Bayesian Method for the Generalized Exponential Model Using Fuzzy Data

Abstract

This paper focuses on Bayesian inference for the parameters of the generalized exponential model under asymmetric and symmetric loss functions when the observations are described in terms of fuzzy numbers. First, a generalized likelihood function based on fuzzy data is derived. Then, considering general entropy, linear exponential and squared error loss functions, the Bayes estimates of the parameters are obtained. Since Bayes estimates could not be expressed in closed forms, Metropolis–Hasting samplers are used to compute the approximate Bayes estimates. For comparison purposes, the maximum likelihood estimates of the parameters are also computed. The proposed inferences are illustrated using three real-world examples. The numerical simulation results demonstrate the superiority of the Bayesian method over the maximum likelihood procedure.

Appendices

Appendix 1: Derivation of the Likelihood Function (10)

The notion of probability was extended to fuzzy events by Zadeh [43]. In this appendix, using the definition of the probability of fuzzy events, we derive the formula (10).

Let $(\mathbb {R}^{n},{\mathcal {A}},P)$ be a probability space in which ${ \mathcal {A}}$ is the $\sigma$-field of Borel sets in $\mathbb {R}^{n}$ and P is a probability measure on $\mathbb {R}^{n}$. A fuzzy event in $\mathbb {R}^{n}$ is a fuzzy subset $\tilde{A}$ of $\mathbb {R}^{n}$ whose membership function $\mu _{\tilde{A}}$ is Borel measurable. The probability of $\tilde{A}$ is defined as

$$\begin{aligned} P(\tilde{A})=\int \mu _{\tilde{A}}(\varvec{x})dP. \end{aligned}$$

(27)

Using Eqs. (2) and (27), the likelihood function of $\gamma$ and $\delta$ is obtained as

$$\begin{aligned} L_{O}(\gamma ,\delta ; \tilde{{\varvec{x}}})= & {} P({\tilde{\varvec{x}}};\gamma ,\delta )=\int f(\varvec{x};\gamma ,\delta )\mu _ {\tilde{{\varvec{x}}}}(\varvec{x})d\varvec{x} \nonumber \\= & {} \prod \limits _{i=1}^{n} \int \gamma \delta e^{-\delta x}(1-e^{-\delta x})^{\gamma -1}\mu _{\tilde{x}_{i}}(x)dx. \end{aligned}$$

(28)

Appendix 2: Calculation of the MLEs

The fuzzy EM algorithm (FEM) proposed by Denoeux [20] is an applicable procedure for the calculation of maximum likelihood estimates in the problems dealing with fuzzy data. Recently, FEM method is widely used in statistical inferences; see for example Khoolenjani and Chatrabgoun [14], Makhdoom et al. [26], and Cannarile et al. [44]. In this approach, the fuzzy sample $\tilde{\mathbf {x}}$ is considered to be as an incomplete specification of a complete data vector $\mathbf {x}$. For implementing the algorithm, the complete-data log-likelihood function of the parameters, say $\ell (\gamma , \delta ;\mathbf {x})$, is obtained. Then, the conditional expectation $Q(\gamma , \delta )=E[\ell (\gamma , \delta ;\mathbf {x})\mid \tilde{\mathbf {x}}]$ is computed. Maximizing $Q(\gamma , \delta )$ with respect to the parameters yields the MLEs $of \gamma$ and $\delta$.

In our case, the complete-data log-likelihood function becomes

$$\begin{aligned} \ell (\gamma , \delta ;\mathbf {x})= & {} n \log \gamma +n\log \delta -\delta \sum _{i=1}^{n}x_i\nonumber \\&+(\gamma -1)\sum _{i=1}^{n} \log (1-e^{-\delta x_i}). \end{aligned}$$

(29)

By equating derivatives of log-likelihood (29) with respect to $\gamma$ and $\delta$ to zero we have

$$\begin{aligned} \frac{n}{\gamma }= & {} -\sum \limits _{i=1}^{n}\log (1-e^{-\delta x_i}), \end{aligned}$$

(30)

$$\begin{aligned} \frac{n}{\delta }= & {} \sum _{i=1}^{n}x_i-(\gamma -1)\sum _{i=1}^{n} \frac{x_ie^{-\delta x_i}}{1-e^{-\delta x_i}}. \end{aligned}$$

(31)

The likelihood Eqs. (30) and (31) is updated by the following steps to compute the MLEs.

• Step 1. Let the initial values of the parameters be $(\gamma ^{0}, \delta ^{0})$ and set $t=0$.

• Step 2. For $i=1,\ldots ,n$, compute the conditional expectations

  $$\begin{aligned} \phi _{1i}= & {} E_{(\gamma ^{t}, \delta ^{t})}[\log (1-e^{-\delta X})\mid \tilde{\mathbf {x}}]\\= & {} \frac{\int e^{-(\delta ^{t} x)}(1-e^{-(\delta ^{t} x)})^{\gamma ^{t} -1}\log (1-e^{-(\delta ^{t} x)})\mu _{\tilde{x}_{i}}(x)dx}{\int e^{-(\delta ^{t} x)}(1-e^{-(\delta ^{t} x)})^{\gamma ^{t} -1}\mu _{\tilde{x}_{i}}(x)dx},\\ \phi _{2i}= & {} E_{(\gamma ^{t}, \delta ^{t})}[\frac{Xe^{-\delta X}}{1-e^{-\delta X}}\mid \tilde{\mathbf {x}}]\\= & {} \frac{\int xe^{-(2\delta ^{t} x)}(1-e^{-(\delta ^{t} x)})^{\gamma ^{t} -2}\mu _{\tilde{x}_{i}}(x)dx}{\int e^{-(\delta ^{t} x)}(1-e^{-(\delta ^{t} x)})^{\gamma ^{t} -1}\mu _{\tilde{x}_{i}}(x)dx}, \end{aligned}$$

  and

  $$\begin{aligned} \phi _{3i}=E_{(\gamma ^{t}, \delta ^{t})}[X\mid \tilde{\mathbf {x}}]=\frac{\int xe^{-(\delta ^{t} x)}(1-e^{-(\delta ^{t} x)})^{\gamma ^{t} -1}\mu _{\tilde{x}_{i}}(x)dx}{\int e^{-(\delta ^{t} x)}(1-e^{-(\delta ^{t} x)})^{\gamma ^{t} -1}\mu _{\tilde{x}_{i}}(x)dx}, \end{aligned}$$

  and replace (30) and (31) with

  $$\begin{aligned} \frac{n}{\gamma }=-\sum \limits _{i=1}^{n}\phi _{1i} \end{aligned}$$

  and

  $$\begin{aligned} \frac{n}{\delta }=\sum _{i=1}^{n}\phi _{3i}-(\gamma -1)\sum _{i=1}^{n} \phi _{2i}, \end{aligned}$$

  respectively.

• Step 3. Obtain the updated values of the parameters as

  $$\begin{aligned} \gamma ^{t+1}=-\frac{n}{\sum \limits _{i=1}^{n}\phi _{1i}} ~ \ \ \text {and}~ \ \ \delta ^{t+1}=\frac{n}{\sum _{i=1}^{n}\phi _{3i}-(\gamma ^{t+1}-1)\sum _{i=1}^{n} \phi _{2i}}. \end{aligned}$$

• Step 4. Let $t=t+1$

• Step 5. Repeat Steps 2–4 until $L(\gamma ^{t+1}, \delta ^{t+1};\tilde{\mathbf {x}})-L(\gamma ^{t}, \delta ^{t};\tilde{\mathbf {x}})/ |L(\gamma ^{t}, \delta ^{t};\tilde{\mathbf {x}})|$ becomes smaller than arbitrary small value $\varrho$. The current values $\gamma ^{t}$ and $\delta ^{t}$ are the ML estimates of the interested parameters.