Linear Functions Of Order Statistics Research Articles

A Conversation with Stephen M. Stigler

Stephen M. Stigler received his Ph.D. in Statistics from the University of California, Berkeley, with a dissertation on the asymptotic distribution of linear functions of order statistics. Starting in 1967, he taught at the University of Wisconsin, Madison, then in 1979 moved to the University of Chicago where he taught from 1979 to 2021. Stigler has worked on a variety of topics in mathematical statistics, ranging from asymptotic theory to the theory of experimental design, and on applications of statistics including in anthropology, forensic science, paleontology, psychology, information transfer and sports. In recent years, he has concentrated on the history of statistics, with inquiries ranging from the development of statistical methods in astronomy and geodesy and their spread to biological and social sciences, to lotteries, to the modern development of statistical theory. He has published four books, The History of Statistics (1986), Statistics on the Table (1999), The Seven Pillars of Statistical Wisdom (2016) and Casanova’s Lottery (2022). A recent research focus has been upon the way the work of Francis Galton on the statistics of inheritance led to the creation of modern multivariate analysis and made a true Bayesian inference possible, and on how R. A. Fisher’s transformation of Karl Pearson’s path breaking research led to a modern period of statistical enlightenment. Stigler is an elected member of the American Academy of Arts and Sciences and of the American Philosophical Society; he has served as President of the Institute of Mathematical Statistics and of the International Statistical Institute. In 2005, he received the Humboldt Foundation Research Award; in 2010, he was elected Membre Associé of the Académie royale de Belgique, Classe des Sciences. Stigler served as Theory and Methods Editor for the Journal of the American Statistical Association 1979–1982. He was a Guggenheim Fellow in 1977, and received awards for undergraduate teaching at the University of Wisconsin (1971) and University of Chicago (1998). This interview with Stigler was conducted remotely in July 2021.

Muth Distribution and Estimation of a Parameter Using Order Statistics

In this work, we have considered a lifetime distribution namely Muth distribution and pointed out instances where it appears as a good model to study the stochastic nature of the variable under consideration. We have derived the best linear unbiased estimator (BLUE) of the scale parameter of the Muth distribution based on order statistics for some known values of the shape parameter.We have further estimated the scale parameter of Muth distribution by U-statistics based on best linear functions of order statistics as kernels. The efficiency of the BLUE relative to the usual unbiased estimator has been also evaluated. An illustration describing the performance of U-statistics estimation method when compared with the classical maximum likelihood method is also given.

Inference on Exponentiated Power Lindley Distribution Based on Order Statistics with Application

Exponentiated power Lindley distribution is proposed as a generalization of some widely well-known distributions such as Lindley, power Lindley, and generalized Lindley distributions. In this paper, the exact explicit expressions for moments of order statistics from the exponentiated power Lindley distribution are derived. By using these relations, the best linear unbiased estimates of the location and scale parameters, based on type-II right-censored sample, are obtained. Next, the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics are calculated and then used to derive the approximate confidence interval for the location and scale parameters using the Edgeworth approximation. Finally, some numerical illustrations and two real data applications are presented.

On the Strong Law of Large Numbers for Linear Combinations of Concomitants

A theorem on the strong law of large numbers for linear functions of concomitants (induced order statistics) for sequences of independent identically distributed two-dimensional random vectors is proved in this paper. The result complements previous work by S.S. Yang (1981) and N. Gribkova and R. Zitikis (2017, 2019). The proof is based on the conditional independence property of the concomitants established by P. K. Bhattacharya (1974); the van Zwet strong law of large numbers for linear functions of order statistics (1980) is used and classical inequalities apply, including the Rosenthal inequality (1970).

BLUEs and Reliability Analysis for General Censored Data Subject to Inverse Gaussian Distribution

Working on product lifetime data is of significant importance for evaluating safety and reliability, predicting remaining useful life and formulating maintenance strategy or replacement policy. In practical applications, observed datasets often consist of failure data and randomly censored data, which are referred as general censored data. Meanwhile, inverse Gaussian (IG) distribution has been widely adopted to depict lifetime data because it not only can possess flexible expression formats but also can explain the mechanism of first hitting time from a soft failure viewpoint. Motivated by these two regards, this paper develops a novel method on best linear unbiased estimations (BLUEs) for general censored data. A three-parameter IG distribution type is adopted. A novel method is established to optimize the skewness parameter. Then, BLUEs of mean and standard deviation can be obtained. The proposed method can construct closed-form parameter estimations in linear functions of order statistics. The computation process has been further simplified for flexible applications. The frequently utilized maximum likelihood estimation method is also introduced as a reference for a better understanding. Comparative results of both comprehensive simulation study and empirical application illustrate that the proposed method can significantly enhance the estimation accuracy and keep a stable performance, because more life information can be extracted and adopted from the censored datasets.

Higher order moments of order statistics from the Lindley distribution and associated inference

ABSTRACTThe Lindley distribution is an important distribution for analysing the stress–strength reliability models and lifetime data. In many ways, the Lindley distribution is a better model than that based on the exponential distribution. Order statistics arise naturally in many of such applications. In this paper, we derive the exact explicit expressions for the single, double (product), triple and quadruple moments of order statistics from the Lindley distribution. Then, we use these moments to obtain the best linear unbiased estimates (BLUEs) of the location and scale parameters based on Type-II right-censored samples. Next, we use these results to determine the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics to develop Edgeworth approximate confidence intervals of the location and scale Lindley parameters. In addition, we carry out some numerical illustrations through Monte Carlo simulations to show the usefulness of the findings. Finally, we apply the findings of the paper to some real data set.

On the Optimum Design of L-Estimators for Phase Offset Estimation in IEEE 1588

In packet-based time synchronization protocols such as IEEE 1588, clock phase offsets are determined via two-way message exchanges between a master and a slave. Since the end-to-end delays in packet networks are inherently stochastic in nature, the recovery of phase offsets from message exchanges must be treated as a statistical estimation problem. Recently, minimax estimators for this problem were proposed by the authors, which are optimum in terms of minimizing the mean-squared estimation error over all values of the unknown parameters. In this paper, we consider a restricted class of estimators referred to as L-estimators, which are linear functions of order statistics. The problem of designing optimum L-estimators is studied under several hitherto unconsidered criteria of optimality. Our results address the case where the queuing delay distributions are fully known, as well as the case where network model uncertainty exists. Optimum L-estimators that utilize information from past observation windows to improve performance are also described. The derived L-estimators have a much lower computational complexity than minimax estimators, and also require lesser statistical knowledge of the queuing delays. Simulation results indicate that L-estimators exhibit a mean-squared estimation error very close to minimax estimators under many network scenarios.

A New Exponential GOF Test for Data Subject to Multiply Type II Censoring

This article presents a new goodness-of-fit (GOF) test statistic for multiply Type II censored Exponential data. The new test also applies to ordinary Type II censored samples and complete samples, since those cases are special cases of multiply Type II censoring. This test statistic is based on a ratio of linear functions of order statistics. Empirical power studies confirm that this ratio test compares favorably to currently available GOF tests for ordinary Type II censored data. Three data analysis examples are provided that demonstrate the usefulness of this new test statistic.

Bayesian analysis of an exponentiated half-logistic distribution under progressively type-II censoring

Abstract This paper develops maximum likelihood estimators (MLEs) of unknown param-eters in an exponentiated half-logistic distribution based on a progressively type-IIcensored sample. We obtain approximate con dence intervals for the MLEs by usingasymptotic variance and covariance matrices. Using importance sampling, we obtainBayes estimators and corresponding credible intervals with the highest posterior densityand Bayes predictive intervals for unknown parameters based on progressively type-IIcensored data from an exponentiated half logistic distribution. For illustration pur-poses, we examine the validity of the proposed estimation method by using real andsimulated data.Keywords: Bayes predictive interval, exponentiated half-logistic distribution, HPD cred-ible interval, importance sampling, progressively type-II censored sample. 1. Introduction Many reliability and survival analysis studies have used half-logistic distributions, partic-ularly for censored data. Several studies have drawn inferences about half logistic distribu-tions. Balakrishnan and Puthenpura (1986) introduce the best linear unbiased estimator oflocation and scale parameters of half-logistic distributions by considering linear functionsof order statistics. Balakrishnan and Wong (1991) obtain approximate maximum likelihoodestimators (AMLEs) of location and scale parameters of half-logistic distributions by usinga type-II right-censored sample. Kang et al. (2008) derive AMLEs and MLEs of scale pa-rameters of half-logistic distributions by using progressively type-II censored samples. Kanget al. (2009) propose AMLEs of scale parameters of half-logistic distributions by consideringdouble-hybrid censored samples. Kang and Seo (2011) recently examine exponentiated half-logistic distributions and propose two types of AMLEs of scale parameters and reliabilityfunctions by using progressively type-II right-censored samples. Kim et al. (2011a) deriveBayesian estimators of shape parameters, reliability functions, and failure rate functions

Estimation of the Scale Parameter of Skew-Normal Distribution Using U-Statistics Based on Order Statistics

Abtsrcat In this work, we obtain the best linear unbiased estimator based on order statistics for the scale parameter of skew-normal distribution for some known values of the shape parameter. We also propose a technique for estimating the scale parameter of skew-normal distribution by U-statistics based on best linear functions of order statistics as kernels. The efficiency comparison of the proposed U-statistics with the maximum likelihood esti- mator is also carried out.

On Uncorrelated Linear Functions of Order Statistics of Independent But not Identically Distributed Random Variables

Abtsrcat In this paper the correlation between the pooled sample mean and certain linear functions of order statistics of independent non-identically distributed(inid) random variables are obtained for pooled sample from pop- ulations with common location parameter μ and common scale parameter σ and the results if the parent distributions are symmetric about the common location parameter μ are derived therefrom.

Some Remarks about Robust Estimation of the Scale Parameter in Weighted Models

Using Zieliński's (1977) concept of robustness, Błażej (2007) obtained the uniformly most bias-robust estimates (UMBREs) of the scale parameter for some statistical models, in a class of linear functions of order statistics. Violations of the models are generated by weight functions. In this article the UMBRE of the scale parameter, based on order statistics, a more general weighted model is derived. Extension of a result of Błażej (2007) is given.

Distributions of Ratios of Two Correlated Skew-Normal Variables and of Ratios of Two Linear Functions of Order Statistics from Bivariate Normal Distribution

In this article, we first derive the distribution of the ratio of two correlated skew-normal variables. Then, with , where |ρ|< 1, and (X 1:2, X 2:2) T denoting the corresponding order statistics, we consider two linear functions of these order statistics, viz., a 1 X 1:2 + a 2 X 2:2 and b 1 X 1:2 + b 2 X 2:2, and derive explicit expressions for the cumulative distribution function (cdf) and probability density function (pdf) of the ratios and .

Simple and Accurate Approximations for Computing Covariance Matrices of Gamma and Weibull Order Statistics

In this article, we propose a simple technique that can be used to calculate accurate approximations for the covariances of Weibull and/or Gamma order statistics, given knowledge of the corresponding means and variances of these order statistics. Ratio formulas for relating the covariances and variances of the order statistics for these two probability distribution functions (pdf's) are intuitively derived from a first-order Mosteller approximation, and then refined using the Blom adjustment technique. The necessary Blom coefficients are in turn found by optimizing the well-known recursive formula for mixed moments using standard nonlinear estimation techniques. Analytical and simulation results confirm that very good approximations to the covariance matrices of Gamma and Weibull order statistics can be obtained for a wide range of shape parameters using this methodology. Four examples are provided which demonstrate how this approach can be used to derive accurate o-BLU weights, precise variance approximations for linear functions of order statistics, and precise power calculations for the Gini test statistic.

A note on skew-elliptical distributions and linear functions of order statistics

Linear functions of order statistics from bivariate, exchangeable, continuous and elliptical random vectors have a skew-elliptical distribution. This result generalizes previous ones on order statistics and skewed distributions.

Estimation of the Parameters of Type-I Generalized Logistic Distribution Using Order Statistics

In this work, we propose a technique of estimating the location parameter μ and scale parameter σ of Type-I generalized logistic distribution by U-statistics constructed by using best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.

Estimation of location and scale parameters of a distribution by U-statistics based on best linear functions of order statistics

In this work we propose a technique of estimating the location parameter μ and scale parameter σ of a distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The method has been illustrated for estimating the location and scale parameters of type-I extreme value distribution. We have computed the asymptotic relative efficiencies of the proposed U-statistics with the appropriate maximum likelihood estimators based on samples drawn from each of type-I extreme value, logistic and normal distributions. In all cases very high asymptotic relative efficiencies are obtained.

Moments of Lorenz Curve and linear measures of inequality

The relationship of moments of Lorenz Curve with family of inequality measures {Ik,k 2} is established. Two families of inequality measures, viz., {Ik,k 2} and {Jk,k 1}, are considered as particular cases of general class of linear measures of income inequality with suitable score functions. Each score function defines a particular linear inequality measure and the basic properties of such measures, using some restrictions on score functions are explored. These linear measures of income inequality can also be expressed as linear functions of order statistics. Asymptotic distribution of some of these measures is derived. Tests of significance for one sample and two samples based on these inequality indices are suggested. The simulation work is carried out to find the power of these tests.

Estimation of the parameters of log-gamma distribution using order statistics

In this work we propose a technique of estimating the location parameter μ and scale parameter σ of log-gamma distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.

Inference Based on Order Statistics from the Generalized Pareto Distribution and Application

ABSTRACT In this article, we derive exact explicit expressions for the single, double, triple, and quadruple moments of order statistics from the generalized Pareto distribution (GPD). Also, we obtain the best linear unbiased estimates of the location and scale parameters (BLUE's) of the GPD. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the generalized Pareto parameters using Edgeworth approximation and compare them with those based on Monte Carlo simulations. To show the usefulness of our results, we also present a numerical example. Finally, we give an application to real data.