Analysis of Frechet Distribution Using Reference Priors

Louzada

Journal of Statistics and Management Systems

et al. 2019

In this article we consider the problem of estimating the parameters of the Fréchet distribution from both frequentist and Bayesian points of view. First we briefly describe different frequentist approaches, namely, maximum likelihood, method of moments, percentile estimators, L-moments, ordinary and weighted least squares, maximum product of spacings, maximum goodness-of-fit estimators and compare them with respect to mean relative estimates, mean squared errors and the 95% coverage probability of the asymptotic confidence intervals using extensive numerical simulations. Next, we consider the Bayesian inference approach using reference priors. The Metropolis-Hasting algorithm is used to draw Markov Chain Monte Carlo samples, and they have in turn been used to compute the Bayes estimates and also to construct the corresponding credible intervals. Five real data sets related to the minimum flow of water on Piracicaba river in Brazil are used to illustrate the applicability of the discussed procedures.

Section: Bayesian Analysismentioning

confidence: 74%

Section: Bayes Estimatormentioning

confidence: 99%

Section: Appendix B Data Setmentioning

confidence: 99%

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The Fréchet distribution: Estimation and application - An overview

Louzada

Journal of Statistics and Management Systems

et al. 2019

“…The probability density function (p.d.f) and the cumulative distribution function (c.d.f) of the Frechet distribution for a random variable X are given by: (1) Where the parameter α > 0determines the shape of the distribution and β > 0is the scale parameter. (8) 2.1.…”

Section: -Component Mixture Of the Frechet Distributionsmentioning

confidence: 99%

“…The prior distributions of the mixing proportions 1 p and 2 p are again taken to be the uniform over the interval   The joint posterior distribution of parameters β1, β2, β3, p1 and p2 given data x assuming the JP is: (12) where …”

Section: The Posterior Distribution Using the Jeffreys' Prior (Jp)mentioning

confidence: 99%

Bayesian Analysis of the Mixture of Frechet Distribution under Different Loss Functions

Sultana¹,

Aslam²,

Shabbir³

2017

Pak.j.stat.oper.res.

This paper has to do with 3-component mixture of the Frechet distributions when the shape parameter is known under Bayesian view point. The type-I right censored sampling scheme is considered due to its extensive use in reliability theory and survival analysis. Taking different non-informative and informative priors, Bayes estimates of the parameter of the mixture model along with their posterior risks are derived under squared error loss function, precautionary loss function and DeGroot loss function. In case, no or little prior information is available, elicitation of hyper parameters is given. In order to study numerically, the execution of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times. A real life data example is also given to illustrate the study.

Appl Stoch Models Bus & Ind

Specifying prior distributions in reliability applications

Tian

Lewis‐Beck

Niemi

et al. 2023

Especially when facing reliability data with limited information (e.g., a small number of failures), there are strong motivations for using Bayesian inference methods. These include the option to use information from physics‐of‐failure or previous experience with a failure mode in a particular material to specify an informative prior distribution. Another advantage is the ability to make statistical inferences without having to rely on specious (when the number of failures is small) asymptotic theory needed to justify non‐Bayesian methods. Users of non‐Bayesian methods are faced with multiple methods of constructing uncertainty intervals (Wald, likelihood, and various bootstrap methods) that can give substantially different answers when there is little information in the data. For Bayesian inference, there is only one method of constructing equal‐tail credible intervals—but it is necessary to provide a prior distribution to fully specify the model. Much work has been done to find default prior distributions that will provide inference methods with good (and in some cases exact) frequentist coverage properties. This paper reviews some of this work and provides, evaluates, and illustrates principled extensions and adaptations of these methods to the practical realities of reliability data (e.g., non‐trivial censoring).