A note on the Hodges–Lehmann estimator

The Hodges‐Lehmann estimator was originally developed as a non‐parametric estimator of a shift parameter. As it is widely used in statistical applications, the question is investigated what it is estimating if the shift model does not hold. It is shown that for data whose distributions are symmetric about their median the Hodges–Lehmann estimator based on the Wilcoxon Rank Sum test estimates the difference between the medians of the distributions. This result does generally not hold if the symmetry assumption is violated. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation in two sample randomly truncated scale model

This paper studies a class of Cramér–von Mises type minimum distance estimators of the scale parameter in the two sample randomly left truncated scale models. The proposed class of estimators includes an analogue of the well-known Hodges–Lehmann estimator. The paper proves the asymptotic normality of these estimators under mild conditions. It also contains a real data application and a simulation study making a comparison of some of the estimators in the class with the ratio of the two means.     

A Note on the Hodges–Lehmann Estimator and the Logistic Distribution

ABSTRACT

It is well known that the Hodges–Lehmann estimator is asymptotically efficient for the location parameter of the logistic distribution. In this article we give a simple and direct proof that this property also characterizes the logistic between all the symmetric location distributions under mild conditions. Using pseudolikelihood, we also show how to find from the Hodges–Lehmann estimator an asymptotically efficient estimator of the scale parameter of the logistic distribution.     

Comparison of estimation methods for the finite population mean in simple random sampling: symmetric super-populations

In this paper, a new estimator combined estimator (CE) is proposed for estimating the finite population mean ¯ Y _N in simple random sampling assuming a long-tailed symmetric super-population model. The efficiency and robustness properties of the CE is compared with the widely used and well-known estimators of the finite population mean ¯ Y _N by Monte Carlo simulation. The parameter estimators considered in this study are the classical least squares estimator, trimmed mean, winsorized mean, trimmed L-mean, modified maximum-likelihood estimator, Huber estimator (W24) and the non-parametric Hodges–Lehmann estimator. The mean square error criteria are used to compare the performance of the estimators. We show that the CE is overall more efficient than the other estimators. The CE is also shown to be more robust for estimating the finite population mean ¯ Y _N , since it is insensitive to outliers and to misspecification of the distribution. We give a real life example.     

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.     

Local Estimation of the Second-Order Parameter in Extreme Value Statistics and Local Unbiased Estimation of the Tail Index

We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data.     

On Asymptotic Deficiency of Estimators

The notion of deficiency was introduced by Hodges and Lehmann. It is known that best asymptotically normal (BAN) estimators are second order asymptotically efficient in the class A₂ of all second order asymptotically median unbiased estimators. In this paper it is shown that the asymptotic deficiency of any two estimators in the restricted class D of the third order asymptotically median unbiased BAN estimators is given by the difference between the coefficients of order n^-1 of the variances of the estimators.     

ROBUST ESTIMATION FOR A SIMPLE EXPONENTIAL MODEL1

A Hodges-Lehmann type estimator (Hodges and Lehmann, 1963) is proposed for the parameter p in the simple exponential model (1) y=1-exp.(-ρx)+ε Some exact and asymptotic properties of the proposed estimator are also given.     

A New Kernel Distribution Function Estimator Based on a Non-parametric Transformation of the Data

Abstract.  A new kernel distribution function (df) estimator based on a non-parametric transformation of the data is proposed. It is shown that the asymptotic bias and mean squared error of the estimator are considerably smaller than that of the standard kernel df estimator. For the practical implementation of the new estimator a data-based choice of the bandwidth is proposed. Two possible areas of application are the non-parametric smoothed bootstrap and survival analysis. In the latter case new estimators for the survival function and the mean residual life function are derived.     

Multivariate generalized Birnbaum—Saunders kernel density estimators

In this article, we first propose the classical multivariate generalized Birnbaum–Saunders kernel estimator for probability density function estimation in the context of multivariate non negative data. Then, we apply two multiplicative bias correction (MBC) techniques for multivariate kernel density estimator. Some properties (bias, variance, and mean integrated squared error) of the corresponding estimators are also investigated. Finally, the performances of the classical and MBC estimators based on family of generalized Birnbaum–Saunders kernels are illustrated by a simulation study.     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

A Robust and Almost Fully Efficient M-Estimator

Simultaneous robust estimates of location and scale parameters are derived from a class of M-estimating equations. A coefficient p ( p > 0), which plays a role similar to that of a tuning constant in the theory of M-estimation, determines the estimating equations. These estimating equations may be obtained as the gradient of a strictly convex criterion function. This article shows that the estimators are uniquely defined, asymptotically bi-variate normal and have positive breakdown for some choices of p . When p = 0.12 and p = 0.3, the estimators are almost fully efficient for normal and exponential distributions: efficiencies with respect to the maximum likelihood estimators are 1.00 and 0.99, respectively. It is shown that the location estimator for known scale has the maximum breakdown point 0.5 independent of p , when the target model is symmetric. Also it is shown that the scale estimator has a positive breakdown point which depends on the choice of p . A simulation study finds that the proposed location estimator has smaller variance than the Hodges–Lehmann estimator, Huber's minimax and bisquare M-estimators.     

A Unified View of Nonparametric Trend-Cycle Predictors Via Reproducing Kernel Hilbert Spaces

We provide a common approach for studying several nonparametric estimators used for smoothing functional time series data. Linear filters based on different building assumptions are transformed into kernel functions via reproducing kernel Hilbert spaces. For each estimator, we identify a density function or second order kernel, from which a hierarchy of higher order estimators is derived. These are shown to give excellent representations for the currently applied symmetric filters. In particular, we derive equivalent kernels of smoothing splines in Sobolev and polynomial spaces. The asymmetric weights are obtained by adapting the kernel functions to the length of the various filters, and a theoretical and empirical comparison is made with the classical estimators used in real time analysis. The former are shown to be superior in terms of signal passing, noise suppression and speed of convergence to the symmetric filter.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

Nonparametric hazard rate estimation by orthogonal wavelet methods

We propose linear and nonlinear wavelet-based hazard rate estimators where the linear estimator is equivalent to a generalized kernel estimator. An asymptotic formula for the mean integrated squared error (MISE) of the nonlinear wavelet-based hazard rate estimator is provided. It is shown that the MISE formula for the nonlinear estimator is available for hazard rates which are smooth only in a piecewise sense, a feature not available for the kernel estimators.     

Nonparametrie recursive estimator for mean residual life and vitality function under dependence conditions

In this paper, we develop a nonparametrie recursive estimator for the vitality and mena residual life function, based on kernel density estimators under mixing dependence conditions. The consistency and asymptotic normality of the estimator are established, under suitable regularity conditions. It is also shown that the Integrated Mean Squared Error converges to zero. The paper is concluyed with some simulation results.     

Hazard function estimation from homogeneous right censored data with missing censoring indicators

The kernel smoothed Nelson–Aalen estimator has been well investigated, but is unsuitable when some of the censoring indicators are missing. A representation introduced by Dikta, however, facilitates hazard estimation when there are missing censoring indicators. In this article, we investigate (i) a kernel smoothed semiparametric hazard estimator and (ii) a kernel smoothed “pre-smoothed” Nelson–Aalen estimator. We derive the asymptotic normality of the proposed estimators and compare their asymptotic variances.     

Bernstein conditional density estimation with application to conditional distribution and regression functions

In this paper we propose a smooth nonparametric estimation for the conditional probability density function based on a Bernstein polynomial representation. Our estimator can be written as a finite mixture of beta densities with data-driven weights. Using the Bernstein estimator of the conditional density function, we derive new estimators for the distribution function and conditional mean. We establish the asymptotic properties of the proposed estimators, by proving their asymptotic normality and by providing their asymptotic bias and variance. Simulation results suggest that the proposed estimators can outperform the Nadaraya–Watson estimator and, in some specific setups, the local linear kernel estimators. Finally, we use our estimators for modeling the income in Italy, conditional on year from 1951 to 1998, and have another look at the well known Old Faithful Geyser data.     

Estimating Percentiles of Time-to-Failure Distribution Obtained from a Linear Degradation Model Using Kernel Density Method

In this article, we propose a nonparametric estimator for percentiles of the time-to-failure distribution obtained from a linear degradation model using the kernel density method. The properties of the proposed kernel estimator are investigated and compared with well-known maximum likelihood and ordinary least squares estimators via a simulation technique. The mean squared error and the length of the bootstrap confidence interval are used as the basis criteria of the comparisons. The simulation study shows that the performance of the kernel estimator is acceptable as a general estimator. When the distribution of the data is assumed to be known, the maximum likelihood and ordinary least squares estimators perform better than the kernel estimator, while the kernel estimator is superior when the assumption of our knowledge of the data distribution is violated. A comparison among different estimators is achieved using a real data set.     

Histogram Model with Plausible Parametric Detection Function for Line Transect Data

This article develops a new model that combines between the histogram and plausible parametric detection function to estimate the population density (abundance) by using line transects technique. A parametric detection function is introduced to improve the properties of the classical histogram estimator. Asymptotic properties of the resulting estimator are derived and an expression for the asymptotic mean square error (AMSE) is given. A general formula for the optimal choice of the histogram bin width based on AMSE is derived. Moreover, other possible alternative procedures to select the bin width are suggested and studied via simulation technique. The results show the superiority of the proposed estimators over both the classical histogram and the usual kernel estimators in most reasonable cases. In addition, the simulation results indicate that the choice of a plausible detection function is less sensitive than the choice of a bin width on the performance of the proposed estimator.