of shift parameter estimation in the two-sample location problem is investigated and compared with existing estimators. The proposed procedure smooths the empirical distribution functions of each random sample and replaces empirical distribution functions in the two-sample Kolmogorov–Smirnov method. The smoothed Kolmogorov–Smirnov is minimized with respect to an arbitrary shift variable in order to find an estimate of the shift parameter. The proposed procedure can be considered the smoothed version of a very little known method of shift parameter estimation from Rao-Schuster-Littell (RSL) [Rao et al., Estimation of shift and center of symmetry based on Kolmogorov–Smirnov statistics, Ann. Stat. 3(4) (1975), pp. 862–873]. Their estimator will be discussed and compared with the proposed estimator in this paper. An example and simulation studies have been performed to compare the proposed procedure with existing shift parameter estimators such as Hodges–Lehmann (H–L) and least squares in addition to RSL's estimator. The results show that the proposed estimator has lower mean-squared error as well as higher relative efficiency against RSL's estimator under normal or contaminated normal model assumptions. Moreover, the proposed estimator performs competitively against H–L and least-squares shift estimators. Smoother function and bandwidth selections are also discussed and several alternatives are proposed in the study.     

k-Step shape estimators based on spatial signs and ranks

In this paper, the shape matrix estimators based on spatial sign and rank vectors are considered. The estimators considered here are slight modifications of the estimators introduced in Dümbgen (1998) and Oja and Randles (2004) and further studied for example in Sirkiä et al. (2009). The shape estimators are computed using pairwise differences of the observed data, therefore there is no need to estimate the location center of the data. When the estimator is based on signs, the use of differences also implies that the estimators have the so called independence property if the estimator, that is used as an initial estimator, has it. The influence functions and limiting distributions of the estimators are derived at the multivariate elliptical case. The estimators are shown to be highly efficient in the multinormal case, and for heavy-tailed distributions they outperform the shape estimator based on sample covariance matrix.     

Bounded influence nonlinear signed‐rank regression

In this paper we consider weighted generalized‐signed‐rank estimators of nonlinear regression coefficients. The generalization allows us to include popular estimators such as the least squares and least absolute deviations estimators but by itself does not give bounded influence estimators. Adding weights results in estimators with bounded influence function. We establish conditions needed for the consistency and asymptotic normality of the proposed estimator and discuss how weight functions can be chosen to achieve bounded influence function of the estimator. Real life examples and Monte Carlo simulation experiments demonstrate the robustness and efficiency of the proposed estimator. An example shows that the weighted signed‐rank estimator can be useful to detect outliers in nonlinear regression. The Canadian Journal of Statistics 40: 172–189; 2012 © 2012 Statistical Society of Canada     

Robust tests for equality of two population means under the normal model

A robust procedure is developed for testing the equality of means in the two sample normal model. This is based on the weighted likelihood estimators of Basu et al. (1993). When the normal model is true the tests proposed have the same asymptotic power as the two sample Student's t-statistic in the equal variance case. However, when the normality assumptions are only approximately true the proposed tests can be substantially more powerful than the classical tests. In a Monte Carlo study for the equal variance case under various outlier models the proposed test using Hellinger distance based weighted likelihood estimator compared favorably with the classical test as well as the robust test proposed by Tiku (1980).     

The asymptotic behaviour of a class of L.-estimators under long-range dependence

This paper obtains asymptotic representations of a class of L-estimators in a linear regression model when the errors are a function of long-range-dependent Gaussian random variables. These representations are then used to address some of the efficiency robustness properties of L-estimators compared to the least-squares estimator. It is observed that under the Gaussian error distribution, each member of the class has the same asymptotic efficiency as that of the least-squares estimator. The results are obtained as a consequence of the asymptotic uniform linearity of some weighted empirical processes based on long-range-dependent random variables.