**Appendix C. The Detailed Derivation of Bayesian Estimates of two Parameters (***β***,***λ***) and the Entropy under the LL Function**

In this case, we take U(*β*, *λ*) = exp(−*hβ*), and then

$$\mu\_1 = -h \exp\left(-h\beta\right), \\ \mu\_{11} = h^2 \exp\left(-h\beta\right), \\ \mu\_{12} = \mu\_{21} = \mu\_{22} = \mu\_2 = 0.$$

Using Equation (26), the Bayesian estimation of parameter *β* is given by

$$\hat{\beta}\_L = -\frac{1}{\hbar} \ln \{ \exp(-\hbar \hat{\beta}) + 0.5 [\mathbf{u}\_{11} \boldsymbol{\tau}\_{11} + \mathbf{u}\_{1} \boldsymbol{\tau}\_{11}^2 \mathbf{z}\_{30} + \mathbf{u}\_{1} \boldsymbol{\tau}\_{21} \boldsymbol{\tau}\_{22} \mathbf{z}\_{43} + 3 \mathbf{u}\_{1} \boldsymbol{\tau}\_{11} \boldsymbol{\tau}\_{12} \mathbf{z}\_{21} + (\mathbf{\bar{\tau}\_{11} \boldsymbol{\tau}\_{22} + 2 \mathbf{u}\_{1} \boldsymbol{\tau}\_{21}^2) \mathbf{u}\_{1} \mathbf{z}\_{12}] + \mathbf{u}\_{1} \boldsymbol{\tau}\_{11} p\_1 + \mathbf{u}\_{1} \boldsymbol{\tau}\_{12} p\_2 \} $$

Similarly, the Bayesian estimation of parameter *λ* is obtained by

$$\hat{\lambda}\_L = -\frac{1}{\hbar} \ln \{ \exp(-\hbar \hat{\lambda}) + 0.5 [\mathbf{u}\_{12} \mathbf{r} \mathbf{z} + \mathbf{u}\_{21} \mathbf{r}\_{12} \mathbf{z} \mathbf{y} + \mathbf{u}\_{22} \mathbf{r} \mathbf{z} \mathbf{y} + (\mathbf{\bar{r}}\_{11} \mathbf{r} \mathbf{z} + 2 \mathbf{r}\_{12}^2) \mathbf{u}\_2 \mathbf{z}\_{21} + 3 \mathbf{u}\_{2} \mathbf{r} \mathbf{z}\_{21} \mathbf{z}\_{21}] + \mathbf{u}\_{2} \mathbf{r}\_{12} \mathbf{p}\_1 + \mathbf{u}\_{2} \mathbf{r} \mathbf{z}\_{22} \mathbf{p}\_2 \}$$

For the Bayesian estimation of the entropy, we have

 $M(\beta,\lambda) = \exp[-hH(f)]$ , $\mu\_1 = \frac{h}{\beta\lambda} \exp[-hH(f)]$ ,  $\mu\_2 = -h\left[-\frac{1}{\lambda} + \frac{1}{\lambda^2}(\ln\beta - \ln(9/8) + \gamma)\right] \exp[-hH(f)]$ ,  $\mu\_{11} = h\left[\frac{-1}{\beta^2\lambda} + \frac{h}{\beta^2\lambda^2}\right] \exp[-hH(f)]$ ,

$$u\_{22} = \left\{-h\left[\frac{1}{\lambda^2} - \frac{2}{\lambda^3}(\ln \beta - \ln(9/8) + \gamma)\right] + h^2[-\frac{1}{\lambda} + \frac{1}{\lambda^2}(\ln \beta - \ln(9/8) + \gamma)]^2\right\} \exp[-hH(f)],$$

$$u\_{12} = u\_{21} = \dots = h[\frac{h-1}{\beta \lambda^2} - h\frac{1}{\beta \lambda^3}(\ln \beta - \ln(9/8) + \gamma)]\exp[-hH(f)].$$

The Bayesian estimation of the entropy under the LL function is given by

*<sup>H</sup>*<sup>ˆ</sup> *<sup>L</sup>*(*f*) = <sup>−</sup> <sup>1</sup> *<sup>h</sup>* ln{exp[−*hH*<sup>ˆ</sup> (*f*)] + 0.5[u11*τ*11+2u12*τ*<sup>12</sup> <sup>+</sup> *<sup>u</sup>*22*τ*22+z30(u1*τ*11+u2*τ*12 *τ*11+z03(u2*τ*22+u1*τ*21)*τ*<sup>22</sup> +z21(3u1*τ*11*τ*12+u2(*τ*11*τ*22+2*τ*<sup>2</sup> <sup>12</sup>)) + z12(3u2*τ*22*τ*21+u1(*τ*11*τ*22+2*τ*<sup>2</sup> <sup>21</sup>))] +*p*1(u1*τ*<sup>11</sup> + u2*τ*21)+*p*2(u2*τ*<sup>22</sup> + u1*τ*12)}

**Appendix D. The Derivation of Bayesian Estimates of two Parameters** (*β***,***λ*) **and the Entropy under the GEL Function**

In this case, we take *<sup>U</sup>*(*β*, *<sup>λ</sup>*) = *<sup>β</sup>*−*<sup>q</sup>* and then *<sup>u</sup>*<sup>1</sup> = −*qβ*−*q*−1, *<sup>u</sup>*<sup>11</sup> = *<sup>q</sup>*(*<sup>q</sup>* + <sup>1</sup>)*β*−*q*−2, and *u*<sup>12</sup> = *u*<sup>21</sup> = *u*<sup>22</sup> = *u*<sup>2</sup> = 0.

Using Equation (26), the Bayesian estimation of parameter *β* is given by

*<sup>β</sup>*ˆ*<sup>E</sup>* <sup>=</sup> {*β*ˆ−*<sup>q</sup>* <sup>+</sup> 0.5[u11*τ*11+u1*τ*<sup>2</sup> <sup>11</sup> z30+u1*τ*21*τ*22z03+3u1*τ*11*τ*12z21 + (*τ*11*τ*22+2u1*τ*<sup>2</sup> <sup>21</sup>)u1z12] + u1*τ*<sup>11</sup> *p*<sup>1</sup> + u1*τ*<sup>12</sup> *p*2} − 1/*q*

Similarly, the Bayesian estimation of parameter *λ* is obtained by

*<sup>λ</sup>*<sup>ˆ</sup> *<sup>L</sup>* <sup>=</sup> {*λ*<sup>ˆ</sup> <sup>−</sup> *<sup>q</sup>* <sup>+</sup> 0.5[u22*τ*22+u2*τ*11*τ*12z30+u2*τ*<sup>2</sup> 22z03+(*τ*11*τ*22+2*τ*<sup>2</sup> <sup>12</sup>)u2z21 + 3u2*τ*22*τ*21*z*21] + u2*τ*<sup>21</sup> *p*<sup>1</sup> + u2*τ*<sup>22</sup> *p*2} − 1/*q*

> For the Bayesian estimation of the entropy under the general EL function, we take *U*(*β*, *λ*)=[*H*(*f*)]−*<sup>q</sup>* , and then

$$\mu\_1 = \frac{q}{\mathbb{R}\Gamma} [H(f)]^{-q-1}, \; \mu\_2 = [\frac{q}{\lambda} - \frac{q}{\lambda^2} (\ln \beta - \ln(9/8) + \gamma)] [H(f)]^{-q-1}, \; \mu\_3 = [\frac{q}{\lambda} - \frac{q}{\lambda^2} (\ln \beta - \ln(9/8) + \gamma)] [H(f)]^{-q-1}$$

$$\begin{split} \mu\_{2} &= \left[\frac{q}{\lambda} - \frac{q}{\lambda^{2}}(\ln \beta - \ln(9/8) + \gamma)\right] [H(f)]^{-q-1}, \mu\_{11} = \frac{q(q+1)}{\beta^{2}\lambda^{2}} [H(f)]^{-q-2} - \frac{q}{\beta^{2}\lambda} [H(f)]^{-q-1}, \\ \mu\_{22} &= \left[\frac{-q}{\lambda^{2}} + \frac{2q}{\lambda^{3}}(\ln \beta - \ln(9/8) + \gamma)\right] [H(f)]^{-q-1} + q(q+1) \left[\frac{1}{\lambda} - \frac{1}{\lambda^{2}}(\ln \beta - \ln(9/8) + \gamma)\right]^{2} [H(f)]^{-q-2}, \\ \mu\_{12} &= u\_{21} = q(q+1) \left[\frac{1}{\beta^{3}\lambda^{2}} - \frac{1}{\beta^{3}\lambda^{3}}(\ln \beta - \ln(9/8) + \gamma)\right] [H(f)]^{-q-2} - \frac{q}{\beta^{2}\lambda^{2}} [H(f)]^{-q-1}. \end{split}$$

Using Equation (26), the approximate Bayesian estimation of the entropy is given by

$$\begin{split} \mathcal{H}\_{\rm E}(f) &= \left\{ [\hat{H}(f)]^{-\theta} + 0.5[(\mathbf{u}\_{11}\tau\_{11} + 2\mathbf{u}\_{12}\tau\_{12} + \mathbf{u}\_{22}\tau\_{22}) + \mathbf{z}\_{30}(\mathbf{u}\_{1}\tau\_{11} + \mathbf{u}\_{2}\tau\_{12}) \right\} \tau\_{11} + \mathbf{z}\_{03}(\mathbf{u}\_{2}\tau\_{22} + \mathbf{u}\_{1}\tau\_{12})\tau\_{22} \right\} \\ &+ \mathbf{z}\_{21}(3\mathbf{u}\_{1}\tau\_{11}\tau\_{12} + \mathbf{u}\_{2}(\tau\_{11}\tau\_{22} + 2\tau\_{12}^{2})) + \mathbf{z}\_{12}(3\mathbf{u}\_{2}\tau\_{22}\tau\_{21} + \mathbf{u}\_{1}(\tau\_{11}\tau\_{22} + 2\tau\_{21}^{2})) \big] \\ &+ p\_{1}(\mathbf{u}\_{1}\tau\_{11} + \mathbf{u}\_{2}\tau\_{21}) + p\_{2}(\mathbf{u}\_{2}\tau\_{22} + \mathbf{u}\_{1}\tau\_{22}))^{-1/q} .\end{split}$$

### **Appendix E.**

**Table A1.** The average 95% approximate confidence intervals and average lengths and coverage probabilities of *β*, *λ* and the entropy (*β* = 1, *λ* = 2, H(*f*) = 0.2448, T = 0.6).



**Table A2.** The average 95% approximate confidence intervals and average lengths and coverage probabilities of *β*, *λ* and the entropy (*β* = 1, *λ* = 2, H(*f*) = 0.2448, T = 1.5).

**Table A3.** The average 95% Bayesian credible intervals and average lengths and coverage probabilities of *β*, *λ* and the entropy (*β* = 1, *λ* = 2 H(*f*) = 0.2448, T = 0.6).



**Table A4.** The average 95% Bayesian credible intervals and average lengths and coverage probabilities of *β*, *λ* and the entropy (*β* = 1, *λ* = 2 H(*f*) = = 0.2448, T = 1.5).

#### **References**

