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.     

Pareto analysis for the lifetime performance index of products on the basis of progressively first-failure-censored batches under balanced symmetric and asymmetric loss functions

One of the most important topics in manufacturing industries is the evaluation of performance lifetimes of products. Based on a given lifetime performance index, this paper deals with evaluating the performance of a process subject to a given lower specification limit. We confine ourselves to the progressively first-failure-censored data coming from a common Pareto distribution. With both the Bayesian and the non-Bayesian approaches being investigated here, we pay more attention to Bayesian estimators under balanced type loss functions. The results are presented under the balanced versions of two well-known loss functions, namely the squared error loss and the Varian's linear-exponential (LINEX) loss. Moreover, based on the Bayesian and the non-Bayesian approaches, the problem of testing hypotheses on the lifetime performance index is studied. Also, a simulation study is performed to assess the obtained results. Finally, two illustrative examples are given.     

An Estimator of Shannon Entropy of Beta-Generated Distributions and a Goodness-of-Fit Test

The aim of this article is twofold: on the one hand to introduce and study some of the statistical properties of an estimator for the Shannon entropy and on the other hand to develop a goodness-of-fit test for beta-generated distributions and the distribution of order statistics. Beta-generated distributions are a broad class of univariate distributions which has received great attention during the last 15 years, as it obeys nice properties and it extends the distribution of order statistics. The proposed estimator of Shannon entropy of beta-generated distributions is motivated by the respective Vasicek’s estimator, as the latter one is tailored to the class of the beta-generated distributions and the distribution of order statistics. The estimator of Shannon entropy is defined and its consistency is studied. It is, moreover, exploited to build a goodness-of-fit test for the beta-generated distribution and the distribution of order statistics. Simulations are performed to examine the small- and moderate-sample properties of the proposed estimator and to compare the power of the proposed test with the power of competitors under a variety of alternatives.     

Generalised Smooth Tests of Goodness of Fit Utilising L‐moments

In this paper we present a semiparametric test of goodness of fit which is based on the method of L‐moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L‐moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L‐moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.     

Modelling skewness and kurtosis with the BCPE density in GAMLSS

This paper illustrates the power of modern statistical modelling in understanding processes characterised by data that are skewed and have heavy tails. Our particular substantive problem concerns film box-office revenues. We are able to show that traditional modelling techniques based on the Pareto–Levy–Mandelbrot distribution led to what is actually a poorly supported conclusion that these data have infinite variance. This in turn led to the dominant paradigm of the movie business that ‘nobody knows anything’ and hence that box-office revenues cannot be predicted. Using the Box–Cox power exponential distribution within the generalized additive models for location, scale and shape framework, we are able to model box-office revenues and develop probabilistic statements about revenues.     

Testing equality of conditionally jfclhwdqw exponential distributions

For the models given V = v (a common random stress), X and Y are independently exponentially distributed with failure rates λ₁and λ₂v, testing H₀λ₁λ₂using a random ‘paired’ sample is considered. It is shown that a uniformly most powerful invariant test does not exist even for one sided alternatives; locally most powerful invariant tests are derived and compared with existing procedures. The method is illustrated with reliability data. Finally, the robustness of the tests when the relationships of the failure rates to V is more complex are established.     

A New Weibull–Pareto Distribution: Properties and Applications

Many distributions have been used as lifetime models. In this article, we propose a new three-parameter Weibull–Pareto distribution, which can produce the most important hazard rate shapes, namely, constant, increasing, decreasing, bathtub, and upsidedown bathtub. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real datasets on Wheaton river flood and bladder cancer. In the two applications, the new model provides better fits than the Kumaraswamy–Pareto, beta-exponentiated Pareto, beta-Pareto, exponentiated Pareto, and Pareto models.     

Bayesian inference for a common scale parameter of several Pareto populations

This article deals with the problem of Bayesian inference concerning the common scale parameter of several Pareto distributions. Bayesian hypothesis testing of, and Bayesian interval estimation for, the common scale parameter is given. Numerical studies including a comparison study, a simulation study, and a practical application study are given in order to illustrate our procedures and to demonstrate the performance, advantages, and merits of the Bayesian procedures over the classical and generalized variable procedures.     

Distributed Decision‐Making in Supply Chains and Private E‐Marketplaces

This paper develops a distributed decision‐making framework for the players in a supply chain or a private e‐marketplace to collaboratively arrive at a global Pareto‐optimal solution. In this model, no player has complete knowledge about all the costs and constraints of the other players. The decision‐making framework employs an iterative procedure, based on the Integer L‐shaped method, in which a master problem is solved to propose global solutions, and each player uses his local problems to construct feasibility and optimality cuts on the master problem. The master problem is modeled as a mixed‐integer program, and the players' local problems are formulated as linear programs. Collaborative planning scenarios in private e‐marketplaces and in supply chains were formulated and solved for test data. The results show that this distributed model is able to achieve near‐optimal solutions considerably faster than the traditional centralized approach.     

Sensitivity Analysis and Estimation of Extreme Tail Behavior in Two‐Dimensional Monte Carlo Simulation

Two‐dimensional Monte Carlo simulation is frequently used to implement probabilistic risk models, as it allows for uncertainty and variability to be quantified separately. In many cases, we are interested in the proportion of individuals from a variable population exceeding a critical threshold, together with uncertainty about this proportion. In this article we introduce a new method that can accurately estimate these quantities much more efficiently than conventional algorithms. We also show how those model parameters having the greatest impact on the probabilities of rare events can be quickly identified via this method. The algorithm combines elements from well‐established statistical techniques in extreme value theory and Bayesian analysis of computer models. We demonstrate the practical application of these methods with a simple example, in which the true distributions are known exactly, and also with a more realistic model of microbial contamination of milk with seven parameters. For the latter, sensitivity analysis (SA) is shown to identify the two inputs explaining the majority of variation in distribution tail behavior. In the subsequent prediction of probabilities of large contamination events, similar results are obtained using the new approach taking 43 seconds or the conventional simulation that requires more than 3 days.