Estimation of stress-strength reliability for Dagum distribution based on progressive type-II censored sample

Abstract

In this paper, classical as well as Bayesian estimation of stress strength reliability $(\eta)$ of Dagum distribution under progressive type-II censored sample is done. Maximum likelihood estimators (MLEs) of $\eta$ are also obtained along with asymptotic, bootstrap-p (boot-p) and bootstrap-t (boot-t) confidence intervals. Bayes estimators of $\eta$ along with highest posterior density (HPD) credible intervals based on informative and non-informative priors are also obtained. Symmetric as well as asymmetric loss functions are considered for the Bayesian estimation. Monte Carlo simulation study is carried out to check the performance of estimators. Real life data sets are considered for illustration purpose.

Keywords

Dagum distribution stress-strength reliability Bayes estimators maximum likelihood estimators

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1. Introduction

Dagum distribution was suggested by Dagum (1977) for the personal income data with an alternative to the Pareto and Log-normal model. The Dagum distribution is also identified as the inverse Burr XII distribution, mainly in the actuarial literature. Kleiber (2008) have showed the application of Dagum distribution in income inequality. Domma (2011) presented the maximum likelihood estimates for the parameters of Dagum distribution under type I right censored and type-II double censored data. Shahzad and Asghar (2013) used TL-moments to estimate the parameters of this distribution. Oluyede and Ye (2014) proposed the weighted Dagum and related distribution with several properties. Dey et al. (2017) provided the properties and different method of estimation of the parameters of Dagum distribution.

This distribution has been widely used in various directions such as, wealth data, income and meterological data, reliability and survival analysis. Domma (2002) has showed that hazard function of Dagum distribution can be monotonically decreasing and bathtub shaped. This behaviour has gained the attention of researchers in different direction such as reliability and survival analysis. Domma et al. (2011) provided the several reliability measures with their characteristics of Dagum distribution. Ahmed (2015) provided the inference about the stress-strength reliability of Dagum distribution for complete sample.

This paper considers the estimation of stress-strength reliability for Dagum distribution under progressive type-II censored sample. Assume that, the strength of a component is represented by $X$ and the stress, applied to this component is represented by $Y$ , then the stress-strength reliability is the probability that stress does not exceed the strength i.e., $P(Y<X)$ . Stress-strength reliability has applications in many fields such as engineering, military, medicine etc. Kundu and Gupta (2006) discussed the maximum likelihood estimators of stress-strength reliability for Weibull distribution. Kim and Chung (2006) considered the Bayesian estimation of $P({Y<X)}$ from Burr-type X model containing spurious observations. Jiang and Wong (2008) have discussed the inference for stress-strength reliability for right truncated exponentially distributed data. Nadar and Kizilaslan (2014) discussed the classical and Bayesian estimation of stress-strength reliability for Kumaraswamy’s distribution using upper record values. Generalized inferences of stress-strength reliability for Pareto distribution have discussed by Gunasekera (2015). Kumari et al. (2019) discussed the Bayesian estimation of $P(Y<X)$ of generalised inverted exponential distribution using upper record values.

The scheme of the paper is as follows: In Section 2, an overview related to model with some distributional properties and stress-strength reliability are given. A brief detail of progressive type-II censoring is also given in Section 2. Section 3 deals with MLEs of stress-strength reliability of the Dagum distribution. The asymptotic confidence interval, bootstrap percentile (boot-p) and bootstrap-t (boot-t) confidence intervals are also defined in Section 3. Section 4 deals with Bayesian estimation of stress-strength reliability using informative as well as non-informative priors. Highest posterior density (HPD) credible intervals are also defined in Section 4. A Monte Carlo simulation is carried out for classical as well as Bayesian estimation of stress strength reliability under different configuration of sample size. The mean square error (MSE) and asymptotic, boot-p and boot-t confidence intervals are obtained in Section 5. The expected loss along with HPD credible intervals are also obtained in Section 5. Real data set are considered for the illustration purpose in Section 6. Section 7 concludes the consequences of simulation study and real data sets.

2. Model

Let a random variable T follows the Dagum distribution with parameters $(\beta,\lambda,\delta)$ , then the cumulative distribution function (cdf) is given by

$\displaystyle F(t)={(1+\lambda t^{-\delta})}^{-\beta};t>0;\beta,\lambda,\delta% >0,$ (1)

and the probability density function (pdf) is

$\displaystyle f(t)=\beta\lambda\delta t^{-\delta-1}{(1+\lambda t^{-\delta})}^{% -\beta-1};t>0;\beta,\lambda,\delta>0.$ (2)

where ${\lambda}$ is the scale parameter and ${\beta}$ , $\delta$ are shape parameters.

Suppose ${X}$ represents the strength of a component and ${Y}$ represents the stress applied to this component. We consider the case when $\lambda$ and $\delta$ are known and $\beta$ is unknown. Let $X$ and $Y$ , two random variables follow Dagum distribution with parameters $({\beta}_{1},\lambda,\delta)$ and $({\beta}_{2},\lambda,\delta)$ respectively, then the stress-strength reliability as defined by Ahmed (2015) is

$\displaystyle\eta=P(Y<X)=\frac{{\beta}_{1}}{{\beta}_{1}+{\beta}_{2}}.$ (3)

The plot of stress-strength reliability for different values of ${\beta}_{1}$ and ${\beta}_{2}$ is given in Fig. 1.

Figure 1.

Stress-strength reliability for Dagum distribution.

2.1 Progressive type-II censoring

When all units cannot be observed up to their failure due to time, cost and other circumstances then requirement of censoring is essential. Type-I and type-II censoring schemes are commonly used censoring schemes. In type-II censoring, the life testing experiment is terminated at a pre fixed number of failures observed while in type-I censoring, the experiment is terminated at a pre fixed time. Hence type-I and type-II censoring schemes are also named as time censoring and failure censoring schemes, respectively. Arora et al. (2019) discussed the Bayesian estimation of reliability measures for Topp-Leone distribution using type-II censoring scheme. These censoring schemes do not provide the facility to remove the units during the experiment. To answer this problem, Cohen (1963) offered a more general censoring scheme named as progressive type-II censoring.

The progressive type-II censoring have an important role in life time data analysis, and reliability field and is a suitable scheme in which units are allowed to be removed from the experiment at each of several ordered failure times (Fernandez (2004)). Balakrishnan and Aggarwala (2000) delivered an excellent review of the progressive censoring. Recent work on progressive censoring can be seen in Rastogi et al. (2012), Ahmed (2014), Tian et al. (2014).

Progressive type-II censoring scheme can be explained as: suppose $n$ units are placed on a life test to observe the failure time and the experiment is ended after the failure of $m(m\leqslant n)$ units in a specific manner. At the failure of first unit, $R_{1}$ units are randomly removed from the remaining $n-1$ units. At the failure of second unit, $R_{2}$ units are randomly removed from the remaining $n-2-R_{1}$ units. Continuing in this manner, at the failure of $m^{\text{th}}$ unit, all remaining $R_{m}=n-m-R_{1}-R_{2}-\dots-R_{m-1}$ units are randomly removed from the experiment. In this way, the obtained $m$ ordered units which are the consequences of this censoring scheme are addressed as progressive type-II censored sample. Type-II censoring scheme i.e., failure censoring can be considered as a particular case of progressive type-II censoring in which $R_{1}=R_{2}=\dots=R_{m-1}=0$ and ${\ R}_{m}=n-m$ . Note that if all $R_{i}$ are zero i.e., $R_{1}=R_{2}=\dots=R_{m}=0$ and $m=n$ , then progressive type-II censoring reduces to complete sample i.e., there is no censoring.

In case of progressive type-II censoring, the likelihood function is given by Balakrishnan and Sandhu (1995) as

$\displaystyle L(\underline{t})=c\prod\limits^{m}_{i=1}{f(t_{i}){\{1-F(t_{i})\}% }^{R_{i}}},$ (4)

where

$\displaystyle c=n(n-R_{1}-1)\dots(n-R_{1}-R_{2}-\dots-R_{m-1}-(m-1)).$

2.1.1 Algorithm to generate a progressive type-II censored sample

To generate the progressive type-II censored sample from Dagum distribution, the procedure given in Balakrishnan and Aggarwala (2000) is followed.

1.
Generate $u_{{1}},u_{{2}},{\dots},u_{m}$ , i.i.d. random variables from $U({0,1}).$
2.
Set $z_{i}{=-}{\text{ln}({1-}u_{i})}$ , so that $z^{\prime}_{i}s$ are i.i.d. standard exponential variables.
3.
Given censoring scheme $\underline{R}=(R_{{1}},R_{{2}},\dots,R_{m})$ . Set $y_{{1}}=\frac{z_{{1}}}{m}$ and for $i=2,3,\dots m$ ,

$\displaystyle y_{i}=y_{i-1}+\frac{z_{i}}{\left(n-\sum\limits^{i-1}_{j=1}R_{j}-% i+1\right)}.$

Now, $y^{\prime}_{i}s$ are progressive type-II censored sample from standard exponential with censoring scheme $\underline{R}=(R_{{1}},R_{{2}},\dots,R_{m})$ .
4.
Set $w_{i}=1-{\exp}(-y_{i})$ so that $w^{\prime}_{i}s$ are progressive type-II censored sample from $U({0,1})$ .
5.
Set $t_{i}=F^{-1}(w_{i})={\left(\frac{w^{-\frac{{1}}{\beta}}_{i}-{1}}{\lambda}% \right)}^{-1/\delta}$ .

where $F(.)$ is the cdf of Dagum $(\beta,\lambda,\delta)$ . Now, $(t_{{1}},t_{{2}},\dots,t_{m})$ is a progressive type-II censored sample from Dagum $(\beta,\lambda,\delta)$ with a censoring scheme $\underline{R}{=}(R_{{1}},R_{{2}},\dots,R_{m})$ .
3. Maximum likelihood estimation of ${\eta}$

Let $\underline{x}=(x_{1:m_{1}:n_{1}},x_{2:m_{1}:n_{1}},\dots,x_{m_{1}:m_{1}:n_{1}})$ and $\underline{y}=(y_{1:m_{2}:n_{2}},y_{2:m_{2}:n_{2}},\dots,y_{m_{2}:m_{2}:n_{2}})$ be progressive type-II censored samples from Dagum $({\beta}_{1},\lambda,\delta)$ and Dagum $({\beta}_{2},\lambda,\delta)$ respectively, then the likelihood function under progressive type-II censoring is given by

$\displaystyle L(\underline{x},\underline{y})=c_{1}\prod\limits^{m_{1}}_{i=1}{f% (x_{i}){\{1-F(x_{i})\}}^{R_{i:x}}\times}c_{2}\prod\limits^{m_{2}}_{j=1}{f(y_{j% }){\{1-F(y_{j})\}}^{R_{j:y}}}=c_{1}c_{2}{(\lambda\delta)}^{(m_{1}+m_{2})}{% \beta}^{m_{1}}_{1}{\beta}^{m_{2}}_{2}\prod\limits^{m_{1}}_{i=1}{x^{-\delta-1}_% {i}}{(1+\lambda x^{-\delta}_{i})}^{-{\beta}_{1}-1}{\{1-{(1+\lambda x^{-\delta}% _{i})}^{-{\beta}_{1}}\}}^{R_{i:x}}\times\prod\limits^{m_{2}}_{j=1}{y^{-\delta-% 1}_{j}}{(1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}-1}{\{1-{(1+\lambda y^{-% \delta}_{j})}^{-{\beta}_{2}}\}}^{R_{j:y}}.$ (5)

where

$\displaystyle c_{1}=n_{1}(n_{1}-R_{1:x}-1)\dots(n_{1}-R_{1:x}-R_{2:x}-\dots-R_% {(m_{1}-1):x}-m_{1}+1)$ $\displaystyle c_{2}=n_{2}(n_{2}-R_{1:y}-1)\dots(n_{2}-R_{1:y}-R_{2:y}-\dots-R_% {{(m}_{2}-1):y}-m_{2}+1)$

From Eq. (5), log-likelihood function of ${\beta}_{1}$ and ${\beta}_{2}$ considering $\lambda$ and $\delta$ to be known is given by

$\displaystyle l({\beta}_{1},{\beta}_{2})=A+m_{1}{\text{ln}({\beta}_{1})}-({% \beta}_{1}+1)\sum\limits^{m_{1}}_{i=1}{{\text{ln}(1+\lambda x^{-\delta}_{i})}+% \sum\limits^{m_{1}}_{i=1}{R_{i:x}{\text{ln}\{1-{(1+\lambda x^{-\delta}_{i})}^{% -{\beta}_{1}}\}}}}{}+m_{2}{\text{ln}({\beta}_{2})}-({\beta}_{2}+1)\sum\limits^% {m_{2}}_{j=1}{\text{ln}(1+\lambda y^{-\delta}_{j})}+\sum\limits^{m_{2}}_{j=1}{% R_{j:y}{\text{ln}\{1-{(1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}}\}}},$ (6)

where $A={\text{ln}(c_{1}c_{2})}+(m_{1}+m_{2}){\text{ln}(\lambda\delta)}-(\delta+1)% \left(\sum\limits^{m_{1}}_{i=1}{{\text{ln}(x_{i})}}+\sum\limits^{m_{2}}_{j=1}{% {\text{ln}(y_{j})}}\right)$ .

Now, MLEs ${\widehat{\beta}}_{1},{\widehat{\beta}}_{2}$ of the parameters ${\beta}_{1},{\beta}_{2}$ are the solutions of the following normal equations

$\displaystyle\frac{\partial l}{\partial{\beta}_{1}}=\frac{m_{1}}{{\beta}_{1}}-% \sum\limits^{m_{1}}_{i=1}{\text{ln}(1+\lambda x^{-\delta}_{i})+\sum\limits^{m_% {1}}_{i=1}{\frac{R_{i:x}{(1+\lambda x^{-\delta}_{i})}^{-{\beta}_{1}}{\text{ln}% (1+\lambda x^{-\delta}_{i})}}{1-{(1+\lambda x^{-\delta}_{i})}^{-{\beta}_{1}}}}% }=0,$ (7) $\displaystyle\frac{\partial l}{\partial{\beta}_{2}}=\frac{m_{2}}{{\beta}_{2}}-% \sum\limits^{m_{2}}_{j=1}{\text{ln}(1+\lambda y^{-\delta}_{j})+}\sum\limits^{m% _{2}}_{j=1}{\frac{R_{j:y}{(1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}}{\text{ln% }(1+\lambda y^{-\delta}_{j})}}{1-{(1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}}}% }=0,$ (8)

After solving Eqs (7) and (8) by numerical methods such as Newton-Raphson method (Dube et al. (2016)), we get the MLEs of ${\beta}_{1}$ and ${\beta}_{2}$ . Then MLE of $\eta$ can be obtained by substituting the values of ${\widehat{\beta}}_{1}$ and ${\widehat{\beta}}_{2}$ as

$\displaystyle\widehat{\eta}=\frac{{\widehat{\beta}}_{1}}{{\widehat{\beta}}_{1}% +{\widehat{\beta}}_{2}}.$ (9)

Figure 2.

Profile log-likelihood function of ${\beta}_{1}$ and ${\beta}_{2}$ .

The profile log-likelihood function of ${\beta}_{1}$ and ${\beta}_{2}$ is plotted in Fig. 2, which shows that the likelihood equations have a unique solution. It can also be seen from Fig. 2 that the likelihood surface has curvature in both ${\beta}_{1}$ and ${\beta}_{2}$ directions, indicating that the MLEs of ${\beta}_{1}$ and ${\beta}_{2}$ exist and are unique.

3.1 Asymptotic confidence interval

It is not easy to find the exact distribution of $\widehat{\eta}$ since Eqs (7) and (8) are not in explicit form, therefore, the exact confidence interval for $\widehat{\eta}$ cannot be obtained. Hence, asymptotic confidence interval is computed using large sample approximation. Let $u=({\beta}_{1},{\beta}_{2})$ and $\hat{u}=({\widehat{\beta}}_{1},{\widehat{\beta}}_{2})$ , then asymptotic $\widehat{\textit{Var}}(\widehat{\eta})$ can be explained by

$\displaystyle\widehat{\textit{Var}}(\widehat{\eta})={[b^{t}I^{-1}(u)b]}_{u=% \hat{u}},$

where

$\displaystyle I(u)=\left[\begin{array}[]{cc}-\left(\frac{{\partial}^{2}l}{% \partial{\beta}^{2}_{1}}\right)&-\left(\frac{{\partial}^{2}l}{\partial{\beta}_% {1}\partial{\beta}_{2}}\right)\\ -\left(\frac{{\partial}^{2}l}{\partial{\beta}_{2}\partial{\beta}_{1}}\right)&-% \left(\frac{{\partial}^{2}l}{\partial{\beta}^{2}_{2}}\right)\end{array}\right]% _{u=\hat{u}}\text{ and }b=\left(\frac{\partial\eta}{\partial{\beta}_{1}},\frac% {\partial\eta}{\partial{\beta}_{2}}\right)^{t}.$

$100(1-\gamma)\%$ asymptotic confidence interval for $\eta$ is given by $\widehat{\eta}\pm z_{\gamma/2}\sqrt{\widehat{\textit{Var}}(\widehat{\eta})}$ , where $z_{\gamma/2}$ is the upper $(\gamma/2)th$ percentile of standard normal distribution $(0<\gamma<1)$ .

3.2 Bootstrap confidence interval

There are two types of confidence intervals obtained: bootstrap percentile (boot-p) and Studentized-t (boot-t) Interval Procedure. For the convenience, let $(x_{1:m_{1}:n_{1}},x_{2:m_{1}:n_{1}},\dots,x_{m_{1}:m_{1}:n_{1}})$ denoted by $(x_{1},x_{2},\dots,x_{m_{1}})$ and $(y_{1:m_{2}:n_{2}},y_{2:m_{2}:n_{2}},\dots,y_{m_{2}:m_{2}:n_{2}})$ denoted by $(y_{1},y_{2},\dots,y_{m_{2}})$ throughout.

(a) Bootstrap percentile (boot-p) Interval Procedure

Efron (1982) proposed the procedure of boot-p confidence interval as follows:

(i)
Generate progressive type-II censored samples $(x_{1},x_{2},\dots,x_{m_{1}})$ from $f({\beta}_{1},\lambda,\delta)$ and $(y_{1},y_{2},\dots,y_{m_{2}})$ from $f({\beta}_{2},\lambda,\delta)$ to calculate the MLEs $({\widehat{\beta}}_{1},{\widehat{\beta}}_{2})$ of ${(\beta}_{1},{\beta}_{2})$ , then calculate the MLE $\widehat{\eta}$ of $\eta$ using Eq. (9).
(ii)
Generate $(x^{}_{1},x^{}_{2},\dots,x^{}_{m_{1}})$ from $f({\widehat{\beta}}_{1},\lambda,\delta)$ and $(y^{}_{1},y^{}_{2},\dots,y^{}_{m_{2}})$ from $f({\widehat{\beta}}_{2},\lambda,\delta)$ to calculate the MLEs $({\widehat{\beta}}^{}_{1},{\widehat{\beta}}^{}_{2})$ of ${(\beta}_{1},{\beta}_{2})$ , then calculate the MLE ${\widehat{\eta}}^{}$ using Eq. (9).
(iii)
Repeat step (ii) B times to calculate MLEs ${\widehat{\eta}}^{}_{(1)},{\widehat{\eta}}^{}_{(2)},\dots,{\widehat{\eta}}^{% }_{(B)}$ .
(iv)
Arrange these MLEs ${\widehat{\eta}}^{}_{(1)},{\widehat{\eta}}^{}_{(2)},\dots,{\widehat{\eta}}^{% }_{(B)}$ in ascending order as ${\widehat{\eta}}^{}_{[1]},{\widehat{\eta}}^{}_{[2]},\dots,{\widehat{\eta}}^{% }_{[B]}$ .
(v)
Two sided $100(1-\gamma)\%$ boot-p confidence interval is given by ${(\widehat{\eta}}^{}_{[(\gamma/2)\times B]},{\widehat{\eta}}^{}_{[(1-\gamma/% 2\ )\times B]})$ , where $[x]$ is the greatest integer of $x$ .

(b) Studentized-t (boot-t) Interval Procedure

Hall (1988) proposed the procedure of boot-t confidence interval as follows:

Step (i) and (ii) are same as in above section (a).

(iii)
Define a statistic as ${T}^{}=\frac{{\widehat{\eta}}^{}-\widehat{\eta}}{\sqrt{\widehat{\textit{Var}% }({\widehat{\eta}}^{})}}$ and this $\widehat{\textit{Var}}({\widehat{\eta}}^{})$ is obtained as in Section 3.1.
(iv)
Repeat step (ii) and (iii) B times and get ${\hat{T}}^{}_{(1)},{\hat{T}}^{}_{(2)},\dots,{\hat{T}}^{}_{(B)}$ . Now, arrange these ${\hat{T}}^{}_{(1)},{\hat{T}}^{}_{(2)},\dots,{\hat{T}}^{}_{(B)}$ in ascending order as ${\hat{T}}^{}_{[1]},{\hat{T}}^{}_{[2]},\dots,{\hat{T}}^{}_{[B]}$ .
(v)
Two sided $100(1-\gamma)\%$ boot-p confidence interval is given by

$\displaystyle\left(\widehat{\eta}+{\hat{T}}^{}_{[(\gamma/2)\times B]}\sqrt{% \widehat{\textit{Var}}(\widehat{\eta})},\widehat{\eta}+{\hat{T}}^{*}_{[(1-% \gamma/2)\times B]}\sqrt{\widehat{\textit{Var}}(\widehat{\eta})}\right).$

4. Bayesian estimation

In this Section, Bayes estimators for $\eta$ are derived using progressive type-II censoring in case of both informative and non-informative priors under symmetric loss function (squared error loss function) and asymmetric loss function (generalized entropy loss function). A brief introduction of loss functions, priors and HPD credible intervals is given below:

Squared Error Loss Function (SELF)

The squared error loss function is defined as $L(\widehat{\xi,}\xi)\propto{(\xi-\widehat{\xi})}^{2}$ , where $\widehat{\xi}$ is the Bayes estimator of unknown parameter $\xi$ . SELF is the simplest symmetric loss function. The Bayes estimator of $\xi$ under SELF is ${\widehat{\xi}}_{s}=E(\xi\mathrel{|\vphantom{\xi\underline{x}}\kern-1.2pt}% \underline{x})$ , where expectation is taken with respect to posterior density.

Generalized Entropy Loss Function (GELF)

Squared error loss function gives equal weights to under estimation and over estimation. However, in many situations under estimation is more serious than over estimations and vice versa. So, in order to overcome this difficulty, another useful asymmetric loss function namely generalized entropy loss function is used here.

Generalized entropy loss function is an asymmetric loss function and defined by Calabria and Pulcini (1996). This loss function is a generalization of the entropy loss function and defined as

$\displaystyle L(\xi,\widehat{\xi})\propto{\left(\frac{\widehat{\xi}}{\xi}% \right)}^{b}-b\text{ln}\left(\frac{\widehat{\xi}}{\xi}\right)-1,$

where $b\neq 0$ . The constant $b$ determines the shape of the loss function. If $b<0$ , then under estimation gets more serious than over estimation and vice-versa. Bayes estimator of $\xi$ under GELF is

$\displaystyle{\widehat{\xi}}_{g}={[E({\xi}^{-b}|\underline{x})]}^{(-1/b)}.$

Gamma Prior

Gamma prior is frequently used informative prior due to its flexibility (Dey & Tanujit, 2014; Kumari et al., 2019). Assuming the parameters ${\beta}_{1}$ and ${\beta}_{2}$ having independent gamma priors with respective probability density functions

$\displaystyle g({\beta}_{1})\propto{\beta}^{a_{1}-1}_{1}e^{-d_{1}{\beta}_{1}};% {\beta}_{1}>0,a_{1},d_{1}>0,$ $\displaystyle g({\beta}_{2})\propto{\beta}^{a_{2}-1}_{2}e^{-d_{2}{\beta}_{2}};% {\beta}_{2}>0,a_{2},d_{2}>0,$

where $a_{1},a_{2},d_{1},d_{2}>0$ are the hyper parameters, the parameters of the informative prior.

The joint prior density of ${\beta}_{1}$ and ${\beta}_{2}$ is given by

$\displaystyle g({\beta}_{1},{\beta}_{2})\propto{\beta}^{a_{1}-1}_{1}{\beta}^{a% _{2}-1}_{2}e^{-(d_{1}{\beta}_{1}+d_{2}{\beta}_{2})};{\beta}_{1},{\beta}_{2}>0,% a_{1},a_{2},d_{1},d_{2}>0.$ (10)

Remark

If the value of all hyper parameters is zero i.e., $a_{i}=d_{i}=0;\forall i=1,2$ , then gamma priors develop non-informative priors and the joint prior density in case of non-informative prior is given by

$\displaystyle p(\alpha,\lambda,\beta,\theta)\propto\frac{1}{{\beta}_{1}{\beta}% _{2}};\alpha,\lambda,\beta,\theta>0.$

4.1 Bayesian estimation of

\bm{\eta}

using gamma prior

The joint posterior distribution of the unknown parameters ${\beta}_{1}$ and ${\beta}_{2}$ given data using Eqs (5) and (10) is

$\displaystyle\pi({\beta}_{1},{\beta}_{2}\mathrel{|\vphantom{{\beta}_{1},{\beta% }_{2}\underline{x},\underline{y}}\kern-1.2pt}\underline{x},\underline{y})=% \frac{L(\underline{x},\underline{y})g({\beta}_{1},{\beta}_{2})}{\int\nolimits^% {\infty}_{0}{\int\nolimits^{\infty}_{0}{L(\underline{x},\underline{y})g({\beta% }_{1},{\beta}_{2})\ d{\beta}_{1}d{\beta}_{2}}}},=K^{-1}{(\lambda\delta)}^{(m_{% 1}+m_{2})}{\beta}^{m_{1}+a_{1}-1}_{1}{\beta}^{m_{2}+a_{2}-1}_{2}e^{-(d_{1}{% \beta}_{1}+d_{2}{\beta}_{2})}\prod\limits^{m_{1}}_{i=1}{x^{-\delta-1}_{i}}{(1+% \lambda x^{-\delta}_{i})}^{-{\beta}_{1}-1}\times{\{1-{(1+\lambda x^{-\delta}_{% i})}^{-{\beta}_{1}}\}}^{R_{i:x}}\prod\limits^{m_{2}}_{j=1}{y^{-\delta-1}_{j}}{% (1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}-1}{\{1-{(1+\lambda y^{-\delta}_{j})% }^{-{\beta}_{2}}\}}^{R_{j:y}}$

where

$\displaystyle K=\int\nolimits^{\infty}_{0}\int\nolimits^{\infty}_{0}\left\{{(% \lambda\delta)}^{(m_{1}+m_{2})}{\beta}^{m_{1}+a_{1}-1}_{1}{\beta}^{m_{2}+a_{2}% -1}_{2}e^{-(d_{1}{\beta}_{1}+d_{2}{\beta}_{2})}\prod\limits^{m_{1}}_{i=1}{x^{-% \delta-1}_{i}}{(1+\lambda x^{-\delta}_{i})}^{-{\beta}_{1}-1}\right.\times\left% .\{1-{(1+\lambda x^{-\delta}_{i})}^{-{\beta}_{1}}\}^{R_{i:x}}\prod\limits^{m_{% 2}}_{j=1}{y^{-\delta-1}_{j}}{(1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}-1}{\{1% -{(1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}}\}}^{R_{j:y}}\right\}$

Hence, joint posterior distribution of the unknown parameters ${\beta}_{1}$ and ${\beta}_{2}$ given data is

$\displaystyle\pi({\beta}_{1},{\beta}_{2}\mathrel{|\vphantom{{\beta}_{1},{\beta% }_{2}\underline{x},\underline{y}}\kern-1.2pt}\underline{x},\underline{y})% \propto{(\lambda\delta)}^{(m_{1}+m_{2})}{\beta}^{m_{1}+a_{1}-1}_{1}{\beta}^{m_% {2}+a_{2}-1}_{2}e^{-(d_{1}{\beta}_{1}+d_{2}{\beta}_{2})}$ $\displaystyle\quad\times\prod\limits^{m_{1}}_{i=1}{x^{-\delta-1}_{i}{(1+% \lambda x^{-\delta}_{i})}^{-{\beta}_{1}-1}}\times\{1-{(1+\lambda x^{-\delta}_{% i})}^{-{\beta}_{1}}\}^{R_{i:x}}$ (11) $\displaystyle\quad\times\prod\limits^{m_{2}}_{j=1}{y^{-\delta-1}_{j}}{(1+% \lambda y^{-\delta}_{j})}^{-{\beta}_{2}-1}\{1-{(1+\lambda y^{-\delta}_{j})}^{-% {\beta}_{2}}\}^{R_{j:y}}.$

Since, the joint posterior distribution of ${\beta}_{1},{\beta}_{2}$ in Eq. (4.1) cannot be obtained analytically, the Markov Chain Monte Carlo (MCMC) technique is adopted to obtain the Bayes estimates and corresponding HPD credible intervals of $\eta$ .

The Metropolis-Hastings (M-H) algorithm can be used to generate random samples from any complex distribution of any dimension that is known up to a normalizing constant. The M-H algorithm was given by Metropolis et al. (1953) and later extended by Hastings (1970). The full conditional posterior distribution of parameters ${\beta}_{1}$ and ${\beta}_{2}$ are defined as

$\displaystyle\pi({\beta}_{1}\mathrel{|\vphantom{{\beta}_{1}{\beta}_{2},% \underline{x},\underline{y}}\kern-1.2pt}{\beta}_{2},\underline{x},\underline{y% })\propto{\beta}^{m_{1}+a_{1}-1}_{1}e^{-d_{1}{\beta}_{1}}\prod\limits^{m_{1}}_% {i=1}{{(1+\lambda x^{-\delta}_{i})}^{-{\beta}_{1}}{\{1-{(1+\lambda x^{-\delta}% _{i})}^{-{\beta}_{1}}\}}^{R_{i:x}}},$ (12) $\displaystyle\pi({\beta}_{2}\mathrel{|\vphantom{{\beta}_{2}{\beta}_{1},% \underline{x},\underline{y}}\kern-1.2pt}{\beta}_{1},\underline{x},\underline{y% })\propto{\beta}^{m_{2}+a_{2}-1}_{2}e^{-d_{2}{\beta}_{2}}\prod\limits^{m_{2}}_% {j=1}{{(1+\lambda y^{-\delta}_{j})}^{-{\beta}_{2}}{\{1-{(1+\lambda y^{-\delta}% _{j})}^{-{\beta}_{2}}\}}^{R_{j:y}}}.$ (13)

The full posterior conditional distributions of ${\beta}_{1}$ and ${\beta}_{2}$ in Eqs (12) and (13) are not in explicit form, hence to generate the random samples from Eqs (12) and (13), the M-H algorithm is used. The MCMC technique of M-H algorithm is as follows:

Step 1:

Start with the initial values ${\beta}^{(0)}_{1},{\beta}^{(0)}_{2}$ .

Step 2:

Set $k=1$ .

Step 3:

Generate proposal values ${\beta}^{*}_{1}$ and ${\beta}^{*}_{2}$ from the normal transition kernels $N({\beta}^{(k-1)}_{1},e_{1})$ and $N({\beta}^{(k-1)}_{2},e_{2})$ , respectively. The standard deviations $e_{1}$ and $e_{2}$ can be chosen on the basis of few trials.

Step 4:

Compute the acceptance ratio

$\displaystyle{\alpha}_{1}={{\min}\left\{1,\frac{\pi({\beta}^{*}_{1}\mathrel{|% \vphantom{{\beta}^{*}_{1}{\beta}_{2},\underline{x},\underline{y}}\kern-1.2pt}{% \beta}_{2},\underline{x},\underline{y})}{\pi({\beta}^{(k-1)}_{1}\mathrel{|% \vphantom{{\beta}^{(k-1)}_{1}{\beta}_{2},\underline{x},\underline{y}}\kern-1.2% pt}{\beta}_{2},\underline{x},\underline{y})}\right\}}\text{ and }{\alpha}_{2}=% {{\min}\left\{1,\frac{\pi({\beta}^{*}_{2}\mathrel{|\vphantom{{\beta}^{*}_{2}{% \beta}_{1},\underline{x},\underline{y}}\kern-1.2pt}{\beta}_{1},\underline{x},% \underline{y})}{\pi({\beta}^{(k-1)}_{2}\mathrel{|\vphantom{{\beta}^{(k-1)}_{2}% {\beta}_{1},\underline{x},\underline{y}}\kern-1.2pt}{\beta}_{1},\underline{x},% \underline{y})}\right\}}.$

Step 5:

Generate two random numbers $r_{1}$ from $U(0,1)$ and $r_{2}$ from $U(0,1)$ .

Step 6:

Accept ${\beta}^{*}_{1}$ as ${\beta}^{(k)}_{1}$ , if $r_{1}\leqslant{\alpha}_{1}$ else ${\beta}^{(k)}_{1}={\beta}^{(k-1)}_{1}$ and accept ${\beta}^{*}_{2}$ as ${\beta}^{(k)}_{2}$ , if $r_{2}\leqslant{\alpha}_{2}$ else ${\beta}^{(k)}_{2}={\beta}^{(k-1)}_{2}$ .

Step 7:

Obtain ${\eta}^{(k)}$ from Eq. (3).

Step 8:

Set $k=k+1$ .

Step 9:

Repeat step 1–8 $N$ times.

The Bayes estimator of $\eta$ under SELF is ${\widehat{\eta}}_{\textit{self}}=\frac{{1}}{N-N_{0}}\sum\limits^{N}_{s=N_{0}+1% }{{\eta}_{{(s)}}}$ . The Bayes estimator of $\eta$ under GELF is ${\widehat{\eta}}_{\textit{gelf}}={\left[\frac{{1}}{N-N_{0}}\sum\limits^{N}_{s=% N_{0}+1}{{\eta}^{-b}_{({s})}}\right]}^{-1/b}$ . Note that $N_{0}$ is the burn-in period.

Remark

Bayes estimators and expected loss function of $\eta$ in case of non-informative prior are obtained using estimation procedure discussed above by taking $a_{i}=0,d_{i}=0;i=1,2$ .

HPD Credible Intervals

Chen and Shao (1999) introduced the algorithm to find the HPD credible intervals. ${100}(1-\gamma){\%}$ HPD credible interval is that ${100}(1-\gamma){\%}$ credible interval which is having smallest width among all possible ${100}(1-\gamma){\%}$ credible intervals.

Once the posterior sample is generated for ${\eta}_{i}(i=1,2,\dots,(N-N_{0}))$ , then ${\eta}_{(1)}\leqslant{\eta}_{(2)}\leqslant\dots\leqslant{\eta}_{(N-N_{0})}$ denotes the ordered values of ${\eta}_{1},{\eta}_{2},\dots,{\eta}_{(N-N_{0})}$ . The ${100}(1-\gamma){\%}$ HPD interval for $\eta$ is defined by $({\eta}_{(j)},{\eta}_{(j+[(1-\gamma)(N-N_{0})])}))$ , where $j$ is chosen such that

$\displaystyle{\eta}_{(j+[(1-\gamma)(N-N_{0})])}-{\eta}_{(j)}={\mathop{\min}_{1% \leqslant j\leqslant M}({\eta}_{(j+[(1-\gamma)(N-N_{0})])}-{\eta}_{(j)})},j=1,% 2,\dots,(N-N_{0}),$

where $[x]$ is the greatest integer of $x$ .

5. Simulation study

In this section, Monte Carlo simulation study using R software is carried out to obtain MLEs and Bayes estimates of $\eta$ of Dagum distribution under progressive type-II censoring as the estimators cannot be obtained theoretically. The Bayesian estimation is done for both informative (gamma prior) as well as non-informative priors under SELF and GELF. For GELF, the value of $b$ is taken 0.5 (for over estimation) and $-$ 0.5 (for under estimation). The values of hyper parameters $(a^{\prime}_{i}s,d^{\prime}_{i}s;i=1,2)$ are chosen such that the true value of parameters is equal to the prior means. Three cases of $\eta$ are considered as small, medium and large values by taking different configurations of the true values and respective hyper-parameters underlying

(i)
$\eta=0.25({\beta}_{1}=1,{\beta}_{2}=3,a_{1}=a_{2}=d_{1}=1,d_{2}=3)$
(ii)
$\eta=0.50({\beta}_{1}=2,{\beta}_{2}=2,a_{1}=a_{2}=2,d_{1}=d_{2}=4)$
(iii)
$\eta=0.75({\beta}_{1}=3,{\beta}_{2}=1,a_{1}=a_{2}=d_{2}=1,d_{1}=3)$ , respectively.

To generate the progressive type-II censored sample from Dagum distribution, the procedure given in Balakrishnan (2000) is followed. This procedure is also discussed by Krishna and Kumar (2013) and they have generated progressive type-II censored sample from generalized inverted exponential distribution. For simulation, different combinations of sample sizes are considered for ${X}$ and ${Y}$ are listed in Table 1.

To understand the notations given in Table 1, consider few cases for ${n=25}$ and ${m=15}$ as

(i)
$(R_{1},R_{2},\dots,R_{15})=(10,014)$

$\Rightarrow R_{1}=10\text{ and }R_{2}=R_{3}=\dots=R_{15}=0$ .
(ii)
$(R_{1},R_{2},\dots,R_{15})=(15,05,15)$

$\Rightarrow R_{1}=R_{2}=R_{3}=R_{4}=R_{5}=1,R_{6}=R_{7}=R_{8}=R_{9}=R_{10}=0,% \text{ and }R_{11}=R_{12}=R_{13}=R_{14}=R_{15}=1$ .
(iii)
$(R_{1},R_{2},\dots,R_{15})=(014,10)$

$\Rightarrow R_{1}=R_{2}=R_{3}=\dots=R_{14}=0\text{ and }R_{15}=10$ .

Table 1
Combinations of different sample sizes and their respective censoring scheme

$n_{1},n_{2}$ $m_{1},m_{2}$ Censoring Scheme (CS) ${(R}_{1},R_{2},\dots,R_{m_{1}}){(R}_{1},R_{2},\dots,R_{m_{2}})$ Censoring Scheme Number (CSN)

$n_{1}=n_{2}=25$ $m_{1}=m_{2}=15$ $(10,014)(10,014)$ [1]

$n_{1}=n_{2}=25$ $m_{1}=m_{2}=15$ $(15,05,15)(15,05,15)$ [2]

$n_{1}=n_{2}=25$ $m_{1}=m_{2}=15$ $(014,10)(014,10)$ [3]

$n_{1}=n_{2}=25$ $m_{1}=m_{2}=20$ $(5,019)(5,019)$ [4]

$n_{1}=n_{2}=25$ $m_{1}=m_{2}=20$ $(13,015,12)(13,015,12)$ [5]

$n_{1}=n_{2}=25$ $m_{1}=m_{2}=20$ $(019,5)(019,5)$ [6]

$n_{1}=n_{2}=40$ $m_{1}=m_{2}=24$ $(16,023)(16,023)$ [7]

$n_{1}=n_{2}=40$ $m_{1}=m_{2}=24$ $(18,08,18)(18,08,18)$ [8]

$n_{1}=n_{2}=40$ $m_{1}=m_{2}=24$ $(023,16)$ [9]

$n_{1}=n_{2}=40$ $m_{1}=m_{2}=32$ $(8,031)(8,031)$ [10]

$n_{1}=n_{2}=40$ $m_{1}=m_{2}=32$ $(14,024,14)(14,024,14)$ [11]

$n_{1}=n_{2}=40$ $m_{1}=m_{2}=32$ $(031,8)(031,8)$ [12]

$n_{1}={25,n}_{2}=40$ $m_{1}=15,m_{2}=24$ $(10,014)(16,023)$ [13]

$n_{1}={25,n}_{2}=40$ $m_{1}=15,m_{2}=24$ $(15,05,15)(18,08,18)$ [14]

$n_{1}={25,n}_{2}=40$ $m_{1}=15,m_{2}=24$ $(014,10)(023,16)$ [15]

$n_{1}={25,n}_{2}=40$ $m_{1}=20,m_{2}=32$ $(5,019)(8,031)$ [16]

$n_{1}={25,n}_{2}=40$ $m_{1}=20,m_{2}=32$ $(13,015,12)(14,024,14)$ [17]

$n_{1}={25,n}_{2}=40$ $m_{1}=20,m_{2}=32$ $(019,5)(031,8)$ [18]

MCMC technique of M-H algorithm used in which a chain of 15,000 observations is generated with 2000 burn-in period i.e., first 2000 observations are discarded as burn-in period from 15,000 observations. This burn-in period decided by cumulative mean plots and the lag value is 20 which is decided by autocorrelation plots i.e., after discarding first 2000 observations as burn-in period from 15,000 observations, take every 20 ${}^{\text{th}}$ observation. The whole procedure is replicated 3000 times.

Tables 2–4 represents mean square error (MSE), length of 95% asymptotic, boot-p and boot-t confidence intervals. The length of asymptotic confidence interval is larger than the others confidence intervals [Tables 2–4]. The length of asymptotic, boot-p and boot-t confidence intervals decreases as the proportion of failure ${m_{1}}/{n_{1}},{m_{2}}/{n_{2}}$ increases.

Tables 5–7 represents expected loss functions, 95% HPD credible intervals length for $\eta$ of Dagum distribution using informative and non-informative priors. As the proportion of failure increases, the expected loss function decreases as is seen from the Tables 5–7. The length of 95% HPD credible intervals decreases as the proportion of failure increases for the Bayes estimates of $\eta$ of Dagum distribution [Tables 5–7]. Also, the expected loss for informative prior is less than the expected loss for the non-informative prior [Tables 5–7]. As seen from the Tables 5–7, the length of credible intervals for gamma prior is less for non-informative prior.

Table 2
MSE and 95% asymptotic, boot-p and boot-t confidence intervals length for $\eta=0.25$

CSN MSE Asymptotic C.I. length Boot-p C.I. length Boot-t C.I. length

$[1]$ 0.00503 1.38546 0.2699 0.29453

$[2]$ 0.00418 1.72149 0.24346 0.29251

$[3]$ 0.00398 1.77019 0.23623 0.29818

$[4]$ 0.00367 0.63725 0.23338 0.25430

$[5]$ 0.00356 0.93224 0.22386 0.24448

$[6]$ 0.00310 1.04551 0.21883 0.24620

$[7]$ 0.00290 1.02206 0.21018 0.27757

$[8]$ 0.00242 1.09671 0.19019 0.27840

$[9]$ 0.00235 1.15724 0.18381 0.27423

$[10]$ 0.00218 0.45489 0.18369 0.23824

$[11]$ 0.00198 0.50524 0.17458 0.23969

$[12]$ 0.00195 0.75987 0.17109 0.23284

$[13]$ 0.00427 1.2084 0.24639 0.26806

$[14]$ 0.00370 0.52043 0.22669 0.26272

$[15]$ 0.00310 0.52108 0.22006 0.27332

$[16]$ 0.00291 0.51797 0.21257 0.22494

$[17]$ 0.00272 0.51696 0.20628 0.22435

$[18]$ 0.00269 0.51225 0.19977 0.2261

Table 3
MSE and 95% asymptotic, boot-p and boot-t confidence intervals length for $\eta=0.50$

CSN MSE Asymptotic C.I. length Boot-p C.I. length Boot-t C.I. length

$[1]$ 0.00777 0.58452 0.34012 0.17678

$[2]$ 0.00652 0.69045 0.31446 0.18027

$[3]$ 0.00595 0.74066 0.30503 0.17721

$[4]$ 0.00563 0.40065 0.29838 0.15318

$[5]$ 0.00576 0.45437 0.28817 0.15178

$[6]$ 0.00562 0.48931 0.28066 0.15537

$[7]$ 0.00511 0.51837 0.27318 0.17636

$[8]$ 0.00404 0.56672 0.24727 0.17180

$[9]$ 0.00367 0.59032 0.23935 0.16851

$[10]$ 0.00351 0.29657 0.23887 0.14577

$[11]$ 0.00333 0.34496 0.22846 0.14631

$[12]$ 0.00336 0.41154 0.22310 0.14406

$[13]$ 0.00631 0.48449 0.30873 0.17889

$[14]$ 0.00571 0.57182 0.28239 0.17923

$[15]$ 0.00502 0.61272 0.27377 0.17973

$[16]$ 0.00484 0.16449 0.26970 0.15161

$[17]$ 0.00498 0.27643 0.26004 0.15217

$[18]$ 0.00428 0.33791 0.25429 0.15258

Table 4
MSE and 95% asymptotic, boot-p and boot-t confidence intervals length for $\eta=0.75$

CSN MSE Asymptotic C.I. length Boot-p C.I. length Boot-t C.I. length

$[1]$ 0.00685 1.39021 0.26638 0.37913

$[2]$ 0.00792 1.55627 0.24645 0.43248

$[3]$ 0.00846 1.50131 0.23809 0.35570

$[4]$ 0.00327 1.02711 0.22939 0.26728

$[5]$ 0.00323 1.38998 0.22399 0.29707

$[6]$ 0.00510 1.44769 0.23082 0.24616

$[7]$ 0.00488 1.02127 0.21712 0.30527

$[8]$ 0.00386 1.49275 0.17160 0.29919

$[9]$ 0.00377 0.88043 0.22705 0.29852

$[10]$ 0.00301 0.54300 0.20641 0.23217

$[11]$ 0.00303 0.89172 0.19453 0.25274

$[12]$ 0.00119 0.41493 0.18486 0.17660

$[13]$ 0.00730 1.14190 0.23487 0.32407

$[14]$ 0.00311 1.38369 0.21376 0.31412

$[15]$ 0.00268 1.50240 0.20630 0.31933

$[16]$ 0.00263 0.79476 0.20495 0.26910

$[17]$ 0.00246 1.32908 0.19682 0.27447

$[18]$ 0.00363 1.50251 0.23147 0.23984

Table 5
Expected loss functions and length of HPD credible intervals for $\eta=0.25$

Informative prior Non-informative prior

CSN SELF GELF $b=+0.5$ GELF $b=-0.5$ HPD length SELF GELF $b=+0.5$ GELF $b=-0.5$ HPD length

$[1]$ 0.00930 0.00765 0.00711 0.19347 0.09325 0.08562 0.08829 0.21791

$[2]$ 0.00835 0.00567 0.00532 0.19889 0.09346 0.07373 0.08703 0.21785

$[3]$ 0.00849 0.00571 0.00544 0.19246 0.08873 0.07708 0.07381 0.21047

$[4]$ 0.00655 0.00420 0.00505 0.17531 0.07147 0.05922 0.06069 0.19868

$[5]$ 0.00761 0.00613 0.00564 0.17098 0.06876 0.06151 0.05924 0.19068

$[6]$ 0.00552 0.00215 0.00292 0.17994 0.06813 0.05543 0.05362 0.20478

$[7]$ 0.00463 0.00389 0.00373 0.15497 0.07291 0.05313 0.05655 0.17094

$[8]$ 0.00487 0.00376 0.00361 0.15293 0.06371 0.04685 0.04873 0.16985

$[9]$ 0.00428 0.00399 0.00387 0.14747 0.06595 0.04553 0.04569 0.16203

$[10]$ 0.00349 0.00201 0.00337 0.14076 0.05301 0.04304 0.04379 0.15265

$[11]$ 0.00340 0.00329 0.00322 0.13561 0.05671 0.03902 0.04170 0.14683

$[12]$ 0.00404 0.00358 0.00334 0.14282 0.05997 0.04079 0.04217 0.15841

$[13]$ 0.00632 0.00577 0.00549 0.17611 0.09418 0.06724 0.06827 0.19137

$[14]$ 0.00551 0.00467 0.00448 0.17607 0.08944 0.05894 0.06629 0.19211

$[15]$ 0.00541 0.00466 0.00452 0.16807 0.07897 0.05778 0.06416 0.18293

$[16]$ 0.00488 0.00431 0.00415 0.16192 0.06979 0.04635 0.05495 0.17399

$[17]$ 0.00432 0.00422 0.00368 0.15512 0.05974 0.05632 0.04762 0.16776

$[18]$ 0.00399 0.00354 0.00346 0.16098 0.06382 0.05604 0.05194 0.17762

Table 6
Expected loss functions and length of HPD credible intervals for $\eta=0.50$

Informative prior Non-informative prior

CSN SELF GELF $b=+0.5$ GELF $b=-0.5$ HPD length SELF GELF $b=+0.5$ GELF $b=-0.5$ HPD length

$[1]$ 0.00416 0.00118 0.00116 025111 0.07373 0.09499 0.08144 0.26381

$[2]$ 0.00328 0.00115 0.00113 0.24538 0.09600 0.09876 0.08883 0.24821

$[3]$ 0.00341 0.00118 0.00118 0.23910 0.08874 0.08733 0.08774 0.24007

$[4]$ 0.00319 0.00110 0.00109 0.22899 0.06997 0.07804 0.07482 0.23928

$[5]$ 0.00315 0.00109 0.00107 0.21960 0.07028 0.08323 0.07837 0.22502

$[6]$ 0.00284 0.00103 0.00101 0.23535 0.07013 0.08506 0.07663 0.22210

$[7]$ 0.00301 0.00901 0.00911 0.19788 0.07120 0.09481 0.09240 0.20498

$[8]$ 0.00268 0.00894 0.00894 0.20048 0.06400 0.08918 0.08754 0.22928

$[9]$ 0.00267 0.00911 0.00902 0.19993 0.06881 0.09575 0.09154 0.20302

$[10]$ 0.00232 0.00802 0.00794 0.18180 0.06091 0.08870 0.08324 0.17889

$[11]$ 0.00245 0.00828 0.00822 0.17652 0.05953 0.08167 0.08569 0.18423

$[12]$ 0.00225 0.00757 0.00756 0.18887 0.06150 0.07939 0.08154 0.19117

$[13]$ 0.00393 0.00158 0.00157 0.21884 0.07105 0.08257 0.08357 0.22928

$[14]$ 0.00352 0.00136 0.00135 0.22389 0.07511 0.07882 0.09508 0.23981

$[15]$ 0.00336 0.00129 0.00128 0.21730 0.07819 0.08768 0.08741 0.22370

$[16]$ 0.00321 0.00129 0.00127 0.20389 0.07022 0.07217 0.08119 0.20895

$[17]$ 0.00284 0.00110 0.00108 0.19903 0.06972 0.07488 0.07850 0.20440

$[18]$ 0.00275 0.00106 0.00104 0.21331 0.06684 0.07449 0.07987 0.21753

Table 7
Expected loss functions and length of HPD credible intervals for $\eta=0.75$

Informative prior Non-informative prior

CSN SELF GELF $b=+0.5$ GELF $b=-0.5$ HPD length SELF GELF $b=+0.5$ GELF $b=-0.5$ HPD length

$[1]$ 0.00865 0.00893 0.00849 0.19033 0.08144 0.07159 0.08224 0.21797

$[2]$ 0.00866 0.00813 0.00863 0.17410 0.08503 0.06962 0.07604 0.19849

$[3]$ 0.00797 0.00871 0.00878 0.16858 0.08774 0.07432 0.08346 0.19158

$[4]$ 0.00787 0.00782 0.00804 0.19448 0.08005 0.06412 0.06424 0.21712

$[5]$ 0.00731 0.00746 0.00777 0.18737 0.07934 0.05975 0.07586 0.21011

$[6]$ 0.00760 0.00722 0.00752 0.17680 0.07169 0.06757 0.06268 0.20620

$[7]$ 0.00860 0.00915 0.00993 0.15634 0.07156 0.08093 0.07816 0.17095

$[8]$ 0.00788 0.00883 0.00890 0.14036 0.06918 0.07503 0.07624 0.15313

$[9]$ 0.00767 0.00786 0.00846 0.13524 0.06742 0.07634 0.06733 0.14701

$[10]$ 0.00738 0.00754 0.00700 0.15328 0.05821 0.07391 0.06453 0.16960

$[11]$ 0.00756 0.00739 0.00674 0.14755 0.05780 0.06943 0.06074 0.16170

$[12]$ 0.00700 0.00714 0.00606 0.14409 0.06350 0.06856 0.05756 0.15740

$[13]$ 0.00942 0.00958 0.00970 0.17668 0.07052 0.08917 0.07967 0.19843

$[14]$ 0.00892 0.00998 0.00919 0.15900 0.07259 0.06284 0.08312 0.17934

$[15]$ 0.00799 0.00950 0.00900 0.15149 0.06513 0.06619 0.07140 0.17280

$[16]$ 0.00724 0.00850 0.00954 0.17937 0.06252 0.05813 0.06822 0.19708

$[17]$ 0.00760 0.00904 0.00895 0.17088 0.06341 0.04820 0.06233 0.18987

$[18]$ 0.00773 0.00742 0.00764 0.16500 0.05840 0.05016 0.05923 0.18531

Results:

•
In case of MLE of $\eta$ , the length of asymptotic confidence interval is larger than the boot-p and boot-t confidence intervals. The length of all the three confidence intervals decreases as the proportion of failure ${m_{1}}/{n_{1}},{m_{2}}/{n_{2}}$ increases.
•
While deriving Bayes estimates of $\eta$ , the expected loss for informative prior is less than the expected loss for the non-informative prior.
•
Also in case of Bayesian estimation, the length of credible intervals for gamma prior is less than in case of non-informative prior. The length of 95% HPD credible intervals decreases as the proportion of failure increases. As the proportion of failure increases, the expected loss function decreases.

Real data analysis

In this Section, a pair of real data set reported by Badar and Priest (1982) is considered for illustration purpose. The data represent the strength measured in GPA of single-carbon fibres and impregnated 1000-carbon fibre tows. Single-carbon fibres were tested under tension at gauge lengths of 1, 10, 20, and 50 mm. Impregnated tows of 1000-carbon fibres were tested at gauge lengths of 20, 50, 150, and 300 mm. Kundu and Gupta (2006) considered the strength data of single fibres of 10 and 20 mm in gauge length, with sample sizes 63 and 69 after subtracting 0.75 from both the data sets, respectively for Weibull distribution in complete sample case. The transformed data sets are given below:

Gauge length 10 mm (X): 1.151, 1.382, 1.453, 1.478, 1.507, 1.600, 1.611, 1.646, 1.647, 1.695, 1.704, 1.724, 1.768, 1.772, 1.775, 1.782, 1.825, 1.864, 1.866, 1.868, 1.874, 1.909, 1.925, 1.988, 1.990, 2.106, 2.167, 2.178, 2.187, 2.187, 2.227, 2.246, 2.280, 2.375, 2.389, 2.395, 2.470, 2.473, 2.485, 2.493, 2.514, 2.522, 2.544, 2.582, 2.596, 2.627, 2.658, 2.685, 2.743, 2.751, 2.787, 2.804, 2.812, 2.878, 3.102, 3.121, 3.136, 3.221, 3.274, 3.277, 3.475, 3.645, 4.270. Gauge length 20 mm (Y): 0.562, 0.564, 0.729, 0.802, 0.950, 1.053, 1.111, 1.115, 1.194, 1.208, 1.216, 1.247, 1.256, 1.271, 1.277, 1.305, 1.313, 1.348, 1.390, 1.429, 1.474, 1.490, 1.503, 1.520, 1.522, 1.524, 1.551, 1.551, 1.609, 1.632, 1.632, 1.676, 1.684, 1.685, 1.728, 1.740, 1.761, 1.764, 1.785, 1.804, 1.816, 1.820, 1.836, 1.879, 1.883, 1.892, 1.898, 1.934, 1.947, 1.976, 2.020, 2.023, 2.050, 2.059, 2.068, 2.071, 2.098, 2.130, 2.204, 2.262, 2.317, 2.334, 2.340, 2.346, 2.378, 2.483, 2.683, 2.835, 2.835.

Kolmogorov-Smirnov (K–S) test is used for each data set to fit the model. It is observed that for data sets $X$ and $Y$ , the K-S test statistic values are 0.08 and 0.10 with corresponding $p$ -values 0.8124 and 0.4142, respectively. It shows that data sets $X$ and $Y$ fit to the model Dagum (5.015335, 4, 4.2) and Dagum (1.62341, 4, 4.2), respectively and the corresponding stress strength reliability is $\eta=$ 0.7554643.

Now, we generate progressive type-II censored samples from the above data sets using R.3.5.2. software following the procedure discussed by Krishna and Kumar (2013). Different types of progressive type-II censoring schemes are considered for real data sets $X$ and $Y$ [Table 8].

Using M-H algorithm, a chain of 15000 observations is generated for the above real data sets. Also the convergence and randomness of observations has been checked by cumulative mean plot and trace plot respectively. The burn-in period in 2000 with a lag value is 20.

Table 8
Different censoring scheme for real data where $n_{1},m_{1}$ are corresponding to $X$ and $n_{2},m_{2}$ are corresponding to $Y$

$n_{1},n_{2}$ $m_{1},m_{2}$ Censoring Scheme (CS) ${(R}_{1},R_{2},\dots,R_{m_{1}}){(R}_{1},R_{2},\dots,R_{m_{2}})$ Censoring Scheme Number (CSN)

$n_{1}={63,n}_{2}=69$ $m_{1}=38,m_{2}=41$ $(25,037)(28,040)$ [1]

$(113,013,112)(114,013,114)$ [2]

$(013,25)(040,28)$ [3]

$m_{1}=50,m_{2}=55$ $(13,049)(14,054)$ [4]

$(17,037,16)(17,041,17)$ [5]

$(049,13)(054,14)$ [6]

Table 9
MSE and 95% asymptotic, boot-p and boot-t confidence intervals length, expected loss functions and length of HPD credible intervals of $\eta$ for real data sets

CSN MSE Asymptotic C.I. length Boot-p C.I. length Boot-t C.I. length SELF GELF $b=+0.5$ GELF $b=-0.5$ HPD length

$[1]$ 0.00798 1.26259 0.35817 0.34929 0.00444 0.00060 0.00049 0.14192

$[2]$ 0.00729 1.66637 0.31143 0.37586 0.00724 0.00091 0.00089 0.14426

$[3]$ 0.00915 1.48911 0.33750 0.37990 0.00567 0.00045 0.00042 0.13879

$[4]$ 0.00418 1.04051 0.25050 0.32237 0.00263 0.00028 0.00022 0.13810

$[5]$ 0.00383 1.13915 0.30030 0.28450 0.00326 0.00035 0.00035 0.11778

$[6]$ 0.00481 1.26372 0.20189 0.21127 0.00353 0.00027 0.00023 0.12179

The classical as well as Bayesian estimation of $\eta$ is done for this real data set and the results are shown in Table 9. As we have no prior information, we apply only the non-informative priors for obtaining expected loss functions for the Bayesian estimates for this real data set and hence no hyper-parameters are considered in this case. The length of asymptotic confidence interval is larger than the others confidence intervals [Table 9]. We can conclude from Table 9, length of asymptotic, boot-p and boot-t confidence intervals decreases as the proportion of failure $({m_{1}}/{n_{1}},{m_{2}}/{n_{2}})$ increases. The length of 95% HPD credible intervals decreases as the proportions of failure increases for the Bayes estimates of $\eta$ of Dagum distribution [Table 9]. As the proportions of failure increases, the expected loss function decreases as is seen from the Table 9. The results are in conformity with results obtained in case of simulation study.
6. Conclusions

$n_{1},n_{2}$	$m_{1},m_{2}$	Censoring Scheme (CS) ${(R}_{1},R_{2},\dots,R_{m_{1}}){(R}_{1},R_{2},\dots,R_{m_{2}})$	Censoring Scheme Number (CSN)
$n_{1}=n_{2}=25$	$m_{1}=m_{2}=15$	$(10,014)(10,014)$	[1]
$n_{1}=n_{2}=25$	$m_{1}=m_{2}=15$	$(15,05,15)(15,05,15)$	[2]
$n_{1}=n_{2}=25$	$m_{1}=m_{2}=15$	$(014,10)(014,10)$	[3]
$n_{1}=n_{2}=25$	$m_{1}=m_{2}=20$	$(5,019)(5,019)$	[4]
$n_{1}=n_{2}=25$	$m_{1}=m_{2}=20$	$(13,015,12)(13,015,12)$	[5]
$n_{1}=n_{2}=25$	$m_{1}=m_{2}=20$	$(019,5)(019,5)$	[6]
$n_{1}=n_{2}=40$	$m_{1}=m_{2}=24$	$(16,023)(16,023)$	[7]
$n_{1}=n_{2}=40$	$m_{1}=m_{2}=24$	$(18,08,18)(18,08,18)$	[8]
$n_{1}=n_{2}=40$	$m_{1}=m_{2}=24$	$(0*23,16)$	[9]
$n_{1}=n_{2}=40$	$m_{1}=m_{2}=32$	$(8,031)(8,031)$	[10]
$n_{1}=n_{2}=40$	$m_{1}=m_{2}=32$	$(14,024,14)(14,024,14)$	[11]
$n_{1}=n_{2}=40$	$m_{1}=m_{2}=32$	$(031,8)(031,8)$	[12]
$n_{1}={25,n}_{2}=40$	$m_{1}=15,m_{2}=24$	$(10,014)(16,023)$	[13]
$n_{1}={25,n}_{2}=40$	$m_{1}=15,m_{2}=24$	$(15,05,15)(18,08,18)$	[14]
$n_{1}={25,n}_{2}=40$	$m_{1}=15,m_{2}=24$	$(014,10)(023,16)$	[15]
$n_{1}={25,n}_{2}=40$	$m_{1}=20,m_{2}=32$	$(5,019)(8,031)$	[16]
$n_{1}={25,n}_{2}=40$	$m_{1}=20,m_{2}=32$	$(13,015,12)(14,024,14)$	[17]
$n_{1}={25,n}_{2}=40$	$m_{1}=20,m_{2}=32$	$(019,5)(031,8)$	[18]

CSN	MSE	Asymptotic C.I. length	Boot-p C.I. length	Boot-t C.I. length
$[1]$	0.00503	1.38546	0.2699	0.29453
$[2]$	0.00418	1.72149	0.24346	0.29251
$[3]$	0.00398	1.77019	0.23623	0.29818
$[4]$	0.00367	0.63725	0.23338	0.25430
$[5]$	0.00356	0.93224	0.22386	0.24448
$[6]$	0.00310	1.04551	0.21883	0.24620
$[7]$	0.00290	1.02206	0.21018	0.27757
$[8]$	0.00242	1.09671	0.19019	0.27840
$[9]$	0.00235	1.15724	0.18381	0.27423
$[10]$	0.00218	0.45489	0.18369	0.23824
$[11]$	0.00198	0.50524	0.17458	0.23969
$[12]$	0.00195	0.75987	0.17109	0.23284
$[13]$	0.00427	1.2084	0.24639	0.26806
$[14]$	0.00370	0.52043	0.22669	0.26272
$[15]$	0.00310	0.52108	0.22006	0.27332
$[16]$	0.00291	0.51797	0.21257	0.22494
$[17]$	0.00272	0.51696	0.20628	0.22435
$[18]$	0.00269	0.51225	0.19977	0.2261

CSN	MSE	Asymptotic C.I. length	Boot-p C.I. length	Boot-t C.I. length
$[1]$	0.00777	0.58452	0.34012	0.17678
$[2]$	0.00652	0.69045	0.31446	0.18027
$[3]$	0.00595	0.74066	0.30503	0.17721
$[4]$	0.00563	0.40065	0.29838	0.15318
$[5]$	0.00576	0.45437	0.28817	0.15178
$[6]$	0.00562	0.48931	0.28066	0.15537
$[7]$	0.00511	0.51837	0.27318	0.17636
$[8]$	0.00404	0.56672	0.24727	0.17180
$[9]$	0.00367	0.59032	0.23935	0.16851
$[10]$	0.00351	0.29657	0.23887	0.14577
$[11]$	0.00333	0.34496	0.22846	0.14631
$[12]$	0.00336	0.41154	0.22310	0.14406
$[13]$	0.00631	0.48449	0.30873	0.17889
$[14]$	0.00571	0.57182	0.28239	0.17923
$[15]$	0.00502	0.61272	0.27377	0.17973
$[16]$	0.00484	0.16449	0.26970	0.15161
$[17]$	0.00498	0.27643	0.26004	0.15217
$[18]$	0.00428	0.33791	0.25429	0.15258

CSN	MSE	Asymptotic C.I. length	Boot-p C.I. length	Boot-t C.I. length
$[1]$	0.00685	1.39021	0.26638	0.37913
$[2]$	0.00792	1.55627	0.24645	0.43248
$[3]$	0.00846	1.50131	0.23809	0.35570
$[4]$	0.00327	1.02711	0.22939	0.26728
$[5]$	0.00323	1.38998	0.22399	0.29707
$[6]$	0.00510	1.44769	0.23082	0.24616
$[7]$	0.00488	1.02127	0.21712	0.30527
$[8]$	0.00386	1.49275	0.17160	0.29919
$[9]$	0.00377	0.88043	0.22705	0.29852
$[10]$	0.00301	0.54300	0.20641	0.23217
$[11]$	0.00303	0.89172	0.19453	0.25274
$[12]$	0.00119	0.41493	0.18486	0.17660
$[13]$	0.00730	1.14190	0.23487	0.32407
$[14]$	0.00311	1.38369	0.21376	0.31412
$[15]$	0.00268	1.50240	0.20630	0.31933
$[16]$	0.00263	0.79476	0.20495	0.26910
$[17]$	0.00246	1.32908	0.19682	0.27447
$[18]$	0.00363	1.50251	0.23147	0.23984

	Informative prior	Non-informative prior
$[1]$	0.00930	0.00765	0.00711	0.19347	0.09325	0.08562	0.08829	0.21791
$[2]$	0.00835	0.00567	0.00532	0.19889	0.09346	0.07373	0.08703	0.21785
$[3]$	0.00849	0.00571	0.00544	0.19246	0.08873	0.07708	0.07381	0.21047
$[4]$	0.00655	0.00420	0.00505	0.17531	0.07147	0.05922	0.06069	0.19868
$[5]$	0.00761	0.00613	0.00564	0.17098	0.06876	0.06151	0.05924	0.19068
$[6]$	0.00552	0.00215	0.00292	0.17994	0.06813	0.05543	0.05362	0.20478
$[7]$	0.00463	0.00389	0.00373	0.15497	0.07291	0.05313	0.05655	0.17094
$[8]$	0.00487	0.00376	0.00361	0.15293	0.06371	0.04685	0.04873	0.16985
$[9]$	0.00428	0.00399	0.00387	0.14747	0.06595	0.04553	0.04569	0.16203
$[10]$	0.00349	0.00201	0.00337	0.14076	0.05301	0.04304	0.04379	0.15265
$[11]$	0.00340	0.00329	0.00322	0.13561	0.05671	0.03902	0.04170	0.14683
$[12]$	0.00404	0.00358	0.00334	0.14282	0.05997	0.04079	0.04217	0.15841
$[13]$	0.00632	0.00577	0.00549	0.17611	0.09418	0.06724	0.06827	0.19137
$[14]$	0.00551	0.00467	0.00448	0.17607	0.08944	0.05894	0.06629	0.19211
$[15]$	0.00541	0.00466	0.00452	0.16807	0.07897	0.05778	0.06416	0.18293
$[16]$	0.00488	0.00431	0.00415	0.16192	0.06979	0.04635	0.05495	0.17399
$[17]$	0.00432	0.00422	0.00368	0.15512	0.05974	0.05632	0.04762	0.16776
$[18]$	0.00399	0.00354	0.00346	0.16098	0.06382	0.05604	0.05194	0.17762

	Informative prior	Non-informative prior
$[1]$	0.00416	0.00118	0.00116	025111	0.07373	0.09499	0.08144	0.26381
$[2]$	0.00328	0.00115	0.00113	0.24538	0.09600	0.09876	0.08883	0.24821
$[3]$	0.00341	0.00118	0.00118	0.23910	0.08874	0.08733	0.08774	0.24007
$[4]$	0.00319	0.00110	0.00109	0.22899	0.06997	0.07804	0.07482	0.23928
$[5]$	0.00315	0.00109	0.00107	0.21960	0.07028	0.08323	0.07837	0.22502
$[6]$	0.00284	0.00103	0.00101	0.23535	0.07013	0.08506	0.07663	0.22210
$[7]$	0.00301	0.00901	0.00911	0.19788	0.07120	0.09481	0.09240	0.20498
$[8]$	0.00268	0.00894	0.00894	0.20048	0.06400	0.08918	0.08754	0.22928
$[9]$	0.00267	0.00911	0.00902	0.19993	0.06881	0.09575	0.09154	0.20302
$[10]$	0.00232	0.00802	0.00794	0.18180	0.06091	0.08870	0.08324	0.17889
$[11]$	0.00245	0.00828	0.00822	0.17652	0.05953	0.08167	0.08569	0.18423
$[12]$	0.00225	0.00757	0.00756	0.18887	0.06150	0.07939	0.08154	0.19117
$[13]$	0.00393	0.00158	0.00157	0.21884	0.07105	0.08257	0.08357	0.22928
$[14]$	0.00352	0.00136	0.00135	0.22389	0.07511	0.07882	0.09508	0.23981
$[15]$	0.00336	0.00129	0.00128	0.21730	0.07819	0.08768	0.08741	0.22370
$[16]$	0.00321	0.00129	0.00127	0.20389	0.07022	0.07217	0.08119	0.20895
$[17]$	0.00284	0.00110	0.00108	0.19903	0.06972	0.07488	0.07850	0.20440
$[18]$	0.00275	0.00106	0.00104	0.21331	0.06684	0.07449	0.07987	0.21753

	Informative prior	Non-informative prior
$[1]$	0.00865	0.00893	0.00849	0.19033	0.08144	0.07159	0.08224	0.21797
$[2]$	0.00866	0.00813	0.00863	0.17410	0.08503	0.06962	0.07604	0.19849
$[3]$	0.00797	0.00871	0.00878	0.16858	0.08774	0.07432	0.08346	0.19158
$[4]$	0.00787	0.00782	0.00804	0.19448	0.08005	0.06412	0.06424	0.21712
$[5]$	0.00731	0.00746	0.00777	0.18737	0.07934	0.05975	0.07586	0.21011
$[6]$	0.00760	0.00722	0.00752	0.17680	0.07169	0.06757	0.06268	0.20620
$[7]$	0.00860	0.00915	0.00993	0.15634	0.07156	0.08093	0.07816	0.17095
$[8]$	0.00788	0.00883	0.00890	0.14036	0.06918	0.07503	0.07624	0.15313
$[9]$	0.00767	0.00786	0.00846	0.13524	0.06742	0.07634	0.06733	0.14701
$[10]$	0.00738	0.00754	0.00700	0.15328	0.05821	0.07391	0.06453	0.16960
$[11]$	0.00756	0.00739	0.00674	0.14755	0.05780	0.06943	0.06074	0.16170
$[12]$	0.00700	0.00714	0.00606	0.14409	0.06350	0.06856	0.05756	0.15740
$[13]$	0.00942	0.00958	0.00970	0.17668	0.07052	0.08917	0.07967	0.19843
$[14]$	0.00892	0.00998	0.00919	0.15900	0.07259	0.06284	0.08312	0.17934
$[15]$	0.00799	0.00950	0.00900	0.15149	0.06513	0.06619	0.07140	0.17280
$[16]$	0.00724	0.00850	0.00954	0.17937	0.06252	0.05813	0.06822	0.19708
$[17]$	0.00760	0.00904	0.00895	0.17088	0.06341	0.04820	0.06233	0.18987
$[18]$	0.00773	0.00742	0.00764	0.16500	0.05840	0.05016	0.05923	0.18531

$n_{1},n_{2}$	$m_{1},m_{2}$	Censoring Scheme (CS) ${(R}_{1},R_{2},\dots,R_{m_{1}}){(R}_{1},R_{2},\dots,R_{m_{2}})$	Censoring Scheme Number (CSN)
$n_{1}={63,n}_{2}=69$	$m_{1}=38,m_{2}=41$	$(25,037)(28,040)$	[1]
		$(113,013,112)(114,013,114)$	[2]
		$(013,25)(040,28)$	[3]
	$m_{1}=50,m_{2}=55$	$(13,049)(14,054)$	[4]
		$(17,037,16)(17,041,17)$	[5]
		$(049,13)(054,14)$	[6]

CSN	MSE	Asymptotic C.I. length	Boot-p C.I. length	Boot-t C.I. length	SELF	GELF $b=+0.5$	GELF $b=-0.5$	HPD length
$[1]$	0.00798	1.26259	0.35817	0.34929	0.00444	0.00060	0.00049	0.14192
$[2]$	0.00729	1.66637	0.31143	0.37586	0.00724	0.00091	0.00089	0.14426
$[3]$	0.00915	1.48911	0.33750	0.37990	0.00567	0.00045	0.00042	0.13879
$[4]$	0.00418	1.04051	0.25050	0.32237	0.00263	0.00028	0.00022	0.13810
$[5]$	0.00383	1.13915	0.30030	0.28450	0.00326	0.00035	0.00035	0.11778
$[6]$	0.00481	1.26372	0.20189	0.21127	0.00353	0.00027	0.00023	0.12179

This paper deals with the estimation of stress strength reliability $\eta=P(Y<X)$ of Dagum distribution under progressive type-II censored samples. We considered that the distribution of $X$ and $Y$ is Dagum having different unknown shape parameters ${\beta}_{1},{\beta}_{2}$ . Classical as well as Bayesian estimation of $\eta$ of Dagum distribution under progressive type-II censored sample is considered. Maximum likelihood estimators of $\eta$ are obtained with asymptotic, boot-p and boot-t confidence intervals. Bayes estimators of $\eta$ with HPD credible intervals is based on informative and non-informative priors. Monte Carlo simulation study is also carried out to check the performance of estimators.

From the obtained results, it can be seen that the length of asymptotic confidence intervals is larger than the others confidence intervals [Tables 2–4]. The length of asymptotic, boot-p and boot-t confidence intervals decreases as the proportion of failure $({m_{1}}/{n_{1}},{m_{2}}/{n_{2}})$ increases. It can be concluded that boot-p and boot-t confidence intervals are perform better that asymptotic confidence intervals.

As the proportions of failure increases, the expected loss function decreases as is seen from the Tables 5–7. The length of 95% HPD credible intervals decreases as the proportion of failure increases for the Bayes estimates of $\eta$ of Dagum distribution [Tables 5–7]. Also the expected loss for informative prior is less than the expected loss for the non-informative prior [Tables 5–7]. As seen from the Tables 5–7, the length of credible intervals for gamma prior is less than for non-informative prior. The results obtained in case of simulation do hold in case of real life data set as well [Table 9].Gamma prior provides better result than non-informative prior.

Footnotes

Acknowledgments

The authors are thankful to the anonymous reviewers and the editor for their valuable suggestions and comments which has led to an improvement in the manuscript. We also like to acknowledge with thanks the financial assistance provided by the UGC, New Delhi, India. All the authors also acknowledge the support provided by DST under PURSE grant.

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