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E-Bayesian and Hierarchical Bayesian Estimation of Rayleigh Distribution Parameter with Type-II Censoring from Imprecise Data


Bayesian estimation methods for Rayleigh distribution parameter affect accurate information. However, in real-world conditions, empirical performance results cannot always be recorded or measured accurately. Thus, we'd like to generalize the estimated methods for real numbers to fuzzy numbers. during this paper, Bayesian, E-Bayesian and Hierarchical Bayesian (H-Bayesian) methods are discussed for Rayleigh distribution parameter on the idea of a Type-II censoring schemes under the squared error loss function. Data is taken into account as imprecise and within the form fuzzy numbers. Then, the efficiency of estimation methods is compared via Monte Carlo simulation. Finally, a true data set for the needs described is analyzed.


Rayleigh distribution was firstly introduced by Rayleigh (1880) within the field of acoustics; since its preface, numerous investigators have used Rayleigh distribution in colorful fields of wisdom and technology. Currently, Rayleigh distribution is extensively employed in statistical model, survival analysis and reliability. Rayleigh distribution is that the foundation of important of the treatment of meteorological radar signals statistics. Also, it's frequently applied in actuarial science and in engineering work to model population lifetimes whose failure rate increases linearly. Several feathers of electro-vacuum devices have this point (Polovko 1968). This statistical law is also generally used for fitting the wind generation and speed measured at a given position over a particular period (Celik 2004; Jowder 2006). turbine manufacturers frequently give standard performance numbers for his or her machines using Rayleigh distribution (Arifujjaman et al. 2008). Operations in meteorology and clinical studies are frequently plant in Marshall and Hitschfeld (1953) and Gross and Clark (1975), respectively. Also, Brillinger (1982) showed that the time between consecutive earthquakes could also be modeled with a Rayleigh law. also, it's been used to describe the ultrasound echo signal when there's a sufficient number of independent roughly original handful (Tuthill et al. 1988; Zagzebski et al. 1999). Likewise, the distribution of surge heights and ridges could be Rayleigh (Tayfuna and Fedeleb 2007). Other aspects of this model are treated in Sinha and Howlader (1983), Howlader (1985), Lalitha & Mishra (1996), Fernández (2000), Aslam (2007), Dey and Das (2007) and Liang (2007).

In this paper, the probability density function (pdf) and cumulative distribution function (cdf) of Rayleigh distribution specified by

$$\begin{array}{*{20}c} {f\left( {x;\sigma } \right) = 2\alpha xe^{{ - \alpha x^{2} }} ,} & {x > 0; \alpha > 0} \\ \end{array}$$
$$F\left( {x;\sigma } \right) = 1 - e^{{ - \alpha x^{2} }} , x > 0; \alpha > 0.$$

H-Bayesian prior distribution was primarily introduced by Lindley and Smith (1972). Han (1997) developed styles to construct hierarchical prior distribution, and E-Bayesian and H-Bayesian techniques were introduced. Lately, E-Bayesian and H-Bayesian techniques are employed by Han (2011) to estimate exponential distribution parameter and Bernoulli distribution reliability estimation, by Jaheen and Okasha (2011) to estimate of reliability of the kind 12 distribution supported Type II progressive censoring samples, by Wang et al. (2012) and Yousefzadeh (2017) to estimate of Pascal distribution parameters. Also, Han (2017) gives the property of E-Bayesian estimation and H-Bayesian estimation of the system reliability parameter.

The researcher might not always gain complete information on failure times for all experimental units. Data attained from similar trials is named censored data. Censoring may be a common issue in lifetime data and reliability researches. There are numerous feathers of censoring like Type-II censoring, twice Type-II censoring, Type-II progressive censoring and random censorship. One among the foremost current censoring schemes is Type II (failure) censoring, where the life testing trial is ended upon the r^th(ris pre-fixed) failure. For farther information, see Cohen (1963) and Balakrishnan and Cohen (1991). The content of Type II censoring has attracted the eye of the numerous authors, a number of which, like Ng et al. (2006), used it to estimate the purpose and distance of the distribution parameters of Birnbaum-Saunders. Singh and Kumar (2007) used it for Bayesian method of exponential distribution parameters, Iliopoulos and Balakrishnan (2011) used it in Laplace distribution, and Kundu and Raqab (2012) used it for Bayesian conclusion of Weibull distribution parameters.

Some experimenters have used fuzzy sets in theory estimation. Coppi et al. (2006) stated some operations of Bayesian styles in statistical analysis. Huang et al. (2006) proposed a replacement method for determining the membership function of parameters and thus the reliability of multi-parametric life distributions. Akbari and Rezaei (2007) proposed a replacement method for estimating fuzzy spot for uniformly minimum variance. Pak et al. (2013) conducted wide studies on deducible procedures for all times distributions supported fuzzy numbers. Then, some applicable definitions are reviewed. The end of this text is to develop the estimation methods for Rayleigh distribution under a Type-II censoring scheme when the available observations are fuzzy data. Bayesian, E-Bayesian and Hierarchical Bayesian estimate of the parameter α under the squared error loss function is obtained. A Monte Carlo simulation study is presented, and a comparison of all estimation methods developed and one real data set is analyzed. The efficiency of estimation methods is compared via Monte Carlo simulation and a true data set is analyzed.

First, the notation and basic definitions of fuzzy set used herein is reviewed. Definitions 15, was firstly proposed by Zadeh (1968), which is defined as:

Definition 1

If \(X\) may be a reference set, likewise a fuzzy set of \(\tilde{A }\) is shown in using the membership function of \({\mu }_{\tilde{A }}\left(x\right):R\to [\mathrm{0,1}]\) during which any \(x\) in \(X\) show \({\mu }_{\tilde{A }}\left(x\right)\) membership degree of \(x\) during \(\tilde{A }\).

Definition 2

Fuzzy set of \(\tilde{A}\) in reference set of \(X\) is normal when and only \(sup_{x \in X} \mu_{{\tilde{A}}} \left( x \right) = 1.\)

Definition 3

The fuzzy set of \(\tilde{A}\) in the reference set of \(X\) is convex, when and if per \(x,y \in X\) and \(\lambda \in \left[ {0,1} \right]\), the following equation is established:

$${\upmu }_{{{\tilde{\text{A}}}}} \left( {{\lambda x} + \left( {1 - {\uplambda }} \right){\text{y}}} \right) \ge inf\left\{ {{\upmu }_{{{\tilde{\text{A}}}}} \left( {\text{x}} \right),{\upmu }_{{{\tilde{\text{A}}}}} \left( {\text{y}} \right)} \right\}$$

Definition 4

If \({\text{X}}\) may be a reference set, likewise the fuzzy set of \({\text{X}}\) may be a fuzzy number if \({\text{X}}\) in \({\text{X}}\) be normal and convex.

Definition 5

Assuming \({\text{L}}:{\text{R}}^{ + } \to \left[ {0,1} \right]\) and \({\text{R}}:{\text{R}}^{ + } \to \left[ {0,1} \right]\) are two continuous functions with the subsequent specifications:

$$L\left( { - x} \right) = L\left( x \right), R\left( { - x} \right) = R\left( x \right)$$
$$L\left( 0 \right) = R\left( 0 \right) = 1$$
$$\mathop {\lim }\limits_{x \to \infty } L\left( x \right) = \mathop {\lim }\limits_{x \to \infty } R\left( x \right) = 0$$

and L and R on \(\left[ {0,\infty } \right)\) is subtractive, likewise the fuzzy number of \(\tilde{\user2{M}}\) may be a quite \({\mathbf{LR}}\) when and if the following equation is established:

$$\mu_{{\tilde{M}}} \left( x \right) = \left\{ {\begin{array}{*{20}c} {L\left( {\frac{m - x}{\alpha }} \right) } & {x \le m, \alpha > 0} \\ {R\left( {\frac{m - x}{\beta }} \right) } & {x \ge m, \beta > 0} \\ \end{array} } \right.$$

where m is that the average of the fuzzy number of \(\tilde{M},\alpha\) and \(\beta\) are the left and right bounds. Also, the fuzzy number of LR type is shown by \(\tilde{M} = \left( {m,\alpha ,\beta } \right)\).

Suppose that \(n\) independent units are placed on a life test with the corresponding life times \(X_{1} ,X_{2} , \ldots ,X_{n}\). It’s assumed that these variables are independent and identically distributed with a probability density function (1). Before the experimentation, variety \(r < n\) is determined, and the experiment is ended after the \(r{\text{th}}\) failure. Now, consider the problem where under the Type II censoring scheme, failure times are not precisely observed and only partial information about them is available in the form of fuzzy numbers \(\tilde{x}_{i} = \left( {m_{i} ,\alpha_{i} ,\beta_{i} } \right), i = 1,2, \ldots ,r\) with the membership function of \(\mu_{{\tilde{x}_{1} }} \left( {x_{1} } \right)\),…,\(\mu_{{\tilde{x}_{r} }} \left( {x_{r} } \right)\). Let the maximum value of the means of these fuzzy numbers be \(m_{\left( r \right)}\).The lifetime of \(n - r\) surviving units, which are removed from the test after the \(r{\text{th}}\) failure, can be coded as fuzzy numbers \(\tilde{x}_{r + 1} ,\tilde{x}_{r + 2} , \ldots ,\tilde{x}_{n}\) with the membership functions

$$\mu _{{\tilde{x}_{j} }} \left( x \right) = \left\{ {\begin{array}{*{20}c} 0 & {x \le m_{{\left( r \right)}} } \\ 1 & {x > m_{{\left( r \right)}} } \\ \end{array} ~\;j = 1,2, \ldots ,r} \right..$$

Suppose that \(X_{1} , \ldots ,X_{n}\) may be a random sample of size \(n\) from Rayleigh distribution with pdf given by (1). Let \({\varvec{X}} = \left( {X_{1} , \ldots ,X_{n} } \right)\) denotes the corresponding random vector. If a realization \({\varvec{x}} = \left( {x_{1} , \ldots ,x_{n} } \right)\) of \({\varvec{X}}\) was known exactly, we could gain the complete-data likelihood function as

$$l\left( {\alpha ,x} \right) = \left( {2\alpha } \right)^{n} \left( {\mathop \prod \limits_{i = 1}^{n} x_{i} } \right)\exp ( - \alpha \mathop \sum \limits_{i = 1}^{n} x_{i}^{2} ) .$$

Now, consider the matter where \({\varvec{x}}\)  isn't observed precisely and only partial information about \({\varvec{x}}\) is out there within the sort of a fuzzy subset \(\tilde{\user2{x}} = \left( {\tilde{x}_{1} , \ldots ,\tilde{x}_{n} } \right)\) with the membership function \(\mu_{{\tilde{\user2{x}}}} \left( {\varvec{x}} \right) = \mu_{{\tilde{x}_{1} }} \left( {x_{1} } \right) \times \ldots \times \mu_{{\tilde{x}_{n} }} \left( {x_{n} } \right)\). Consistent with Zadeh’s definition of the probability of a fuzzy event (Zadeh 1968), the corresponding observed-data likelihood function can also be attained as

$$\begin{aligned} ~L\left( {\alpha ,\tilde{x}} \right) & = \int {f\left( {x,\alpha } \right)\mu _{{\tilde{x}}} \left( x \right){\text{d}}x} = \mathop \prod \limits_{{j = 1}}^{n} \mathop \smallint \limits_{0}^{\infty } 2\alpha xe^{{ - \alpha x^{2} }} \mu _{{\tilde{x}_{j} }} \left( x \right){\text{d}}x \\ & = \left( {\mathop \prod \limits_{{j = 1}}^{r} \int\limits_{0}^{\infty } {2\alpha xe^{{ - \alpha x^{2} }} \mu _{{\tilde{x}_{j} }} \left( x \right){\text{d}}x} } \right)\left( {\mathop \prod \limits_{{j = r + 1}}^{n} \int\limits_{0}^{\infty } {2\alpha xe^{{ - \alpha x^{2} }} \mu _{{\tilde{x}_{j} }} \left( x \right){\text{d}}x} } \right) \\ & = \left( {\mathop \prod \limits_{{j = 1}}^{r} \int\limits_{0}^{\infty } {2\alpha xe^{{ - \alpha x^{2} }} \mu _{{\tilde{x}_{j} }} \left( x \right){\text{d}}x} } \right)\left[ {\mathop \prod \limits_{{j = r + 1}}^{n} \left( {\int\limits_{0}^{{m_{{\left( r \right)}} }} {2\alpha xe^{{ - \alpha x^{2} }} \mu _{{\tilde{x}_{j} }} \left( x \right){\text{d}}x + \int\limits_{{m_{{\left( r \right)}} }}^{\infty } {2\alpha xe^{{ - \alpha x^{2} }} \mu _{{\tilde{x}_{j} }} \left( x \right){\text{d}}x} } } \right)} \right] \\ & = \left( {2\alpha } \right)^{r} e^{{ - ~\left( {n - r} \right)\alpha \left( {m_{{\left( r \right)}} } \right)^{2} }} \left( {\mathop \prod \limits_{{j = 1}}^{r} \int\limits_{0}^{\infty } {xe^{{ - \alpha x^{2} }} \mu _{{\tilde{x}_{j} }} \left( x \right){\text{d}}x} } \right) \\ \end{aligned}$$

Assume the prior distribution of \(\alpha\)  is that the Gamma distribution of density function as follows:

$$\begin{array}{*{20}c} {\pi \left( {\alpha {|}a,b} \right) = \frac{{b^{a} }}{\Gamma \left( a \right)}\alpha^{a - 1} e^{ - b\alpha } ,} & {\alpha > 0,a,b > 0} \\ \end{array}$$

The derivative of \(\pi \left( {\alpha {|}a,b} \right)\) with reference to \(\alpha\) is attained as

$$\frac{{d\pi \left( {\alpha {|}a,b} \right)}}{d\alpha } = \frac{{b^{a} \alpha^{a - 2} e^{ - b\alpha } }}{\Gamma \left( a \right)}\left( {\left( {a - 1} \right) - b\alpha } \right)$$

According to Han (1997), \(a\) and \(b\) should be chosen to ensure that \(\pi \left( {\alpha {|}a,b} \right)\)  may be a decreasing function of \(\alpha\). Therefore, \(b > 0\) and \(0 < a < 1\). Given \(0 < a < 1\), the larger value of \(b\), the tail of the density function becomes thinner. Berger (1985) showed that the thinner tailed prior distribution generally reduces the robustness of Bayesian estimation. Therefore, the hyper-parameter b should be chosen under the restriction \(0 < b < c\), where \(c\) may be a given boundary. Hyper-parameters \(a\) and \(b\) satisfy \(D = \left\{ {\left. {\left( {a,b} \right)|0 < a < 1,{ }0 < b < c} \right\}} \right.\). Suppose that the prior distribution of \(a\) is uniform distribution on \(\left( {0,1} \right)\), and therefore the prior distribution of \(b\) is uniform distribution on \(\left( {0,c} \right)\), when \(a\) and \(b\) are independent, the joint density of \(a\) and \(b\) is given by

$$\begin{array}{*{20}c} {\pi \left( {a, b} \right) = \frac{1}{c} , } & {0 < a < 1, 0 < b < c } \\ \end{array}$$

Following is that the introduction, the sections of this paper are as follows: In Sect. 2, E-Bayesian and H-Bayesian methods for Rayleigh distribution parameter on the idea of a kindII censoring schemes and supported fuzzy data using SE loss function as \(L\left( {\hat{\theta },\theta } \right) = \left( {\hat{\theta } - \theta } \right)^{2}\) are described. A numerical example and a Monte Carlo simulation are expressed to match the estimators, in Sect. 3. Also, we illustrate the estimation methods, during this Section. Eventually, in Sect. 4, the conclusion is handed.

E-Bayesian and H-Bayesian estimates of \({\varvec{\upalpha}}\) Parameter

According to (3) and (4), the posterior distribution of providing fuzzy data is as follows:

$$\pi \left( {\alpha {|}\tilde{\user2{x}}} \right) = \frac{{\alpha^{r + a - 1} e^{{ - \alpha \left[ {\left( {n - r} \right)\left( {m_{\left( r \right)} } \right)^{2} + b} \right]}} u\left( \alpha \right)}}{{\mathop \smallint \nolimits_{0}^{\infty } \alpha^{r + a - 1} e^{{ - \alpha \left[ {\left( {n - r} \right)\left( {m_{\left( r \right)} } \right)^{2} + b} \right]}} u\left( \alpha \right){\text{d}}\alpha }}$$

where \(u\left( \alpha \right) = \mathop \prod \limits_{j = 1}^{r} \left( {\mathop \smallint \limits_{0}^{\infty } xe^{{ - \alpha x^{2} }} \mu_{{x_{j} }} \left( x \right){\text{d}}x} \right)\). Therefore, Bayesian estimate of \(\alpha\), under SE loss function is as follows:

$$\hat{\alpha }_{Bay} \left( {a,b} \right) = E_{{\alpha |\tilde{x}}} \left( \alpha \right) = \frac{{\mathop \smallint \nolimits_{0}^{\infty } \alpha^{r + a} e^{{ - \alpha \left[ {\left( {n - r} \right)\left( {m_{\left( r \right)} } \right)^{2} + b} \right]}} u\left( \alpha \right){\text{d}}\alpha }}{{\mathop \smallint \nolimits_{0}^{\infty } \alpha^{r + a - 1} e^{{ - \alpha \left[ {\left( {n - r} \right)\left( {m_{\left( r \right)} } \right)^{2} + b} \right]}} u\left( \alpha \right){\text{d}}\alpha }}$$

In general, it's insolvable to calculate it. Therefore, Lindley (1980) approximation is employed to calculate it. Assume:

$$w\left( \alpha \right) = L\left( \alpha \right) + \vartheta \left( \alpha \right)$$

where \(L\left( \alpha \right) = n\ln \alpha - \alpha \left( {n - r} \right)\left( {m_{\left( r \right)} } \right)^{2} + \ln u\left( \alpha \right)\) and \(\vartheta \left( \alpha \right) = \left( {a - 1} \right)\ln \alpha - b\alpha\). So, Eq. (7) is converted as follows.

$$\hat{\alpha }_{Bay} \left( {a,b} \right) = \frac{{\mathop \smallint \nolimits_{0}^{\infty } \alpha e^{w\left( \alpha \right)} {\text{d}}\alpha }}{{\mathop \smallint \nolimits_{0}^{\infty } e^{w\left( \alpha \right)} {\text{d}}\alpha }} = \hat{\alpha } + \vartheta_{1} \left( {\hat{\alpha }} \right)\delta_{11} \left( {\hat{\alpha }} \right) + \frac{1}{2}w_{3} \left( {\hat{\alpha }} \right)\delta_{11}^{2} \left( {\hat{\alpha }} \right)$$


$$\vartheta_{1} = \frac{d\vartheta \left( \alpha \right)}{{d\alpha }} = \frac{a - b\alpha - 1}{\alpha }$$


$$\delta _{{11}} = \left[ { - \frac{{d^{2} w\left( \alpha \right)}}{{d\alpha ^{2} }}} \right]^{{ - 1}} = \left\{ {\frac{n}{{\alpha ^{2} }} + \mathop \sum \limits_{{j = 1}}^{r} \left( {\frac{{I_{{3j}}^{2} - I_{{5j}} I_{{1j}} }}{{I_{{1j}}^{2} }}} \right)} \right\}^{{ - 1}} ~$$


$$w_{3} = \frac{{{\text{d}}^{3} w\left( \alpha \right)}}{{{\text{d}}\alpha^{3} }} = - \frac{2n}{{\alpha^{3} }} + \mathop \sum \limits_{j = 1}^{r} \left( {\frac{{I_{7j} }}{{I_{1j} }} + \frac{{2I_{3j}^{4} - 3I_{1j} I_{3j} I_{5j} }}{{I_{1j}^{3} }}} \right)$$

where \(I_{nj} = \mathop \smallint \limits_{0}^{\infty } x^{n} e^{{ - \alpha x^{2} }} \mu_{{x_{j} }} \left( x \right){\text{d}}x , n \ge 1\).

The definition of E-Bayesian estimation was firstly proposed by Han (2009), relate as follows.

Definition 6

With \(\hat{\theta }_{Bay} \left( {a,b} \right)\) being continuous,

$$~\hat{\theta }_{{EBay}} = \int {\int\limits_{D}^{{}} {\hat{\theta }_{{Bay}} \left( {a,b} \right)\pi \left( {a,b} \right){\text{d}}b{\text{d}}a} }$$

is called the expected Bayesian estimation of \(\theta\) (E- Bayesian estimation of\(\theta\)), which is assumed to be finite, where \(D\) is that the domain of \(a\) and \(b\), \(\hat{\theta }_{Bay} \left( {a,b} \right)\) is Bayesian estimation of \(\theta\) with hyper parameters \(a\) and \(b\), and \(\pi \left( {a,{ }b} \right)\) is that the density function of \(a\) and \(b\) over \(D\). Definition 6 indicates that E- Bayesian estimation of \(\theta\)  is simply the expectation of Bayesian estimation of \(\theta\) for all the hyper parameters.

If the prior distribution of \(a\) and \(b\) are uniform distributions in domain \(D\), then consistent with (8) and (11), the expected Bayesian estimation of \(\alpha\) is given by

$$\hat{\alpha }_{{EBay}} = \frac{1}{c}\int\limits_{0}^{1} {\int\limits_{0}^{c} {\left( {\alpha + \vartheta _{1} \delta _{{11}} + \frac{1}{2}w_{3} \delta _{{11}}^{2} } \right){\text{d}}a{\text{d}}b} } = \frac{{2\alpha ^{2} + \alpha \delta _{{11}} \left( {w_{3} \delta _{{11}} - c^{2} } \right) - c\delta _{{11}} }}{{2c\alpha }}~~~$$

where \(\delta_{11}\) and \(w_{3}\) are given in (9) and (10), respectively.

Lindley and Smith (1972) addressed a thought of hierarchical prior distribution, relate as follows.

Definition 7

If \(a\) and \(b\) are hyper parameters within the parameter of \(\theta\), the prior density function of \(\theta\) is \(\pi \left( {\theta {|}a,b} \right),\)  and therefore the prior density function of the hyper parameters of \(a\) and \(b\) is \(\pi \left( {a, b} \right)\), then the hierarchical prior density function of \(\theta\) is defined as follows:

$$~\pi \left( \theta \right) = \int {\int\limits_{D}^{{}} {\pi \left( {\theta {\text{|}}a,b} \right)\pi \left( {a,~b} \right){\text{d}}b{\text{d}}a.~} ~}$$

According to (4), (5) and (13), the hierarchical prior density distribution of \(\alpha\) is given by

$$\pi \left( \alpha \right) = \mathop \int \limits_{0}^{1} \mathop \int \limits_{0}^{c} \pi \left( {\alpha {|}a,b} \right)\pi \left( {a, b} \right){\text{d}}b{\text{d}}a = \frac{1}{{\alpha^{2} c}}\mathop \int \limits_{0}^{1} \frac{{\Gamma \left( {a + 1, c\alpha } \right)}}{\Gamma \left( a \right)}{\text{d}}a , \alpha > 0,$$

where \(\Gamma \left( {t,s} \right) = \mathop \smallint \limits_{0}^{s} u^{t - 1} e^{ - u} du\).

According to (3) and (14), the hierarchical posterior density function of \(\alpha\) is as follows:

$$\pi^{*} \left( {\alpha |\tilde{x}} \right) = \frac{{\alpha^{r - 2} e^{{ - \alpha \left( {n - r} \right)\beta m_{\left( r \right)} }} S\left( \alpha \right)u\left( \alpha \right)}}{{\mathop \smallint \nolimits_{0}^{\infty } \alpha^{r - 2} e^{{ - \alpha \left( {n - r} \right)\beta m_{\left( r \right)} }} S\left( \alpha \right)u\left( \alpha \right){\text{d}}\alpha }}$$

where \(S\left( \alpha \right) = \mathop \smallint \limits_{0}^{1} \frac{{\Gamma \left( {a + 1, c\alpha } \right)}}{\Gamma \left( a \right)}da\).

Now, using (15), H-Bayesian estimate of \({\upalpha }\) under the SE loss function is as follows:

$$\hat{\alpha }_{HBay} = \frac{{\mathop \smallint \nolimits_{0}^{\infty } \alpha^{r - 1} e^{{ - \alpha \left( {n - r} \right)\beta m_{\left( r \right)} }} S\left( \alpha \right)u\left( \alpha \right){\text{d}}\alpha }}{{\mathop \smallint \nolimits_{0}^{\infty } \alpha^{r - 2} e^{{ - \alpha \left( {n - r} \right)\beta m_{\left( r \right)} }} S\left( \alpha \right)u\left( \alpha \right){\text{d}}\alpha }}$$

The estimation of which is generally insolvable. Thus, Lindley (1980) approximation method is employed for estimation. Assume \(w^{*} \left( \alpha \right) = L\left( \alpha \right) + \vartheta^{*} \left( \alpha \right)\), where, \(L\left( \alpha \right) = n\ln \alpha - \alpha \left( {n - r} \right)m_{\left( r \right)} + \ln u\left( \alpha \right)\) and \(\vartheta^{*} \left( \alpha \right) = \ln S\left( \alpha \right) - 2\ln \alpha\).


$$\hat{\alpha }_{HBay} = \frac{{\mathop \smallint \nolimits_{0}^{\infty } \alpha e^{{w^{*} \left( \alpha \right)}} d\alpha }}{{\mathop \smallint \nolimits_{0}^{\infty } e^{{w^{*} \left( \alpha \right)}} d\alpha }} = \alpha + \vartheta_{1}^{*} \delta_{11}^{*} + \frac{1}{2}w_{3}^{*} \left( {\delta_{11}^{*} } \right)^{2}$$


$$\vartheta_{1}^{*} = \frac{{d\vartheta^{*} \left( \alpha \right)}}{d\alpha } = \frac{{ce^{ - c\alpha } K_{0} \left( \alpha \right)}}{S\left( \alpha \right)} - \frac{2}{\alpha }$$


$$\delta_{11}^{*} = \left[ { - \frac{{d^{2} w^{*} \left( \alpha \right)}}{{d\alpha^{2} }}} \right]^{ - 1} = \left\{ {\delta_{11}^{ - 1} + \frac{{c^{2} e^{ - c\alpha } \left[ {e^{ - c\alpha } K_{0}^{2} \left( \alpha \right) - S\left( \alpha \right)\left( {K_{1} \left( \alpha \right) - K_{0} \left( \alpha \right)} \right)} \right]}}{{S^{2} \left( \alpha \right)}}} \right\}^{ - 1}$$
$$w_{3}^{*} = \frac{{d^{3} w^{*} \left( \alpha \right)}}{{d\alpha^{3} }} = w_{3} + M_{1} \left( \alpha \right) + M_{2} \left( \alpha \right) - \frac{{2K_{1} \left( \alpha \right)K_{2} \left( \alpha \right)c^{4} e^{ - 2c\alpha } - 2c^{5} e^{ - 2c\alpha } K_{1}^{2} \left( \alpha \right)}}{{S^{2} \left( \alpha \right)}} - \frac{{2K_{0} \left( \alpha \right)K_{1}^{2} \left( \alpha \right)c^{5} e^{ - 3c\alpha } }}{{S^{3} \left( \alpha \right)}}$$


$$M_{n} \left( \alpha \right) = \frac{{c^{2} e^{ - c\alpha } K_{n} \left( \alpha \right) - c^{3} e^{ - c\alpha } K_{n - 1} \left( \alpha \right)}}{S\left( \alpha \right)} - \frac{{c^{4} e^{ - 2c\alpha } K_{n - 1} \left( \alpha \right)K_{1} \left( \alpha \right)}}{{S^{2} \left( \alpha \right)}} , n = 1, 2$$


$$\begin{array}{*{20}c} {K_{0} \left( \alpha \right) = \mathop \smallint \limits_{0}^{1} \frac{{\left( {c\alpha } \right)^{a} }}{{{\Gamma }\left( a \right)}} {\text{d}}a,} & {K_{n} \left( \alpha \right) = \mathop \smallint \limits_{0}^{1} \left( {\mathop \prod \limits_{i = 0}^{n - 1} \left( {a - i} \right)} \right)\frac{{\left( {c\alpha } \right)^{a - n} }}{{{\Gamma }\left( a \right)}} {\text{d}}a, } & { n = 1, 2} \\ \end{array}$$

and \(\delta_{11}\) and \(w_{3}\) are given in (9) and (10), respectively.

Numerical Experiments

In this section, a numerical example and a Monte Carlo simulation are performed to assess the performance of the estimation methods described within the preceding sections.

Simulation Study

Bayesian, E-Bayesian and H-Bayesian estimates of α parameter are compared during this subsection. The simulation structure are frequently described below.

  • Step 1: \(b\) is produced any specific \(c\) and using the prior distribution of\({\pi }_{1}\left(b\right)=\frac{1}{c} , 0<b<c\).

  • Step 2: \(a\) is produced by using the prior distribution of\(\pi \left(a\right)=1 , 0<a<1\).

  • Step 3: \(\mathrm{\alpha }\)  is produced using b estimated within the step 1 and employing a estimated within the step 2 using (4).

  • Step 4: Using the \(\mathrm{\alpha }\) estimated within the step 3, the samples of Type II censoring with different \((n,r)\) of Rayleigh distribution and thus the possibility density function (1) are produced using the equation below, where \(U\) may be a continuous uniform distribution on\((\mathrm{0,1})\). The produced samples are as \({\varvec{X}} = \left( {X_{1} , \ldots ,X_{r} } \right)\).

    $$X = \left\{ { - \frac{1}{\alpha }\log \left( {1 - u} \right)} \right\}^{\frac{1}{2}}$$

Each sample of Type II censoring X produced within the step 4 is taken into account supported fuzzy system of Pak et al. (2013) using the membership functions below as a fuzzy sample.

\(\mu _{{\tilde{x}_{1} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {x \le 0.25} \hfill \\ {\frac{{0.5 - x}}{{0.25}}} \hfill & {0.25 \le x \le 0.5} \hfill \\ 0 \hfill & {{\text{~~otherwise}}} \hfill \\ \end{array} } \right.~\;\;\mu _{{\tilde{x}_{2} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - 0.25}}{{0.25}}} \hfill & {0.25 \le x \le 0.5} \hfill \\ {\frac{{0.75 - x}}{{0.25}}} \hfill & {0.5 \le x \le 0.75} \hfill \\ 0 \hfill & {{\text{~otherwise}}} \hfill \\ \end{array} } \right.\).

$$\mu _{{\tilde{x}_{3} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - 0.5}}{{0.25}}} \hfill & {~~0.5 \le x \le 0.75} \hfill \\ {\frac{{1 - x}}{{0.25}}} \hfill & {0.75 \le x \le 1} \hfill \\ 0 \hfill & {{\text{~otherwise}}} \hfill \\ \end{array} } \right.~\;\;\mu _{{\tilde{x}_{4} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - 0.75}}{{0.25}}} \hfill & {0.75 \le x \le 1} \hfill \\ {\frac{{1.25 - x}}{{0.25}}} \hfill & {1 \le x \le 1.25} \hfill \\ 0 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
$$\mu _{{\tilde{x}_{5} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - 1}}{{0.25}}} \hfill & {1 \le x \le 1.25} \hfill \\ {\frac{{1.5 - x}}{{0.25}}} \hfill & {1.25 \le x \le 1.5} \hfill \\ 0 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.~\;\mu _{{\tilde{x}_{6} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - 1}}{{0.25}}~} \hfill & {1 \le x \le 1.25} \hfill \\ {\frac{{1.5 - x}}{{0.25}}} \hfill & {1.25 \le x \le 1.5} \hfill \\ 0 \hfill & {~{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
$$\mu _{{\tilde{x}_{7} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - 1.5}}{{0.25}}} \hfill & {1.5 \le x \le 1.75} \hfill \\ {\frac{{2 - x}}{{0.25}}} \hfill & {1.75 \le x \le 2} \hfill \\ 0 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} ~~} \right.\;\mu _{{\tilde{x}_{8} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {x - 1.75} \hfill & {~1.75 \le x \le 2} \hfill \\ 1 \hfill & {x \ge 2} \hfill \\ 0 \hfill & {~{\text{otherwise}}} \hfill \\ \end{array} } \right.$$

Then, Bayesian, E-Bayesian and H-Bayesian estimates of \({\upalpha }\) parameter are attained using (8), (12), and (17). The simulation is completed using R software. The performance of all estimates are compared numerically of the mean square error (MSE) value. Steps 1 to 4 are irritated 1000 times. The typical estimation and its mean square error are attained and are listed in Table 1 and a couple of. These estimates under the SE loss function are obtained for \(c = 1,{ }1.5,{ }2,{ }2.5\), respectively. Supported tabulated MSE values, the subsequent conclusions are often drawn from Tables 1 and 2. Indicating that the performance of H-Bayesian estimate of \(\alpha\) under SE loss function is best.

Table 1 Average value (AV) and mean squared error (MSE) of the estimates of \(\alpha\) for \(c=1, 1.5\)
Table 2 Average value (AV) and mean squared error (MSE) of the estimates of \(\alpha\) for c = 2, 2.5

Application With Real Data Set

A real data set for instance estimation methods, proposed during this paper. This data set represent repair times (in h) for an airborne communication transceiver. They were first anatomized by Von Alven (1964). The info include:

0.2, 0.3, 0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.7,0.7, 0.7, 0.7, 0.8,1.0, 1.0, 1.0, 1.0, 1.1, 1.3, 1.5, 1.5, 1.5, 1.5, 2.0, 2.0, 2.2, 2.5, 2.7, 3.0 3.0, 3.3, 3.3, 4.0, 4.0, 4.5, 4.7, 5.0, 5.4, 5.4, 7.0, 7.5, 8.8, 9.0, 10.3, 22.0, 24.5.

For calculating Bayesian estimation, E-Bayesian estimation and H-Bayesian estimation, we do not have any prior information. Therefore, assumed that \(a=3,\) \(b=2\). Bayesian, E-Bayesian and H-Bayesian estimations for \(c=3\), \(r=20\) are obtained as 0.09125, 0.078964 and 0.054658, respectively. The sorts of estimations of the probability density function (1) for the estimation methods are drawn in Fig. 1. This represents the prevalence of H-Bayesian estimation toward other estimates.

figure 1

Chart probability density function estimation (1) of Bayesian, E-Bayesian and H-Bayesian methods


Grounded on complete and censored samples, some researches on parameter estimation of Rayleigh distribution have formerly been conducted. But, it’s assumed that the available data are performed in precise numbers. Still, a number of the collected data might be possibly displayed within the sort of imprecise numbers and fuzzy information. Consequently, during this study, Bayesian estimation, E-Bayesian estimation and H-Bayesian estimation of Rayleigh distribution parameter under the sort II censoring scheme were estimated supported fuzzy data with squared error loss function. Eventually, these estimates were compared using Monte Carlo simulation and a true data set indicating that H- Bayesian method had better performance.


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Correspondence to Einollah Deiri.

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Heidari, K.F., Deiri, E. & Jamkhaneh, E.B. E-Bayesian and Hierarchical Bayesian Estimation of Rayleigh Distribution Parameter with Type-II Censoring from Imprecise Data. J Indian Soc Probab Stat 23, 63–76 (2022).

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  • E-Bayesian estimation
  • Hierarchical Bayesian estimation
  • Rayleigh distribution
  • Type II censoring schemes
  • Fuzzy data