Improved reduced-bias tail index and quantile estimators

In this paper, we deal with bias reduction techniques for heavy tails, trying to improve mainly upon the performance of classical high quantile estimators. High quantiles depend strongly on the tail index γ$\gamma$, for which new classes of reduced-bias estimators have recently been introduced, where the second-order parameters in the bias are estimated at a level _k1$k_1$ of a larger order than the level k at which the tail index is estimated. Doing this, it was seen that the asymptotic variance of the new estimators could be kept equal to the one of the popular Hill estimators. In a similar way, we now introduce new classes of tail index and associated high quantile estimators, with an asymptotic mean squared error smaller than that of the classical ones for all k in a large class of heavy-tailed models. We derive their asymptotic distributional properties and compare them with those of alternative estimators. Next to that, an illustration of the finite sample behavior of the estimators is also provided through a Monte Carlo simulation study and the application to a set of real data in the field of insurance.     

On the uth geometric conditional quantile

Motivated by Chaudhuri's work [1996. On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91, 862–872] on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions, in high-dimensional spaces. We establish a Bahadur-type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality including bias on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.     

Estimating the distribution function using k-tuple ranked set samples

The basic assumption underlying the concept of ranked set sampling is that actual measurement of units is expensive, whereas ranking is cheap. This may not be true in reality in certain cases where ranking may be moderately expensive. In such situations, based on total cost considerations, k-tuple ranked set sampling is known to be a viable alternative, where one selects k units (instead of one) from each ranked set. In this article, we consider estimation of the distribution function based on k-tuple ranked set samples when the cost of selecting and ranking units is not ignorable. We investigate estimation both in the balanced and unbalanced data case. Properties of the estimation procedure in the presence of ranking error are also investigated. Results of simulation studies as well as an application to a real data set are presented to illustrate some of the theoretical findings.     

Bayesian mixture modeling for spatial Poisson process intensities,with applications to extreme value analysis

We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial nonhomogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mixture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clustering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.     

Convergence rates for smoothing spline estimators in varying coefficient models

We consider the estimation of a multiple regression model in which the coefficients change slowly in “time”, with “time” being an additional covariate. Under reasonable smoothness conditions, we prove the usual expected mean square error bounds for the smoothing spline estimators of the coefficient functions.     

Adaptation over parametric families of symmetric linear estimators

This paper treats an abstract parametric family of symmetric linear estimators for the mean vector of a standard linear model. The estimator in this family that has smallest estimated quadratic risk is shown to attain, asymptotically, the smallest risk achievable over all candidate estimators in the family. The asymptotic analysis is carried out under a strong Gauss–Markov form of the linear model in which the dimension of the regression space tends to infinity. Leading examples to which the results apply include: (a) penalized least squares fits constrained by multiple, weighted, quadratic penalties; and (b) running, symmetrically weighted, means. In both instances, the weights define a parameter vector whose natural domain is a continuum.     

Empirical likelihood for a partially linear model with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. Empirical likelihood ratios for the regression coefficients and the baseline function are investigated, the empirical log-likelihood ratios are proven to be asymptotically chi-squared and the corresponding confidence regions for the parameters of interest are then constructed. The finite sample behavior of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial dataset.     

Consistency of maximum likelihood estimators in a large class of deconvolution models

We consider the maximum likelihood estimator $\hat{F}_n$ of a distribution function in a class of deconvolution models where the known density of the noise variable is of bounded variation. This class of noise densities contains in particular bounded, decreasing densities. The estimator $\hat{F}_n$ is defined, characterized in terms of Fenchel optimality conditions and computed. Under appropriate conditions, various consistency results for $\hat{F}_n$ are derived, including uniform strong consistency. The Canadian Journal of Statistics 41: 98–110; 2013 © 2012 Statistical Society of Canada     

Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.     

Weibull tail-distributions revisited: A new look at some tail estimators

In this paper, we propose to include Weibull tail-distributions in a more general family of distributions. In particular, the considered model also encompasses the whole Fréchet maximum domain of attraction as well as log-Weibull tail-distributions. The asymptotic normality of some tail estimators based on the log-spacings between the largest order statistics is established in a unified way within the considered family. This result permits to understand the similarity between most estimators of the Weibull tail-coefficient and the Hill estimator. Some different asymptotic properties, in terms of bias, rate of convergence, are also highlighted.     

Limit theory of generalized order statistics

Generalized order statistics (gos) were introduced by Kamps [1995. A Concept of Generalized Order Statistics. Teubner, Stuttgart] to unify several models of ordered random variables (rv's), e.g., (ordinary) order statistics (oos), records, sequential order statistics (sos). In a wide subclass of gos that includes oos and sos, the possible limit distribution functions (df's) of the maximum gos are obtained in Nasri-Roudsari [1996. Extreme value theory of generalized order statistics. J. Statist. Plann. Inference 55, 281–297]. In this paper, for this subclass, necessary and sufficient conditions of weak convergence, as well as the form of the possible limit df's of extreme, intermediate and central gos are derived. These results are extended to a wider subclass.     

Estimation and goodness-of-fit procedures for Farlie–Gumbel–Morgenstern bivariate copula of order statistics

In this study, we provide the Farlie–Gumbel–Morgenstern bivariate copula of rth and sth order statistics. The main emphasis in this study is on the inference procedure which is based on the maximum pseudo-likelihood estimate for the copula parameter. As for the methodology, goodness-of-fit test statistic for copulas which is based on a Cramér–von Mises functional of the empirical copula process is applied for selecting an appropriate model by bootstrapping. An application of the methodology to simulated data set is also presented.     

A Comparison of Survival Function Estimators in the Koziol–Green Model

In this paper, three competing survival function estimators are compared under the assumptions of the so-called Koziol– Green model, which is a simple model of informative random censoring. It is shown that the model specific estimators of Ebrahimi and Abdushukurov, Cheng, and Lin are asymptotically equivalent. Further, exact expressions for the (noncentral) moments of these estimators are given, and their biases are analytically compared with the bias of the familiar Kaplan–Meier estimator. Finally, MSE comparisons of the three estimators are given for some selected rates of censoring.     

Parametric inference based on judgment post stratified samples

In this paper, we consider a judgment post stratified (JPS) sample of set size $H$ from a location and scale family of distributions. In a JPS sample, ranks of measured units are random variables. By conditioning on these ranks, we derive the maximum likelihood (MLEs) and best linear unbiased estimators (BLUEs) of the location and scale parameters. Since ranks are random variables, by considering the conditional distributions of ranks given the measured observations we construct Rao-Blackwellized version of MLEs and BLUEs. We show that Rao-Blackwellized estimators always have smaller mean squared errors than MLEs and BLUEs in a JPS sample. In addition, the paper provides empirical evidence for the efficiency of the proposed estimators through a series of Monte Carlo simulations.     

Estimating conditional tail expectation with actuarial applications in view

We develop statistical inferential tools for estimating and comparing conditional tail expectation (CTE) functions, which are of considerable interest in actuarial science. In particular, we construct estimators for the CTE functions, develop the necessary asymptotic theory for the estimators, and then use the theory for constructing confidence intervals and bands for the functions. Both parametric and non-parametric approaches are explored. Simulation studies illustrate the performance of estimators in various situations. Results are obtained under minimal assumptions, and the general Vervaat process plays a crucial role in achieving these goals.     

On the enumeration of some D-optimal designs

Two matrices with elements taken from the set {-1,1}$\{-1,1\}$ are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old.     

Generalized partially linear varying-coefficient models

Generalized varying-coefficient models are useful extensions of generalized linear models. They arise naturally when investigating how regression coefficients change over different groups characterized by certain covariates such as age. In this paper, we extend these models to generalized partially linear varying-coefficient models, in which some coefficients are constants and the others are functions of certain covariates. Procedures for estimating the linear and non-parametric parts are developed and their associated statistical properties are studied. The methods proposed are illustrated using some simulations and real data analysis.