Unbiased Estimation of the Distribution Function of a Two-Parameter Exponential Population Using Order Statistics

In this article, we consider the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population. We give necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics and suggest unbiased estimators in some situations where unbiased estimators exist. A few properties of the suggested estimators for some special cases have also been discussed.     

Standard and goodness-of-fit parameter estimation methods for the three-parameter lognormal distribution

A class of goodness-of-fit estimators is found to provide a useful alternative in certain situations to the standard maximum likelihood method which has some undesirable estimation characteristics for estimation from the three-parameter lognormal distribution. The class of goodness-of-fit tests considered include the Shapiro-Wilk and Filliben tests which reduce to a weighted linear combination of the order statistics that can be maximized in estimation problems. The weighted order statistic estimators are compared to the standard procedures in Monte Carlo simulations. Robustness of the procedures are examined and example data sets analyzed.     

Estimating pareto quantiles using two order statistics

Asymptotically best linear unbiased estimators of the population quantiles for the location-scale Pareto distribution with fixed shape parameter are obtained using two suitably chosen order statistics. Formulae for the appropriate order statistics, coefficients, variances, and asymptotic relative efficiencies (relative to the usual non-parametric estimator for quantiles) are given     

New estimators based on order statistics in some families of scale distributions

In this study some new unbiased estimators based on order statistics are proposed for the scale parameter in some family of scale distributions. These new estimators are suitable for the cases of complete (uncensored) and symmetric doubly Type-II censored samples. Further, they can be adapted to Type II right or Type II left censored samples. In addition, unbiased standard deviation estimators of the proposed estimators are also given. Moreover, unlike BLU estimators based on order statistics, expectation and variance-covariance of relevant order statistics are not required in computing these new estimators.

Simulation studies are conducted to compare performances of the new estimators with their counterpart BLU estimators for small sample sizes. The simulation results show that most of the proposed estimators in general perform almost as good as the counterpart BLU estimators; even some of them are better than BLU in some cases. Furthermore, a real data set is used to illustrate the new estimators and the results obtained parallel with those of BLUE methods.     

The global minimum variance unbiased estimator of the parameter for a truncated parameter family under the optimal ranked set sampling

The minimum variance unbiased estimators (MVUEs) of the parameters for various distributions are extensively studied under ranked set sampling (RSS). However, the results in existing literatures are only locally MVUEs, i.e. the MVUE in a class of some unbiased estimators is obtained. In this paper, the global MVUE of the parameter in a truncated parameter family is obtained, that is to say, it is the MVUE in the class of all unbiased estimators. Firstly we find the optimal RSS according to the character of a truncated parameter family, i.e. arrange RSS based on complete and sufficient statistics of independent and identically distributed samples. Then under this RSS, the global MVUE of the parameter in a truncated parameter family is found. Numerical simulations for some usual distributions in this family fully support the result from the above two-step optimizations. A real data set is used for illustration.     

Generalized Kernel Estimators for the Weibull-Tail Coefficient

We introduce new families of estimators for the Weibull-tail coefficient, obtained from a weighted sum of a power transformation of excesses over a high random threshold. Asymptotic normality of the estimators is proven for an intermediate sequence of upper order statistics, and under classical regularity conditions for L-statistics and a second-order condition on the tail behavior of the underlying distribution. The small sample performance of two specific examples of kernel functions is evaluated in a simulation study.     

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     

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.     

Kernel estimators for the second order parameter in extreme value statistics

We develop and study in the framework of Pareto-type distributions a general class of kernel estimators for the second order parameter ρ$\rho$, a parameter related to the rate of convergence of a sequence of linearly normalized maximum values towards its limit. Inspired by the kernel goodness-of-fit statistics introduced in Goegebeur et al. (2008), for which the mean of the normal limiting distribution is a function of ρ$\rho$, we construct estimators for ρ$\rho$ using ratios of ratios of differences of such goodness-of-fit statistics, involving different kernel functions as well as power transformations. The consistency of this class of ρ$\rho$ estimators is established under some mild regularity conditions on the kernel function, a second order condition on the tail function 1−F of the underlying model, and for suitably chosen intermediate order statistics. Asymptotic normality is achieved under a further condition on the tail function, the so-called third order condition. Two specific examples of kernel statistics are studied in greater depth, and their asymptotic behavior illustrated numerically. The finite sample properties are examined by means of a simulation study.     

Generalized order statistics from two parameter uniform distribution

In this paper some distributional properties of the generalized order statistics from uniform distribution are given. The minimum variance linear unbiased as well best ( in the sense of minimum mean squared error) invariant estimators of the parameters of the two parameter uniform distribution based on the first m generalized order statistics are presented.     

Linear hypothesis testing for weighted functional data with applications

In socioeconomic areas, functional observations may be collected with weights, called weighted functional data. In this paper, we deal with a general linear hypothesis testing (GLHT) problem in the framework of functional analysis of variance with weighted functional data. With weights taken into account, we obtain unbiased and consistent estimators of the group mean and covariance functions. For the GLHT problem, we obtain a pointwise F-test statistic and build two global tests, respectively, via integrating the pointwise F-test statistic or taking its supremum over an interval of interest. The asymptotic distributions of test statistics under the null and some local alternatives are derived. Methods for approximating their null distributions are discussed. An application of the proposed methods to density function data is also presented. Intensive simulation studies and two real data examples show that the proposed tests outperform the existing competitors substantially in terms of size control and power.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

History and role of order statistics

This paper starts with an itroduction which gives definitions of order statistics,makes a clear distinction between order statistics and rank statistics, and gives an overview of the problems in which methods based on order statstics are useful.The following sections deal with the history and role of order sta¬tistics in selected areas: central tendency, dispersion, and regression; treatment of outliers and robust estimation; maximum likelihood estimators; best linear unbiased estimators; extreme-values; multiple comparisons and studentized range. The last two sections discuss a basic coverage property and some of its conse¬quences and describe the author's chronological annotated bibliography of order statistics. Finally, there is a list of references.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

Bias reduction in the estimation of a shape second-order parameter of a heavy-tailed model

In extreme value theory, the shape second-order parameter is a quite relevant parameter related to the speed of convergence of maximum values, linearly normalized, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter in statistics of extremes. In this article, we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right-tailed models. These estimators, based on the largest order statistics, depend on a real tuning parameter, which makes them highly flexible and possibly unbiased for several underlying models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. The performance of the methodology for the adaptive choice of parameters is evaluated through a Monte Carlo simulation study.     

Second‐order analysis of marked inhomogeneous spatiotemporal point processes: Applications to earthquake data

To analyze interactions in marked spatiotemporal point processes (MSTPPs), we introduce marked second‐order reduced moment measures and K‐functions for inhomogeneous second‐order intensity‐reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark‐based classifications of points, depend on the specific mark space and mark reference measure chosen. Unbiased and consistent minus‐sampling estimators are derived for all statistics considered, and a test for random labeling is indicated. In addition, we treat Voronoi intensity estimators for MSTPPs. These new statistics are finally employed to analyze an Andaman Sea earthquake data set.     

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.     

Estimating variance components by using survey data

Summary. Inflation-type weighted estimators for variance components can be badly biased. Modified weighted estimators suggested in the literature are also badly biased for certain sampling designs. We propose new estimators for variance components, some of which are approximately unbiased regardless of the sampling design. These estimators require knowledge of the joint inclusion probabilities of the observations. The small sample properties of the estimators are studied via simulation for the simple one-way random-effects model. An application is given by using data from the US Hispanic Health and Nutrition Examination Survey.     

On concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and its application in estimation

In this paper, we consider concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and develop its distribution theory. Using ranked set sample obtained from the above distribution, unbiased estimators of the parameters associated with the study variate involved in it are generated. The best linear unbiased estimators (BLUEs) based on observations in the ranked set sample of those parameters as well have been derived. The efficiencies of the BLUEs relative to the respective unbiased estimators generated also have been evaluated.