Improved Bayes estimators and prediction for the Wilson-Hilferty distribution

Abstract

In this paper, we revisit the Wilson-Hilferty distribution and presented its mathematical properties such as the r-th moments and reliability properties. The parameters estimators are discussed using objective reference Bayesian analysis for both complete and censored data where the resulting marginal posterior intervals have accurate frequentist coverage. A simulation study is presented to compare the performance of the proposed estimators with the frequentist approach where it is observed a clear advantage for the Bayesian method. Finally, the proposed methodology is illustrated on three real datasets.

A Reparameterized Weighted Lindley Distribution: Properties, Estimation and Applications

“…In this subsection, we reanalyzed the data set extracted from Lawless (2011), which consists of a number of cycles, divided by 1,000, up to the failure for 60 electrical appliances in a life test (see Table 3). Many authors have analyzed these uncensored data, including Reed (2011), Khan (2018) and Ramos et al (2019). Such data are known to have a bathtub-shaped hazard rate function.…”

“…As a second application, in this subsection we reanalyzed the data related to the times up to corrective maintenance of an agricultural machine, presented by Ramos et al (2019). This data set includes two censored observations, both in 13 days.…”

Analysis of Wilson‐Hilferty distribution under progressive Type‐II censoring

“…Recently, re‐parameterize form of WH distribution has been suggested by 3 which has the following simple probability density function (PDF), $$ f(\cdot )$$, and cumulative distribution function (CDF), $$ F(\cdot )$$, respectively as $$\begin{equation} f(x;\theta ,\beta )=\frac{3{x^{3\theta -1}}}{\Gamma (\theta )}{\left(\frac{\theta }{\beta }\right)}^{\theta }\exp {\left({-\frac{\theta }{\beta }x^{3}}\right)}; x>0, \theta ,\beta >0, \end{equation}$$and $$\begin{equation} F(x;\theta ,\beta )=\frac{1}{\Gamma (\theta )}\gamma {\left(\frac{\theta }{\beta }x^{3},\theta \right)}; x>0, \theta ,\beta >0, \end{equation}$$where θ and β are the shape and scale parameters, respectively, $$\Gamma (\theta )=\int _{0}^{\infty }t^{\theta -1}e^{-t}dt$$ and $$\gamma (x,y)=\int _{0}^{x}w^{y-1}e^{-w}dw$$ are the complete and lower incomplete gamma functions, respectively. Replacing 3 by …”

“…Replacing 3 by $$ p (> 0)$$ in (), the WH model becomes a special case of the Stacy distribution proposed by reference, 4 for more details see reference 3 . The reliability function (RF) and hazard rate function (HRF) at time t , denoted as $$ R(\cdot )$$ and $$ h(\cdot )$$, are respectively given by $$\begin{equation} R(t;\theta ,\beta )=\frac{1}{\Gamma (\theta )}\Gamma {\left(\frac{\theta }{\beta }t^{3},\theta \right)}; t>0, \theta ,\beta >0, \end{equation}$$and $$\begin{equation} h(t;\theta ,\beta )={3}t^{3\theta -1}{\left({\Gamma {\left(\frac{\theta }{\beta }t^{3},\theta \right)}}\right)}^{-1}{\left(\frac{\theta }{\beta }\right)}^{\theta } \exp {\left({-\frac{\theta }{\beta }t^{3}}\right)}; t>0, \theta ,\beta >0, \end{equation}$$where $$\Gamma (x,y)=\int _{x}^{\infty }w^{y-1}e^{-w}dw$$ is an upper incomplete gamma function 3 . stated that...…”

“…They also showed that it is superior to earlier transformation √ 2𝜒 suggested by. 2 Recently, re-parameterize form of WH distribution has been suggested by 3 which has the following simple probability density function (PDF), 𝑓(⋅), and cumulative distribution function (CDF), 𝐹(⋅), respectively as 𝑓(𝑥; 𝜃, 𝛽) = 3𝑥 3𝜃−1 Γ(𝜃)…”