Shannon Information Properties of the Endpoints of Record Coverage

This paper addresses the largest and the smallest observations, at the times when a new record of either kind (upper or lower) occurs, which are it called the current upper and lower record, respectively. We examine the entropy properties of these statistics, especially the difference between entropy of upper and lower bounds of record coverage. The results are presented for some common parametric families of distributions. Several upper and lower bounds, in terms of the entropy of parent distribution, for the entropy of current records are obtained. It is shown that mutual information, as well as Kullback–Leibler distance between the endpoints of record coverage, Kullback–Leibler distance between data distribution, and current records, are all distribution-free.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

Properties of Entropies of Record Values in Reliability and Life Testing Context

In this article, the entropies of record value distributions from some continuous probability models are computed and their properties are investigated, both analytically when feasible, and numerically. In an attempt to establish relationships between entropies of the parent and the corresponding record value distributions, the entropies of record value distributions associated with the uniform, exponential, Weibull, classical Pareto, normal, gamma, beta, and Cauchy distributions are considered in this article. The entropy of record value distributions associated with the uniform, exponential, Weibull, and Pareto distributions, have tractable closed forms. The entropies of record value distributions associated with the normal, gamma, beta, and Cauchy distributions do not have tractable closed forms and need further investigations. Some general conclusions are drawn in the final section.     

Distributions and moments of record values in a sequence of maximally dependent random variables

A sequence of possibly dependent random variables is maximally dependent if all the sample maxima in the sequence have stochastically maximal distributions in the class of all distributions with the same marginals. For a sequence of maximally dependent standard uniform random variables, we determine the distribution functions of record times and values. We show that the distribution of the record occurrence times coincides with the respective distribution for the i.i.d. sequence, and the distributions of the record values are stochastically maximal in the class of sequences with the same record times distributions, containing all the exchangeable sequences. We also derive analytic formulae for the moments of record values from the maximally dependent sequence, and compare them with those of the i.i.d. case.     

Entropy properties of record statistics

Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations. They are closely connected with the occurrence times of a corresponding non-homogenous Poisson process and reliability theory. In this paper, the information properties of record values are presented based on Shannon information. Several upper and lower bounds for the entropy of record values are obtained. It is shown that, the mutual information between record values is distribution free and is computable using the distribution of the record values of the sequence from the uniform distribution.     

Estimation of Pr(X < Y) Using Record Values in the One and Two Parameter Exponential Distributions

We consider the problem of estimating the stress-strength reliability when the available data is in the form of record values. The one parameter and two parameters exponential distribution are considered. In the case of two parameters exponential distributions we considered the case where the location parameter is common and the case where the scale parameter is common. The maximum likelihood estimators and the associated confidence intervals are derived.     

Inference and prediction for the burr type x model based on records

Based on record values, the maximum likelihood, minimum variance unbiased and Bayes estimators of the one parameter of the Burr type X distribution are computed and compared. The Bayesian and non-Bayesian confidence intervals for this parameter are also presented. A Bayesian prediction interval for the sth future record is obtained in a closed form. Based on simulated record values, numerical computations and comparisons between the different estimators are given     

Nonparametric confidence and tolerance intervals from record values data

In a number of situations only observations that exceed or only those that fall below the current extreme value are recorded. Examples include meteorology, hydrology, athletic events and mining. Industrial stress testing is also an example in which only items that are weaker than all the observed items are destroyed. In this paper, it is shown that, how record values can be used to provide distribution-free confidence intervals for population quantiles and tolerance intervals. We provide some tables that help one choose the appropriate record values and present a numerical example. Also universal upper bounds for the expectation of the length of the confidence intervals are derived. The results may be of interest in situation where only record values are stored.     

A note on characterization based on shannon entropy of record statistics

In this note we compare the Shannon entropy of record statistics with the Shannon entropy of the original data and give an application to characterization of the generalized Pareto distribution,     

Prediction intervals for the future record values from exponential distribution: comparative study

!n this paper we consider the predicf an problem of the future nth record value based an the first m (m < n) observed record values from one-parameter exponential distribution. We introduce four procedures for obtaining prediction intervals for the nth record value. The performance of the so obtained intervals is assessed through numerical and simulation studies. In these studies, we provide the means and standard errors of lower limits. upper limits and lengths of prediction intervals. Further, we check the validation of these intervals based on some point predictors.     

On a Characterization of Generalized Pareto Distribution Based on Generalized Order Statistics

In recent years, several attempts have been made to characterize the generalized Pareto distribution (GPD) based on the properties of order statistics and record values. In the present article, we give a characterization result on GPD based on the spacing of generalized order statistics.     

Estimation and Prediction for Exponentiated Family of Distributions Based on Records

In this article, a family of distributions, namely the exponentiated family of distributions, is defined and for the unknown parameters, different point estimates are derived based on record statistics. Prediction for future record values is presented from a Bayesian view point. Two numerical examples and a Monte Carlo simulation study are presented to illustrate the results.     

Pitman Closeness of kth Records Based on a Two-Sequence Case

Suppose upper kth records were observed from an X-sequence of iid continuous random variables, and kth upper records from another independent Y-sequence of iid variables from the same distribution are to be observed. The Pitman closeness probabilities of these statistics are derived. For symmetric distribution, the Pitman closeness probabilities of kth record statistics to the population median, are also examined and it is shown that these probabilities are distribution free. Numerical computations are conducted to illustrate the results developed here.     

Evidential inference based on record data and inter-record times

Science looks to statistics for an objective measure of the strength of evidence in a given body of observations. In this paper, we shall use a criterion defined as a combination of probabilities of weak and strong misleading evidence to do the comparison between only record values and the same number of record values and inter-record times. Also, a simulation is presented to illustrate the results.     

On the Influence of Record Terms in the Addition of Independent Random Variables

We discuss some problems connected with the role of record values and maximal values generated by sequences of random variables X₁, X₂,…, X _n in the process of the growth of sums X₁ +···+ X_n, n = 1, 2,….     

Closeness of k-records to Progressive Type-II Censored Order Statistics for Location-scale Families

In this article, the Pitman closeness of upper and lower k-records to progressive Type-II censored order statistics for location-scale families is investigated. In each case, the special properties of the probability of Pitman closeness are obtained and the corresponding monotonicity properties are discussed. Moreover, the closest k-record to a specific progressive Type-II censored data is obtained. Finally, for the standard exponential and standard uniform distributions, explicit expressions for the probability of Pitman closeness are derived. For various censoring schemes, the results of the numerical computations are displayed in tables. Most of the results in Ahmadi and Balakrishnan (2013) Ahmadi, J., Balakrishnan, N. (2013). On the nearness of record values to order statistics from Pitman measure of closeness. Metrika 76:521–541.[Crossref], [Web of Science ®] , [Google Scholar] can be achieved as special cases.     

Estimation of parameters of bivariate normal distribution using concomitants of record values

In this paper, we discuss the concomitants of record values arising from the well-known bivariate normal distribution BVND(μ₁, μ₂,σ₁,σ₂, ρ). We have obtained the best linear unbiased estimators of μ₂ and σ₂ when ρ is known and derived some unbiased linear estimators of ρ when μ₂ and σ₂ are known, based on the concomitants of first n record values. The variances of these estimators have been obtained.     

Relations for m-generalized order statistics via an Opial-type inequality

An Opial-type inequality is applied to obtain relations for expectations of functions of m-generalized order statistics (m-gOSs), their distribution functions, as well as moment-generating functions. Respective inequalities for common order statistics and record values are contained as particular cases.     

The probability of identification: applying ideas from forensic statistics to disclosure risk assessment

Summary.  The paper establishes a correspondence between statistical disclosure control and forensic statistics regarding their common use of the concept of 'probability of identification'. The paper then seeks to investigate what lessons for disclosure control can be learnt from the forensic identification literature. The main lesson that is considered is that disclosure risk assessment cannot, in general, ignore the search method that is employed by an intruder seeking to achieve disclosure. The effects of using several search methods are considered. Through consideration of the plausibility of assumptions and 'worst case' approaches, the paper suggests how the impact of search method can be handled. The paper focuses on foundations of disclosure risk assessment, providing some justification for some modelling assumptions underlying some existing record level measures of disclosure risk. The paper illustrates the effects of using various search methods in a numerical example based on microdata from a sample from the 2001 UK census.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.