Pareto analysis based on records

Estimation of the parameters of an exponential distribution based on record data has been treated by Samaniego and Whitaker [On estimating population characteristics from record-breaking observations, I. Parametric results, Naval Res. Logist. Q. 33 (1986), pp. 531–543] and Doostparast [A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. Recently, Doostparast and Balakrishnan [Optimal record-based statistical procedures for the two-parameter exponential distribution, J. Statist. Comput. Simul. 81(12) (2011), pp. 2003–2019] obtained optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning location and scale parameters based on record data from a two-parameter exponential model. In this paper, we derive optimal statistical procedures including point and interval estimation as well as most powerful tests based on record data from a two-parameter Pareto model. For illustrative purpose, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

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.     

Estimation with the lognormal distribution on the basis of records

Record scheme is a method to reduce the total time on test of an experiment. In this scheme, items are sequentially observed and only values smaller than all previous ones are recorded. In some situations, when the experiments are time-consuming and sometimes the items are lost during the experiment, the record scheme dominates the usual random sample scheme [M. Doostparast and N. Balakrishnan, Optimal sample size for record data and associated cost analysis for exponential distribution, J. Statist. Comput. Simul. 80(12) (2010), pp. 1389–1401]. Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [On estimating population characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Q. 33 (1986), pp. 531–543] and Doostparast [A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. The lognormal distribution is used in a wide range of applications when the multiplicative scale is appropriate and the log-transformation removes the skew and brings about symmetry of the data distribution [N.T. Longford, Inference with the lognormal distribution, J. Statist. Plann. Inference 139 (2009), pp. 2329–2340]. In this paper, point estimates as well as confidence intervals for the unknown parameters are obtained. This will also be addressed by the Bayesian point of view. To carry out the performance of the estimators obtained, a simulation study is conducted. For illustration proposes, a real data set, due to Lawless [Statistical Models and Methods for Lifetime Data, 2nd ed., John Wiley & Sons, New York, 2003], is analysed using the procedures obtained.     

Concomitant record ranked set sampling

In this work, we define a new method of ranked set sampling (RSS) which is suitable when the characteristic (variable) Y of primary interest on the units is jointly distributed with an auxiliary characteristic X on which one can take its measurement on any number of units, so that units having record values on X alone are ranked and retained for making measurement on Y. We name this RSS as concomitant record ranked set sampling (CRRSS). We propose estimators of the parameters associated with the variable Y of primary interest based on observations of the proposed CRRSS which are applicable to a very large class of distributions viz. Morgenstern family of distributions. We illustrate the application of CRRSS and our estimation technique of parameters, when the basic distribution is Morgenstern-type bivariate logistic distribution. A primary data collected by CRRSS method is demonstrated and the obtained data used to illustrate the results developed in this work.     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

On estimation of the exponentiated Pareto distribution under different sample schemes

Bayes and classical estimators have been obtained for a two-parameter exponentiated Pareto distribution for when samples are available from complete, type I and type II censoring schemes. Bayes estimators have been developed under a squared error loss function as well as under a LINEX loss function using priors of non-informative type for the parameters. It has been seen that the estimators obtained are not available in nice closed forms, although they can be easily evaluated for a given sample by using suitable numerical methods. The performances of the proposed estimators have been compared on the basis of their simulated risks obtained under squared error as well as under LINEX loss functions.     

Comparisons of several Pareto distributions based on record values

In order to avoid wrong conclusions in any further analysis, it is of importance to conduct a formal comparison for characteristic quantities of the distributions. These characteristic quantities we are familiar with include mean, quantity and reliability function, and so on. In this paper, we consider two tests aiming at the comparisons for function of parameters in Pareto distribution based on record values. They are generalized p-value-based test and parametric bootstrap-based test, respectively. The resulting procedures are easy to compute and are applicable to small samples. A simulation study is conducted to investigate and compare the performance of the proposed tests. A phenomenon we note is that generalized p-value-based test almost uniformly outperforms the parametric bootstrap-based test.     

Projection Mean–Variance Bounds on Expectations of kth Record Values from Restricted Families

We present sharp mean–variance bounds for expectations of kth record values based on distributions coming from restricted families of distributions. These families are defined in terms of convex or star ordering with respect to generalized Pareto distribution. The bounds for expectations of kth record values from DD, DFR, DDA, and DFRA families are special cases of our results. The bounds are derived by application of the projection method.     

Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times

ABSTRACT

The maximum likelihood and Bayesian approaches for estimating the parameters and the prediction of future record values for the Kumaraswamy distribution has been considered when the lower record values along with the number of observations following the record values (inter-record-times) have been observed. The Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. The Bayes and the maximum likelihood estimates are compared by using the estimated risk through Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record values arising from the Kumaraswamy distribution based on record values with their corresponding inter-record times and only record values. The comparison of the derived predictors are carried out by using Monte Carlo simulations. Real data are analysed for an illustration of the findings.     

Bayesian prediction of order statistics with fixed and random sample sizes based on k-record values from Pareto distribution

In this paper, the two-parameter Pareto distribution is considered and the problem of prediction of order statistics from a future sample and that of its geometric mean are discussed. The Bayesian approach is applied to construct predictors based on observed k-record values for the cases when the future sample size is fixed and when it is random. Several Bayesian prediction intervals are derived. Finally, the results of a simulation study and a numerical example are presented for illustrating all the inferential procedures developed here.     

Structural inference on the parameters of the pareto distribution from complete and censored life test data

Recently, the two-parameter Pareto distribution has been recognized as a useful model for survival populations associated with life test experiments. In this paper we apply the structural approach to derive the structural densities of the parameters, from considerations of the group structure of the Pareto density. The structural densities, based on complete and censored samples, are plotted and the corresponding shortest confidence intervals of the parameters are obtained. Numerical examples are given to illustrate our results.     

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO,GENERALIZED PARETO,AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.     

Comparing the Fisher information in record data and random observations

We compare the Fisher information (FI) contained in the firstn record values and record times with the FI inn i. i. d. observations. General results are established for exponential family and Weibull type setups, and a summary table is provided listing several common distributions. We show that the FI in record data improves notably once the record times are included, often changing from being less to being equal or greater than the FI in a random sample of the same size. The behavior in the Weibull case is surprising. There it depends onn, whether the record or the i.i. d. observations have more FI. We propose new estimators based on record data. The results may be of interest in some life testing situations. Supported in part by Fondo Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grant # 1010222 of Chile.     

Optimum constant-stress life test plans for Pareto distribution under type-I censoring

The paper considers the case of constant-stress partially accelerated life testing (CSPALT) when two stress levels are involved under type-I censoring. The lifetimes of test items are assumed to follow a two-parameter Pareto lifetime distribution. Maximum-likelihood method is used to estimate the parameters of CSPALT model. Confidence intervals for the model parameters are constructed. Optimum CSPALT plans that determine the best choice of the proportion of test units allocated to each stress are developed. Such optimum test plans minimize the generalized asymptotic variance of the maximum-likelihood estimators of the model parameters. For illustration, Monte Carlo simulation studies are presented.     

小样本下贝叶斯参数估计法对操作风险的度量

操作风险损失事件的数据一般较为匮乏,这会影响到模型参数估计的准确性,进而导致经济资本配置的偏差和风险控制能力的降低。在损失分布法的框架下,运用基于MCMC模拟的贝叶斯方法,借助WinBUGS软件包通过Gibbs抽样构造出负二项分布和帕累托分布的稳态马尔可夫链,以分别动态模拟操作风险损失频率和强度的后验分布,计算出操作风险所要求的经济资本。对比极大似然估计法,实证结果表明,在小样本条件下此方法可以取得较好的结果。     

Inference based on progressively censored sample from Pareto population

Abstract

In this paper, we assume that the lifetimes have a two-parameter Pareto distribution and discuss some results of progressive Type-II censored sample. We obtain maximum likelihood estimators and Bayes estimators of the unknown parameters under squared error loss and a precautionary loss functions in progressively Type-II censored sample. Robust Bayes estimation of unknown parameters over three different classes of priors under progressively Type-II censored sample, squared error loss, and precautionary loss functions are obtained. We discuss estimation of unknown parameters on competing risks progressive Type-II censoring. Finally, we consider the problem of estimating the common scale parameter of two Pareto distributions when samples are progressively Type-II censored.     

Bayesian Analysis of a Queueing System with a Long-Tailed Arrival Process

Internet traffic data is characterized by some unusual statistical properties, in particular, the presence of heavy-tailed variables. A typical model for heavy-tailed distributions is the Pareto distribution although this is not adequate in many cases. In this article, we consider a mixture of two-parameter Pareto distributions as a model for heavy-tailed data and use a Bayesian approach based on the birth-death Markov chain Monte Carlo algorithm to fit this model. We estimate some measures of interest related to the queueing system k-Par/M/1 where k-Par denotes a mixture of k Pareto distributions. Heavy-tailed variables are difficult to model in such queueing systems because of the lack of a simple expression for the Laplace Transform (LT). We use a procedure based on recent LT approximating results for the Pareto/M/1 system. We illustrate our approach with both simulated and real data.     

Tests for the Validity of the Assumption that the Underlying Distribution of Life is Pareto

This article considers the problem of testing the validity of the assumption that the underlying distribution of life is Pareto. For complete and censored samples, the relationship between the Pareto and the exponential distributions could be of vital importance to test for the validity of this assumption. For grouped uncensored data the classical Pearson χ² test based on the multinomial model can be used. Attention is confined in this article to handle grouped data with withdrawals within intervals. Graphical as well as analytical procedures will be presented. Maximum likelihood estimators for the parameters of the Pareto distribution based on grouped data will be derived.     

Optimal sample size for record data and associated cost analysis for exponential distribution

Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] and Doostparast [M. Doostparast, A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. When a random sample Y ₁, …, Y _n is examined sequentially and successive minimum values are recorded, Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] obtained a maximum likelihood estimator of the mean of the population and showed its convergence in probability. We establish here its convergence in mean square error, which is stronger than the convergence in probability. Next, we discuss the optimal sample size for estimating the mean based on a criterion involving a cost function as well as the Fisher information based on records arising from a random sample. Finally, a comparison between complete data and record is carried out and some special cases are discussed in detail.