Nonparametric regression estimation of conditional tails: the random covariate case

We present families of nonparametric estimators for the conditional tail index of a Pareto-type distribution in the presence of random covariates. These families are constructed from locally weighted sums of power transformations of excesses over a high threshold. The asymptotic properties of the proposed estimators are derived under some assumptions on the conditional response distribution, the weight function and the density function of the covariates. We also introduce bias-corrected versions of the estimators for the conditional tail index, and propose in this context a consistent estimator for the second-order tail parameter. The finite sample performance of some specific examples from our classes of estimators is illustrated with a small simulation experiment.     

A synthesis of stein-rule and mixed regression procedures inlinear regression models under non-normality

Stein-rule philosophy and mixed regression technique are combined to develop two families of improved estimators of regression coefficients in the linear regression model under incomplete prior information. The properties of these estimators are studied when disturbances are small and non-normal. Conditions for their dominance over mixed regression estimator are derived taking risk as the criterion for performance.     

Order preserving property of moment estimators

Balakrishnan and Mi (2001) considered order preserving property of maximum likelihood estimators. In this paper there are given conditions under which the moment estimators have the property of preserving stochastic orders. The preserving property for the usual stochastic order as well as for likelihood ratio one is considered. Sufficient conditions are established for some parametric families of distributions.     

Convergence rates for empirical bayes estimators of parameters in linear regressin models

In this paper, the convergence rates of the EB estimators of the regression coefficients and the error variance in a linear model are obtained. The rates can approximate to O(n¹) arbitrarily. The convergency of the EB estimators of the regression coefiicients and the variance components in a variance component model is also investigated. The investigation makes use of the results concerning the convergence rates of the EB estimators of the parameters in multi-parameter exponential families.     

Componentwise B-spline estimation for varying coefficient models with longitudinal data

A componentwise B-spline method is proposed for estimating the unknown functions in the varying-coefficient models with longitudinal data. Different amounts of smoothing are used for different individual coefficient functions and the estimators of different coefficient functions are obtained by different minimization operations. The local asymptotic bias and variance of the estimators are derived. It is shown that our estimators achieve the local and global optimal convergence rates even if the coefficient functions belong to different smoothness families. The asymptotic distributions of the estimators are also established and are used to construct approximate pointwise confidence intervals for coefficient functions. Finite sample properties of our procedures are studied through Monte Carlo simulations.     

Time series models with asymmetric innovations

We consider AR(q) models in time series with asymmetric innovations represented by two families ofdistributions: (i) gamma with support IR : (0, ∞), and (ii) generalized logistic with support IR:(-∞,∞). Since the ML (maximum likelihood) estimators are intractable, we derive the MML (modified maximum likelihood) estimators of the parameters and show that they are remarkably efficient besides being easy to compute. We investigate the efficiency properties of the classical LS (least squares) estimators. Their efficiencies relative to the proposed MML estimators are very low.     

Estimation of quantiles of exponential and double exponential distributions based on two order statistics

Asymptotically best linear unbiased estimators (ABLUE) of quantiles, x^., in the two-parameter (location-scale) exponential and double exponential families are obtained as linear combinations of two suitably chosen order statistics. Exact formulae for the linear combinations are given as functions of £. The derived estimators in both cases compare favorably with the usual nonparametric estimator. Also, in the exponential case the derived estimator compares favorably with the Sarhan-Greenberg BLUE based on a complete sample     

Une condition nécessaire d'admissibilité et ses conséquences sur les estimateurs à rétrécisseur de la moyenne d'un vecteur normal

Considering exponential families of distributions, we estimate parameters which are not the natural parameters. We prove that the admissible estimators of these parameters are limits of Bayes estimators and can be expressed through a given functional form. An important particular case of this model pertains to the estimation of the mean of a multidimensional normal distribution when the variance is known up to a multiplicative factor. We deduce from the main result a necessry condition for the admissibility of matricial shrinkage estimators.     

Linear Estimators and Predictors Based on Generalized Order Statistics from Generalized Pareto Distributions

Linear estimation and prediction based on several samples of generalized order statistics from generalized Pareto distributions is considered. Representations of best linear unbiased estimators (BLUEs) and best linear equivariant estimators in location-scale families are derived, as well as corresponding optimal linear predictors. Moreover, we study positivity of the linear estimators of the scale parameter. An example illustrates that the BLUE may attain negative values with positive probability in certain situations.     

Model-based study of families of exponential-type estimators in presence of non response

Among other types of non sampling errors, non response error (NRE) is an inherent component of any sample survey, which is supposed to be given much attention during the designing and execution stages. With increasing awareness of these estimators, therefore, there is an urge for the development of suitable techniques for controlling them.

This article proposes two families of estimators for population mean in the presence of non response and discuses various properties under model approach, namely polynomial regression model. The families include some existing estimators. Comparison of efficiencies along with the robustness of the estimators under misspecification of models has been empirically discussed.     

Ratio and product estimators in stratified random sampling

Khoshnevisan et al. [2007. A general family of estimators for estimating population mean using known value of some population parameter(s). Far East Journal of Theoretical Statistics 22, 181–191] have introduced a family of estimators using auxiliary information in simple random sampling. They have showed that these estimators are more efficient than the classical ratio estimator and that the minimum value of the mean square error (MSE) of this family is equal to the value of MSE of regression estimator. In this article, we adapt the estimators in this family to the stratified random sampling and motivated by the estimator in Searls [1964. Utilization of known coefficient of kurtosis in the estimation procedure of variance. Journal of the American Statistical Association 59, 1225–1226], we also propose a new family of estimators for the stratified random sampling. The expressions of bias and MSE of the adapted and proposed families are derived in a general form. Besides, considering the minimum cases of these MSE equations, the efficient conditions between the adapted and proposed families are obtained. Moreover, these theoretical findings are supported by a numerical example with original data.     

On Existence of Maximum Likelihood Estimators in Exponential Families

We propose a simple necessary and sufficient condition for existence of maximum likelihood estimators in a large class of canonical exponential families. We give an application to log-spline families.     

Some recurrence relation-based estimators of the parameters of a generalized log-series distribution

Two families of closed form estimators are proposed for estimating the single parameter of the log-series distribution(LSD)and for estimating the two parameters of a generalization of the LSD distribution(GLSD)presented by Tripathi and Gupta(1985). These families are based on the recurrence relations obtained from these distributions, are of closed form, and have very high asymptotic relative effi¬ciencies. Some two-stage procedures are suggested.     

Using pseudometrics in kernel density estimation

Common kernel density estimators (KDE) are generalised, which involve that assumptions on the kernel of the distribution can be given. Instead of using metrics as input to the kernels, the new estimators use parameterisable pseudometrics. In general, the volumes of the balls in pseudometric spaces are dependent on both the radius and the location of the centre. To enable constant smoothing, the volumes of the balls need to be calculated and analytical expressions are preferred for computational reasons. Two suitable parametric families of pseudometrics are identified. One of them has common KDE as special cases. In a few experiments, the proposed estimators show increased statistical power when proper assumptions are made. As a consequence, this paper describes an approach, where partial knowledge about the distribution can be used effectively. Furthermore, it is suggested that the new estimators are adequate for statistical learning algorithms such as regression and classification.     

SEPARATE VERSUS SYSTEM METHODS OF STEIN-RULE ESTIMATION IN SEEMINGLY UNRELATED REGRESSION MODELS

ABSTRACT

Despite the sizeable literature associated with the seemingly unrelated regression models, not much is known about the use of Stein-rule estimators in these models. This gap is remedied in this paper, in which two families of Stein-rule estimators in seemingly unrelated regression equations are presented and their large sample asymptotic properties explored and evaluated. One family of estimators uses a shrinkage factor obtained solely from the equation under study while the other has a shrinkage factor based on all the equations of the model. Using a quadratic loss measure and Monte-Carlo sampling experiments, the finite sample risk performance of these estimators is also evaluated and compared with the traditional feasible generalized least squares estimator.     

Set Compound Estimation Under Entropy Loss

Set compound estimation has been studied for nearly half a century. This paper explores, for the first time, set compound estimation under entropy (Kullback-Leibler information) loss for a k -dimensional standard exponential family with a compact parameter space. It makes detailed investigation of entropy loss with the exponential family and related proper-ties. Asymptotically optimal set compound estimators with rates O ( n ^−1/2) under this loss are established for some discrete exponential families by using power series, representing the Bayes estimators in terms of a mixture density and applying the Singh-Datta Lemma. Poisson, negative binomial families and a two-dimensional model serve as examples.     

Estimating the Expectation of the Log-Likelihood with Censored Data for Estimator Selection

A criterion for choosing an estimator in a family of semi-parametric estimators from incomplete data is proposed. This criterion is the expected observed log-likelihood (ELL). Adapted versions of this criterion in case of censored data and in presence of explanatory variables are exhibited. We show that likelihood cross-validation (LCV) is an estimator of ELL and we exhibit three bootstrap estimators. A simulation study considering both families of kernel and penalized likelihood estimators of the hazard function (indexed on a smoothing parameter) demonstrates good results of LCV and a bootstrap estimator called ELL_bboot . We apply the ELL_bboot criterion to compare the kernel and penalized likelihood estimators to estimate the risk of developing dementia for women using data from a large cohort study.     

Empirical Bayes estimation in continuous one-parameter exponential families under associated samples

In this paper, we study the empirical Bayes (EB) estimation in continuous one-parameter exponential families under negatively associated (NA) samples and positively associated (PA) samples. Under certain regularity conditions, it is shown that the convergence rates of proposed EB estimators under NA or PA samples are the same as those of EB estimators under independent observations, which significantly improve the existing results in EB estimation under associated samples.     

Estimation of population mean based on dual use of auxiliary information in non response

Whenever there is auxiliary information available in any form, the researchers want to utilize it in the method of estimation to obtain the most efficient estimator. When there exists enough amount of correlation between the study and the auxiliary variables, and parallel to these associations, the ranks of the auxiliary variables are also correlated with the study variable, which can be used a valuable device for enhancing the precision of an estimator accordingly. This article addresses the problem of estimating the finite population mean that utilizes the complementary information in the presence of (i) the auxiliary variable and (ii) the ranks of the auxiliary variable for non response. We suggest an improved estimator for estimating the finite population mean using the auxiliary information in the presence of non response. Expressions for bias and mean squared error of considered estimators are derived up to the first order of approximation. The performance of estimators is compared theoretically and numerically. A numerical study is carried out to evaluate the performances of estimators. It is observed that the proposed estimator is more efficient than the usual sample mean and the regression estimators, and some other families of ratio and exponential type of estimators.     

NONNORMAL REGRESSION. I. SKEW DISTRIBUTIONS

In a linear regression model of the type y= θ X+e, it is often assumed that the random error eis normally distributed. In numerous situations, e.g., when ymeasures life times or reaction times, etypically has a skew distribution. We consider two important families of skew distributions, (a) Weibull with support IR: (0, ∞) on the real line, and (b) generalised logistic with support IR: (?∞, ∞). Since the maximum likelihood estimators are intractable in these situations, we derive modified likelihood estimators which have explicit algebraic forms and are, therefore, easy to compute. We show that these estimators are remarkably efficient, and robust. We develop hypothesis testing procedures and give a real life example. Symmetric families of distributions, both long and short tailed, will be considered in a future paper.