Robust joint estimation of location and scale parameters in ranked set samples

A robust estimator is developed for the location and scale parameters of a location-scale family. The estimator is defined as the minimizer of a minimum distance function that measures the distance between the ranked set sample empirical cumulative distribution function and a possibly misspecified target model. We show that the estimator is asymptotically normal, robust, and has high efficiency with respect to its competitors in literature. It is also shown that the location estimator is consistent within the class of all symmetric distributions whereas the scale estimator is Fisher consistent at the true target model. The paper also considers an optimal allocation procedure that does not introduce any bias due to judgment error classification. It is shown that this allocation procedure is equivalent to Neyman allocation. A numerical efficiency comparison is provided.     

Asymptotically efficient estimation in response adaptive trials

We examine the issue of asymptotic efficiency of estimation for response adaptive designs of clinical trials, from which the collected data set contains a dependency structure. We establish the asymptotic lower bound of exponential rates for consistent estimators. Under certain regularity conditions, we show that the maximum likelihood estimator achieves the asymptotic lower bound for response adaptive trials with dichotomous responses. Furthermore, it is shown that the maximum likelihood estimator of the treatment effect is asymptotically efficient in the Bahadur sense for response adaptive clinical trials.     

Comparisons of asymptotic biases and variances of m-estimators of scale under asymmetric contamination

We study robustness properties of two types of M-estimators of scale when both location and scale parameters are unknown: (i) the scale estimator arising from simultaneous M-estimation of location and scale; and (ii) its symmetrization about the sample median. The robustness criteria considered are maximal asymptotic bias and maximal asymptotic variance when the known symmetric unimodal error distribution is subject to unknown, possibly asymmetric, £-con-tamination. Influence functions and asymptotic variance functionals are derived, and computations of asymptotic biases and variances, under the normal distribution with ε-contamination at oo, are presented for the special subclass arising from Huber's Proposal 2 and its symmetrized version. Symmetrization is seen to reduce both asymptotic bias and variance. Some complementary theoretical results are obtained, and the tradeoff between asymptotic bias and variance is discussed.     

Location and Scale Parameter Family of Distributions Attaining the Bhattacharyya Bound

It is well known that, under appropriate regularity conditions, the variance of an unbiased estimator of a real-valued function of an unknown parameter can coincide with the Cramér–Rao lower bound only if the family of distributions is a one-parameter exponential family. But it seems that the necessary conditions about the probability distribution for which there exists an unbiased estimator whose variance coincides with the Bhattacharyya lower bound are not completely known. The purpose of this paper is to specify the location, scale, and location-scale parameter family of distributions attaining the general order Bhattacharyya bound in certain class.     

Universal admissibility in second order asymptotics

Some sufficient conditions for an estimator to be universally second order admissible are derived. Those sufficient conditions consist of the elementary integrals with respect to the Fisher information and the limits of some functions characterized by the dealt statistical model, and thus can be checked with comparative ease. In location model and scale model, the sufficient condition for the linear estimator with respect to the maximum likelihood estimator (MLE) to be universally second order admissible is given. Furthermore, a guide for classifying any estimator into either the universal admissibility or the non-universal admissibility is proposed.     

First‐order bias correction for fractionally integrated time series

Most of the long memory estimators for stationary fractionally integrated time series models are known to experience non‐negligible bias in small and finite samples. Simple moment estimators are also vulnerable to such bias, but can easily be corrected. In this article, the authors propose bias reduction methods for a lag‐one sample autocorrelation‐based moment estimator. In order to reduce the bias of the moment estimator, the authors explicitly obtain the exact bias of lag‐one sample autocorrelation up to the order n⁻¹. An example where the exact first‐order bias can be noticeably more accurate than its asymptotic counterpart, even for large samples, is presented. The authors show via a simulation study that the proposed methods are promising and effective in reducing the bias of the moment estimator with minimal variance inflation. The proposed methods are applied to the northern hemisphere data. The Canadian Journal of Statistics 37: 476–493; 2009 © 2009 Statistical Society of Canada     

Estimation of the scale parameter of the half-logistic distribution with multiply type II censored sample

In this paper, we consider the maximum likelihood and Bayes estimation of the scale parameter of the half-logistic distribution based on a multiply type II censored sample. However, the maximum likelihood estimator(MLE) and Bayes estimator do not exist in an explicit form for the scale parameter. We consider a simple method of deriving an explicit estimator by approximating the likelihood function and discuss the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney & Kadane, 1986) is used to obtain the Bayes estimator. In order to compare the MLE, approximate MLE and Bayes estimates of the scale parameter, Monte Carlo simulation is used.     

On bagging and nonlinear estimation

We propose an elementary model for the way in which stochastic perturbations of a statistical objective function, such as a negative log-likelihood, produce excessive nonlinear variation of the resulting estimator. Theory for the model is transparently simple, and is used to provide new insight into the main factors that affect performance of bagging. In particular, it is shown that if the perturbations are sufficiently symmetric then bagging will not significantly increase bias; and if the perturbations also offer opportunities for cancellation then bagging will reduce variance. For the first property it is sufficient that the third derivative of a perturbation vanish locally, and for the second, that second and fourth derivatives have opposite signs. Functions that satisfy these conditions resemble sinusoids. Therefore, our results imply that bagging will reduce the nonlinear variation, as measured by either variance or mean-squared error, produced in an estimator by sinusoid-like, stochastic perturbations of the objective function. Analysis of our simple model also suggests relationships between the results obtained using different with-replacement and without-replacement bagging schemes. We simulate regression trees in settings that are far more complex than those explicitly addressed by the model, and find that these relationships are generally borne out.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Maximum-likelihood estimation for the removal method

We prove that the profile log-likelihood function for the removal method of estimating population size is unimodal. The result is obtained by a variation-diminishing property of the Laplace transform. An implication of this result is that the likelihood-ratio confidence region for the population size is always an interval. Necessary and sufficient conditions for the existence of a finite maximum-likelihood estimator are presented. We also present evidence that the likelihood-ratio confidence interval for the population size has acceptable small-sample coverage properties.     

Does bias reduction with external estimator of second order parameter work for endpoint?

Bias reduction estimation for tail index has been studied in the literature. One method is to reduce bias with an external estimator of the second order regular variation parameter; see Gomes and Martins [2002. Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31]. It is known that negative extreme value index implies that the underlying distribution has a finite right endpoint. As far as we know, there exists no bias reduction estimator for the endpoint of a distribution. In this paper, we study the bias reduction method with an external estimator of the second order parameter for both the negative extreme value index and endpoint simultaneously. Surprisingly, we find that this bias reduction method for negative extreme value index requires a larger order of sample fraction than that for positive extreme value index. This finding implies that this bias reduction method for endpoint is less attractive than that for positive extreme value index. Nevertheless, our simulation study prefers the proposed bias reduction estimators to the biased estimators in Hall [1982. On estimating the endpoint of a distribution. Ann. Statist. 10, 556–568].     

SMALL SAMPLE BIAS CORRECTION FOR HUBER'S PROPOSAL-2 SCALE M-ESTIMATOR

The most popular and perhaps universal estimator of location and scale in robust estimation, where the population is normal with possible small departures, is Huber's Proposal‐2 M‐estimator. This paper gives the first‐order small sample bias correction for the scale estimator, verifying the calculation through theory and simulation. Other ways of reducing small sample bias, say by jackknifing or bootstrapping, can be computationally intensive, and would not be routinely used with this iteratively derived estimator. It is suggested that bias‐reduced estimates of scale are most useful when forming confidence intervals for location and/or scale based on the asymptotic distribution.     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.     

A comparison of estimation techniques for the three parameter pareto distribution

This paper compares minimum distance estimation with best linear unbiased estimation to determine which technique provides the most accurate estimates for location and scale parameters as applied to the three parameter Pareto distribution. Two minimum distance estimators are developed for each of the three distance measures used (Kolmogorov, Cramer‐von Mises, and Anderson‐Darling) resulting in six new estimators. For a given sample size 6 or 18 and shape parameter 1(1)4, the location and scale parameters are estimated. A Monte Carlo technique is used to generate the sample sets. The best linear unbiased estimator and the six minimum distance estimators provide parameter estimates based on each sample set. These estimates are compared using mean square error as the evaluation tool. Results show that the best linear unbaised estimator provided more accurate estimates of location and scale than did the minimum estimators tested.     

Bias-reduced extreme quantile estimators of Weibull tail-distributions

In this paper, we consider the problem of estimating an extreme quantile of a Weibull tail-distribution. The new extreme quantile estimator has a reduced bias compared to the more classical ones proposed in the literature. It is based on an exponential regression model that was introduced in Diebolt et al. [2007. Bias-reduced estimators of the Weibull-tail coefficient. Test, to appear]. The asymptotic normality of the extreme quantile estimator is established. We also introduce an adaptive selection procedure to determine the number of upper order statistics to be used. A simulation study as well as an application to a real data set is provided in order to prove the efficiency of the above-mentioned methods.     

Sequential and two-stage procedures for selecting the better exponential population covering the case of unknown and unequal scale parameters

The problems of selecting the larger location parameter of two exponential distributions are discussed. When the scale parameters are the same but unknown, we consider the procedure of Desu et al. (1977) in detail, and study some of its exact and asymptotic properties. We indicate how this procedure can be modified along the lines of Mukhopadhyay (1979, 1980) to achieve first-order asymptotic efficiency. We then propose a sequential procedure for this set-up and show that it is asymptotically second-order efficient according to Ghosh and Mukhopadhyay (1981). In case the scale parameters are completely unknown and unequal, we propose a two-stage procedure that guarantees the probability of correct selection to exceed the prescribed nominal level in the preference zone. We do not need any new tables to implement this particular procedure other than those in Krishnaiah and Armitage (1964), Gupta and Sobel (1962), Guttman and Milton (1969). We also propose a sequential method in this case and derive some of its asymptotic properties.     

Estimation of an exponential quantile under pitman's measure of closeness

We consider the problem of estimating a quantile of an exponential distribution with unknown location and scale parameters under Pitman's measure of closeness (PMC). The loss function is required to satisfy some mild conditions but is otherwise arbitrary. An optimal estimator is obtained in the class of location-scale-equivariant estimators, and its admissibility in the sense of PMC is investigated.     

Lifetesting and estimation with arbitrary distribution function

We consider the problem of estimating the life–distribution F from censored lifetimes. The observation scheme is renewal testing over a long time horizon although the results can apply to survival testing with repetitions. We exhibit a product–limit estimator of F which is shown to be consistent and to converge weakly to a GAUSsian process. To do this we first extend these properties of the NELSON-AALEN martingale estimator to the family of PoissoN–type counting processes. Our proof of weak convergence is based on the general functional central limit theorems for semimartingales as developed by .JACOB, SHIRYAYEV and others     

Simultaneous robust estimation of location and scale parameters: A minimum-distance approach

Simultaneous robust estimates of location and scale parameters are derived from minimizing a minimum-distance criterion function. The criterion function measures the squared distance between the pth power (p > 0) of the empirical distribution function and the pth power of the imperfectly determined model distribution function over the real line. We show that the estimator is uniquely defined, is asymptotically bivariate normal and for p > 0.3 has positive breakdown. If the scale parameter is known, when p = 0.9 the asymptotic variance (1.0436) of the location estimator for the normal model is smaller than the asymptotic variance of the Hodges-Lehmann (HL)estimator (1.0472). Efficiencies with respect to HL and maximum-likelihood estimators (MLE) are 1.0034 and 0.9582, respectively. Similarly, if the location parameter is known, when p = 0.97 the asymptotic variance (0.6158) of the scale estimator is minimum. The efficiency with respect to the MLE is 0.8119. We show that the estimator can tolerate more corrupted observations at oo than at – for p < 1, and vice versa for p > 1.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.