Search

For time and cost considerations, high censoring rate is quite common in life tests, which is a critical issue in lifetime monitoring. Conventional control charts designed for highly censored data are commonly based on the Weibull distribution. However, the distribution assumption may not be valid in practice, which brings challenges to the monitoring procedures. Motivated by this, a semi-parametric exponential weighted moving average (EWMA) control charting procedure is developed for highly censored lifetime data of any distribution. The control scheme uses the Kaplan–Meier estimator to construct the cumulative distribution function (CDF) and generalized Pareto distribution to improve the tail estimation. Then a Kolmogorov–Smirnov statistic defined by the differences between the in-control CDF and the empirical CDF is integrated into an EWMA charting scheme to monitor the Type I right-censored sample. We use simulation studies and a real-data analysis to show the efficiency of the proposed control chart. [ABSTRACT FROM AUTHOR]

Bivariate Pareto distributions have proven very useful in modelling lifetime data, hydrology, competing risk data, and many other datasets. In this paper, we explore Kendall tau and Gini correlations in the bivariate Pareto distributions, comparing with the popular Pearson correlation and robust quadrant correlation. It is interesting to establish the fact that zero of those correlations mutually imply independence in this family. We derive the variance of the asymptotic normality for each sample correlation via the influence function approach. When the second moment is finite, we demonstrate that the symmetric Gini correlation is asymptotically efficient as well as relatively efficient among finite samples. However, Kendall tau is more appealing in terms of compromising efficiency and robustness. [ABSTRACT FROM AUTHOR]

One of the common goals of time series analysis is to use the observed series to inform predictions for future observations. In the absence of any actual new data to predict, cross-validation can be used to estimate a model's future predictive accuracy, for instance, for the purpose of model comparison or selection. Exact cross-validation for Bayesian models is often computationally expensive, but approximate cross-validation methods have been developed, most notably methods for leave-one-out cross-validation (LOO-CV). If the actual prediction task is to predict the future given the past, LOO-CV provides an overly optimistic estimate because the information from future observations is available to influence predictions of the past. To properly account for the time series structure, we can use leave-future-out cross-validation (LFO-CV). Like exact LOO-CV, exact LFO-CV requires refitting the model many times to different subsets of the data. Using Pareto smoothed importance sampling, we propose a method for approximating exact LFO-CV that drastically reduces the computational costs while also providing informative diagnostics about the quality of the approximation. [ABSTRACT FROM AUTHOR]

Two-sample tests are probably the most commonly used tests in statistics. These tests generally address one aspect of the samples' distribution, such as mean or variance. When the null hypothesis is that two distributions are equal, the Anderson–Darling (AD) test, which is developed from the Cramer–von Mises (CvM) test, is generally employed. Unfortunately, we find that the AD test often fails to identify true differences when the differences are complex: they are not only in terms of mean, variance and/or skewness but also in terms of multi-modality. In such cases, we find that Cramer test, a modification of the CvM test, performs well. However, the adaptation of the Cramer test in routine analysis is hindered by the fact that the mean, variance and skewness of the test statistic are not available, which resulted in the problem of calculating the associated p-value. For this purpose, we propose a new method for obtaining a p-value by approximating the distribution of the test statistic by a generalized Pareto distribution. By approximating the distribution in this way, the calculation of the p-value is much faster than e.g. bootstrap method, especially for large n. We have observed that this approximation enables the Cramer test to have proper control of type-I error. A simulation study indicates that the Cramer test is as powerful as other tests in simple cases and more powerful in more complicated cases. [ABSTRACT FROM AUTHOR]

This paper studies the E-Bayesian estimations and their E-MSE of Pareto distribution parameter under different loss functions. In order to measure the estimated error, in the case of the two hyper parameters, the definition of E-MSE (expected mean square error) is introduced based on the definition of E-Bayesian estimation, and the formulas of E-Bayesian estimation and the formulas of E-MSE are given, respectively. Monte Carlo simulations are performed to analysis the performances of the proposed methods, results are compared on the basis of E-MSE. Finally, combined with the golfers income problem are performed to calculated, the maximum likelihood estimations of the shape parameter and scale parameter are given, respectively, and that compared and analysed with the E-Bayesian estimations; moreover, the Bayesian credible intervals of the shape parameter and scale parameter are also given, respectively, by MCMC method. [ABSTRACT FROM AUTHOR]

In this paper, based on progressively Type-II censored samples, the problem of estimation of multicomponent stress–strength reliability under generalized half-normal (GHN) distribution is considered. The reliability of a k-component stress-strength system is estimated when both stress and strength variates are assumed to have a GHN distribution with various cases of same and different shape and scale parameters. Different methods such as the maximum likelihood estimates (MLEs) and Bayes estimation are discussed. The expectation maximization algorithm and approximate maximum likelihood methods are proposed to compute the MLE of reliability. The Lindley's approximation method, as well as Metropolis–Hastings algorithm, are applied to compute Bayes estimates. The performance of the proposed procedures is also demonstrated via a Monte Carlo simulation study and an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new statistics for detecting outliers in the Pareto distribution. These new statistics are the extension of the statistics for detecting outliers in exponential and gamma distributions. In fact, we compare the power of our test statistics with the other statistics and select the best test statistic for detecting outliers in the Pareto distribution. Finally, numerical examples of different insurance claims are used to see the performance of the test. [ABSTRACT FROM AUTHOR]

A common approach to modelling extreme data are to consider the distribution of the exceedance value over a high threshold. This approach is based on the distribution of excess, which follows the generalized Pareto distribution (GPD) and has shown to be adequate for this type of situation. As with all data involving analysis in time, excesses above a threshold may also vary and suffer from the influence of covariates. Thus, the GPD distribution can be modelled by entering the presence of these factors. This paper presents a new model for extreme values, where GPD parameters are written on the basis of a dynamic regression model. The estimation of the model parameters is made under the Bayesian paradigm, with sampling points via MCMC. As with environmental data, behaviour data are related to other factors such as time and covariates such as latitude and distance from the sea. Simulation studies have shown the efficiency and identifiability of the model, and applying real rain data from the state of Piaui, Brazil, shows the advantage in predicting and interpreting the model against other similar models proposed in the literature. [ABSTRACT FROM AUTHOR]

In this article, the general test statistic introduced by Alizadeh Noughabi and Balakrishnan [Goodness of fit using a new estimate of Kullback-Leibler information based on Type II censored data. IEEE Trans Reliab. 2015;64:627–635.] is applied for testing goodness of fit of lifetime distributions based on Type II censored data. The test statistic is constructed based on an estimate of Kullback–Leibler (KL) information. We investigate the properties of the proposed test statistic such as the test statistic is nonnegative, just like KL information. We apply this test statistic to following distributions: Exponential, Weibull, Log-normal and Pareto. The critical values and Type I error of the proposed tests are obtained. It is shown that the proposed tests have an excellent Type I error and hence can be used confidently in practice. Then, by Monte Carlo simulations, the power values of the proposed tests are computed against several alternatives and compared with those of the existing tests. Finally, some real-world reliability data are used for illustrative purpose. [ABSTRACT FROM PUBLISHER]

In this paper, the reliability of a system is discussed when the strength of the system and the stress imposed on it are independent, non-identical exponentiated Pareto distributed random variables. Different point estimations and interval estimations are proposed. The point estimators obtained are maximum likelihood, uniformly minimum variance unbiased and Bayesian estimators. The interval estimations obtained are approximate, exact, bootstrap-p and bootstrap-t confidence intervals and Bayesian credible interval. Different methods and the corresponding confidence intervals are compared using Monte-carlo simulations. [ABSTRACT FROM PUBLISHER]

In this paper, we study the E-Bayesian and hierarchical Bayesian estimations of the parameter derived from Pareto distribution under different loss functions. The definition of the E-Bayesian estimation of the parameter is provided. Moreover, for Pareto distribution, under the condition of the scale parameter is known, based on the different loss functions, formulas of the E-Bayesian estimation and hierarchical Bayesian estimations for the shape parameter are given, respectively, properties of the E-Bayesian estimation – (i) the relationship between of E-Bayesian estimations under different loss functions are provided, (ii) the relationship between of E-Bayesian and hierarchical Bayesian estimations under the same loss function are also provided, and using the Monte Carlo method simulation example is given. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application. [ABSTRACT FROM PUBLISHER]

This paper considers problems of interval estimation and hypotheses testing for the generalized Lorenz curve under the Pareto distribution. Our approach is based on the concepts of generalized test variables and generalized pivotal quantities. The merits of the proposed procedures are numerically carried out and compared with asymptotic and bootstrap methods. Empirical evidence shows that the coverage accuracy of the proposed confidence intervals and the type I error control of the proposed exact tests are satisfactory. For illustration purposes, a real data set on median income of the 20 occupations in the United States Census of Population is analysed. [ABSTRACT FROM PUBLISHER]

An attribute control chart is developed in this manuscript for the Pareto distribution of second kind under a time truncated life test. We assume that the lifetime of the product of interest follows this distribution with known or unknown shape parameter. The performance evaluation of the proposed control chart is studied using the average run length (ARL). The tables are presented for various values of shape parameter, sample size, specified ARL and shift constants. A simulation study is given to demonstrate the performance of the proposed control chart for monitoring of non-conforming items in the industries. [ABSTRACT FROM PUBLISHER]

In this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-II hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here. [ABSTRACT FROM AUTHOR]

A ranked set sampling procedure with unequal samples for positively skew distributions (RSSUS) is proposed and used to estimate the population mean. The estimators based on RSSUS are compared with the estimators based on ranked set sampling (RSS) and median ranked set sampling (MRSS) procedures. It is observed that the relative precisions of the estimators based on RSSUS are higher than those of the estimators based on RSS and MRSS procedures. [ABSTRACT FROM PUBLISHER]

Most of the data arising in reliability can be modelled by Pareto distribution. In the present paper, we proposed the Bayes estimation of dynamic cumulative residual entropy for the classical Pareto distribution under upper record values. This measure performs important roles in reliability and survival analysis to model and analyse the data. A class of informative and non-informative priors has been assumed to derive the corresponding posterior distributions. The Bayes estimators (BEs) and the associated posterior risks have been calculated under different symmetric and asymmetric loss functions. We demonstrate the use of the proposed Bayesian estimation procedure with the average July temperatures data of Neuenburg, Switzerland, during the period 1864–1993. The performance of the BEs has been evaluated and compared under a comprehensive simulation study. The purpose is to find out the combination of a loss function and a prior having the minimum Bayes risk and hence producing the best results. [ABSTRACT FROM PUBLISHER]

Accelerated life testing is widely used in product life testing experiments since it provides significant reduction in time and cost of testing. In this paper, assuming that the lifetime of items under use condition follow the two-parameter Pareto distribution of the second kind, partially accelerated life tests based on progressively Type-II censored samples are considered. The likelihood equations of the model parameters and the acceleration factor are reduced to a single nonlinear equation to be solved numerically to obtain the maximum-likelihood estimates (MLEs). Based on normal approximation to the asymptotic distribution of MLEs, the approximate confidence intervals (ACIs) for the parameters are derived. Two bootstrap CIs are also proposed. The classical Bayes estimates cannot be obtained in explicit form, so we propose to apply Markov chain Monte Carlo method to tackle this problem, which allows us to construct the credible interval of the involved parameters. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayes estimates with MLEs and to compare the performance of different corresponding CIs considered. [ABSTRACT FROM AUTHOR]

From the class of extreme value distributions, we focus on the set of heavy-tailed distributions which produce low-frequency, high-cost events. The regular Pareto distribution is the basic model of choice, being the simplest heavy-tailed distribution. Real data suggest that modifications of the Pareto distribution may be a better fit; an alternative model is the truncated Pareto distribution (TPD). For further study, this paper proposed a TPD Sieve class of distributions. The properties and estimation on the Sieve class are also discussed. We fit the models to the largest Black Sea bass caught in Buzzard's Bay, MA, USA and the costliest Atlantic hurricanes from 1900 to 2005. Using measures of model adequacy, the TPD Sieve model is generally found to be the best-fitting model. [ABSTRACT FROM PUBLISHER]

Wu et al. [Computational comparison for weighted moments estimators and BLUE of the scale parameter of a Pareto distribution with known shape parameter under type II multiply censored sample, Appl. Math. Comput. 181 (2006), pp. 1462–1470] proposed the weighted moments estimators (WMEs) of the scale parameter of a Pareto distribution with known shape parameter on a multiply type II-censored sample. They claimed that some WMEs are better than the best linear unbiased estimator (BLUE) based on the exact mean-squared error (MSE). In this paper, the general WME (GWME) is proposed and the computational comparison of the proposed estimator with the WMEs and BLUE is done on the basis of the exact MSE for given sample sizes and different censoring schemes. As a result, the GWME is performing better than the best estimator among 12 WMEs and BLUE for all cases. Therefore, GWME is recommended for use. At last, one example is given to demonstrate the proposed GWME. [ABSTRACT FROM AUTHOR]

Estimators of the quantiles of the normal and log-normal distributions are derived. They are more efficient than the established estimators by a wide margin for small samples and high quantiles of the log-normal distribution. Although their evaluation is iterative, it requires only moderate amount of computing, which is not related to the sample size. The method is also applied to the quantiles of the Pareto distribution, but the resulting estimator is more efficient only in some settings. An application to financial statistics, estimating the return on a unit investment in equity markets over a long term, is presented. [ABSTRACT FROM AUTHOR]

In this paper, the design of reliability sampling plans for the Pareto lifetime model under progressive Type-II right censoring is considered. Sampling plans are derived using the decision theoretic approach with a suitable loss or cost function that consists of sampling cost, rejection cost, and acceptance cost. The decision rule is based on the estimated reliability function. Plans are constructed within the Bayesian context using the natural conjugate prior. Simulations for evaluating the Bayes risk are carried out and the optimal sampling plans are reported for various sample sizes, observed number of failures and removal probabilities. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Doostparast and Balakrishnan (Pareto record-based analysis, Statistics, under review) recently developed optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning shape and scale parameters, for the two-parameter Pareto distribution based on record data. In this paper, on the basis of record values and inter-record times from the two-parameter Pareto distribution, maximum-likelihood and Bayes estimators as well as credible regions are developed for the two parameters of the Pareto distribution. For illustrative purposes, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures. [ABSTRACT FROM PUBLISHER]

The exact distributions of X+Y, X Y and X/(X+Y) are studied when X and Y are independent Pareto and gamma random variables. Applications are discussed, to real problems in clinical trials, computer networks and economics. [ABSTRACT FROM AUTHOR]

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191-197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper. [ABSTRACT FROM AUTHOR]

Progressively censored data from a classical Pareto distribution are to be used to make inferences about its shape and precision parameters and the reliability function. An approximation form due to Tierney and Kadane (1986) is used for obtaining the Bayes estimates. Bayesian prediction of further observations from this distribution is also considered. When the Bayesian approach is concerned, conjugate priors for either the one or the two parameters cases are considered. To illustrate the given procedures, a numerical example and a simulation study are given. [ABSTRACT FROM AUTHOR]