A restricted subset selection procedure for Weibull populations

Consider k(k ≥ 2) two-parameter Weibull populations. We want to select a subset of the populations not exceeding m in size such that the subset contains at least ? of the t best populations. We have proposed a procedure which uses either the maximum likelihood estimators or ‘simplified’ linear estimators of the parameters. The estimators are based on type II censored data. The ranking of the populations is done by comparing their reliabilities at a certain fixed time. In selected cases the constants for the procedure are tabulated using Monte Carlo methods.     

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.     

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

Abstract

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.     

On quantiles estimation based on different stratified sampling with optimal allocation

This work considers the problem of estimating a quantile function based on different stratified sampling mechanism. First, we develop an estimate for population quantiles based on stratified simple random sampling (SSRS) and extend the discussion for stratified ranked set sampling (SRSS). Furthermore, the asymptotic behavior of the proposed estimators are presented. In addition, we derive an analytical expression for the optimal allocation under both sampling schemes. Simulation studies are designed to examine the performance of the proposed estimators under varying distributional assumptions. The efficiency of the proposed estimates is further illustrated by analyzing a real data set from CHNS.     

Nonnegative Unbiased Estimation of Scale Parameters and Associated Quantiles Based on a Ranked Set Sample

In this article, we are interested in estimating the scale parameter in location and scale families. It is well known that the best linear unbiased estimator (BLUE) of scale parameter based on a simple random sample (SRS) is nonnegative. However, the BLUE of scale parameter based on a ranked set sample (RSS) can assume negative values. We suggest various modifications of BLUE of scale parameter based on RSS so that the resulting estimators are unbiased as well as nonnegative. Their performances in terms of relative efficiencies are compared and some recommendations are made for normal, logistic, double exponential, two-parameter exponential and Weibull distributions. We also briefly discuss an application of the proposed nonnegative BLUE of scale parameter for quantile estimation for the above populations.     

The generalized odd log-logistic family of distributions: properties,regression models and applications

We propose a new class of continuous distributions with two extra shape parameters named the generalized odd log-logistic family of distributions. The proposed family contains as special cases the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, two entropy measures and order statistics are obtained. We derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behaviour of the estimators by means of Monte Carlo simulations. We introduce the log-odd log-logistic Weibull regression model with censored data based on the odd log-logistic-Weibull distribution. The importance of the new family is illustrated using three real data sets. These applications indicate that this family can provide better fits than other well-known classes of distributions. The beauty and importance of the proposed family lies in its ability to model different types of real data.     

Nonparametric estimation of mean residual quantile function under right censoring

In this paper, we develop non-parametric estimation of the mean residual quantile function based on right-censored data. Two non-parametric estimators, one based on the empirical quantile function and the other using the kernel smoothing method, are proposed. Asymptotic properties of the estimators are discussed. Monte Carlo simulation studies are conducted to compare the two estimators. The method is illustrated with the aid of two real data sets.     

Quantile Estimation in the Presence of Auxiliary Information under Negatively Associated Samples

In this article, we apply the empirical likelihood technique to propose a new class of quantile estimators in the presence of some auxiliary information under negatively associated samples. It is shown that the proposed quantile estimators are asymptotically normally distributed with smaller asymptotic variances than those of the usual quantile estimators. It is also shown that blocking technique is an useful tool in estimating asymptotic variance under negatively associated samples, which makes it possible to construct normal approximation based confidence intervals for quantiles.     

Multiple-output quantile regression through optimal quantization

A new nonparametric quantile regression method based on the concept of optimal quantization was developed recently and was showed to provide estimators that often dominate their classical, kernel-type, competitors. In the present work, we extend this method to multiple-output regression problems. We show how quantization allows approximating population multiple-output regression quantiles based on halfspace depth. We prove that this approximation becomes arbitrarily accurate as the size of the quantization grid goes to infinity. We also derive a weak consistency result for a sample version of the proposed regression quantiles. Through simulations, we compare the performances of our estimators with (local constant and local bilinear) kernel competitors. The results reveal that the proposed quantization-based estimators, which are local constant in nature, outperform their kernel counterparts and even often dominate their local bilinear kernel competitors. The various approaches are also compared on artificial and real data.     

Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.     

Interquantile shrinkage in additive models

In this paper, we investigate the commonality of nonparametric component functions among different quantile levels in additive regression models. We propose two fused adaptive group Least Absolute Shrinkage and Selection Operator penalties to shrink the difference of functions between neighbouring quantile levels. The proposed methodology is able to simultaneously estimate the nonparametric functions and identify the quantile regions where functions are unvarying, and thus is expected to perform better than standard additive quantile regression when there exists a region of quantile levels on which the functions are unvarying. Under some regularity conditions, the proposed penalised estimators can theoretically achieve the optimal rate of convergence and identify the true varying/unvarying regions consistently. Simulation studies and a real data application show that the proposed methods yield good numerical results.     

Estimating Quantiles of Normal Populations with a Common Mean

Suppose independent random samples are available from two normal populations with a common mean and unequal variances. Estimation of a quantile of the first population is considered with respect to the quadratic loss. Some new estimators for the quantile are proposed using some previously known estimators of a common mean. Inadmissibility results are proved for estimators which are equivariant under affine and location groups of transformations. Risk values of various estimators of a quantile are compared numerically using a detailed simulation study.     

A weighted linear quantile regression

In this article, we introduce a new weighted quantile regression method. Traditionally, the estimation of the parameters involved in quantile regression is obtained by minimizing a loss function based on absolute distances with weights independent of explanatory variables. Specifically, we study a new estimation method using a weighted loss function with the weights associated with explanatory variables so that the performance of the resulting estimation can be improved. In full generality, we derive the asymptotic distribution of the weighted quantile regression estimators for any uniformly bounded positive weight function independent of the response. Two practical weighting schemes are proposed, each for a certain type of data. Monte Carlo simulations are carried out for comparing our proposed methods with the classical approaches. We also demonstrate the proposed methods using two real-life data sets from the literature. Both our simulation study and the results from these examples show that our proposed method outperforms the classical approaches when the relative efficiency is measured by the mean-squared errors of the estimators.     

Asymptotic Normality of Extreme Quantile Estimators Based on the Peaks-Over-Threshold Approach

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over thresholds. In this article, we establish the asymptotic normality of the well-known extreme quantile estimators based on this POT method, under very general assumptions. As an illustration, from this result, we deduce the asymptotic normality of the POT extreme quantile estimators in the case where the maximum likelihood (ML) or the generalized probability-weighted moments (GPWM) methods are used. Simulations are provided in order to compare the efficiency of these estimators based on ML or GPWM methods with classical ones proposed in the literature.     

Distribution estimation with auxiliary information for missing data

There is much literature on statistical inference for distribution under missing data, but surprisingly very little previous attention has been paid to missing data in the context of estimating distribution with auxiliary information. In this article, the auxiliary information with missing data is proposed. We use Zhou, Wan and Wang's method (2008) to mitigate the effects of missing data through a reformulation of the estimating equations, imputed through a semi-parametric procedure. Whence we can estimate distribution and the τth quantile of the distribution by taking auxiliary information into account. Asymptotic properties of the distribution estimator and corresponding sample quantile are derived and analyzed. The distribution estimators based on our method are found to significantly outperform the corresponding estimators without auxiliary information. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.     

Weighted quantile regression and testing for varying-coefficient models with randomly truncated data

This paper develops a varying-coefficient approach to the estimation and testing of regression quantiles under randomly truncated data. In order to handle the truncated data, the random weights are introduced and the weighted quantile regression (WQR) estimators for nonparametric functions are proposed. To achieve nice efficiency properties, we further develop a weighted composite quantile regression (WCQR) estimation method for nonparametric functions in varying-coefficient models. The asymptotic properties both for the proposed WQR and WCQR estimators are established. In addition, we propose a novel bootstrap-based test procedure to test whether the nonparametric functions in varying-coefficient quantile models can be specified by some function forms. The performance of the proposed estimators and test procedure are investigated through simulation studies and a real data example.     

Ordered partially ordered judgment subset sampling with applications to parametric inference

ABSTRACT

In this paper, we use the idea of order statistics from independent and non-identically distributed random variables to propose ordered partially ordered judgment subset sampling (OPOJSS) and then develop optimal linear parametric inferences. The best linear unbiased and invariant estimators of the location and scale parameters of a location-scale family are developed based on OPOJSS. It is shown that, despite the presence or absence of ranking errors, the proposed estimators with OPOJSS are uniformly better than the existing estimators with simple random sampling (SRS), ranked set sampling (RSS), ordered RSS (ORSS) and partially ordered judgment subset sampling (POJSS). Moreover, we also derive the best linear unbiased estimators (BLUEs) of the unknown parameters of the simple linear regression model with replicated observations using POJSS and OPOJSS. It is found that the BLUEs with OPOJSS are more precise than the BLUEs based on SRS, RSS, ORSS and POJSS.     

Parameter estimation based upon nonparametric function estimators

By considering the solution to a linear approximation of a nonlinear regression problem, a procedure for developing a para¬meter estimator, based upon a nonpammetric estimator of a para¬metric function, is given. The resulting estimators, which are determinable in closed form, are asymptotically normally distri¬buted and are optimal among the class of estimators based upon the function estimator. Further, in many cases, the estimator will have the same asymptotic distribution theory as the correspond¬ing maximum likelihood estimator. Estimators based upon the Kaplan-Meier quantile function are developed for randomly censored samples.     

Subsampling quantile estimators and uniformity criteria

Several asymptotically equivalent quantile estimators recently have been proposed as alternative to the conventional sample quantile. A variety of weight functions have been obtained either by subsampling considerations or by a kernel approach, analogous to density estimation techniques. Focusing on the former approach, a unified treatment of quantile estimators derived by subsampling is developed. Closely related to the generalized Harrell-Davis (HD) and Kaigh-Lachenbruch (KL) estimators, a new statistic performed well in Monte Carlo effiency comparisons presented here. Moreover, the new estimator shares certain desirable computational and finite-sample theeoretical properties with the KL estimator to yield convenient components representations for tests of uniformity and goodness-of-fit criteria. Similar analytic treatment for the HD statistics and kernel quantile estimators, however, is precluded by intractable eigenvalue problems.     

A two-stage procedure to pool information across quantile levels in linear quantile regression

In linear quantile regression, the regression coefficients for different quantiles are typically estimated separately. Efforts to improve the efficiency of estimators are often based on assumptions of commonality among the slope coefficients. We propose instead a two-stage procedure whereby the regression coefficients are first estimated separately and then smoothed over quantile level. Due to the strong correlation between coefficient estimates at nearby quantile levels, existing bandwidth selectors will pick bandwidths that are too small. To remedy this, we use 10-fold cross-validation to determine a common bandwidth inflation factor for smoothing the intercept as well as slope estimates. Simulation results suggest that the proposed method is effective in pooling information across quantile levels, resulting in estimates that are typically more efficient than the separately obtained estimates and the interquantile shrinkage estimates derived using a fused penalty function. The usefulness of the proposed method is demonstrated in a real data example.