\right]$$

Generalized Entropy
-------------------

The generalized entropy is obtained by; $$G_E=\dfrac{\nu_\lambda\mu^{-\lambda}-1}{\lambda(\lambda-1)}~~~~~;\lambda\ne 0,1$$ where, $\nu_\lambda=\int_{x=0}^{\infty}x^{\lambda}f_w(x,\alpha,\theta^{})dx$ and $\mu=E(X)$. The value of $\nu_\lambda$ is calculated as

$$\begin{split}
\nu_\lambda&=\dfrac{2}{\sqrt{\pi}}\left( \dfrac{1}{\alpha}\right)^{\frac{\lambda}{2\beta}}\Gamma\left( \dfrac{3\beta+\lambda}{2\beta}\right)  
\end{split}$$

After using (4.6) and (2.8), we get $$G_E=\left( \dfrac{4}{\pi}\right)^{\frac{1-\lambda}{2}} \left[ \dfrac{\Gamma\left( \dfrac{3\beta+\lambda}{2\beta}\right)  \left\lbrace \Gamma\left( \dfrac{3\beta+1}{2\beta}\right)  \right\rbrace  ^{-\lambda}}{\lambda(\lambda-1)}\right] ~~~~~;\lambda\ne0,1$$

Parameter Estimation
====================

Here, we describe maximum likelihood estimation method and Bayes estimation method for estimating the unknown parameters $\alpha, \beta$ of the PMaD. The estimators obtained under these methods are not in nice closed form; thus, numerical approximation techniques are used to get the solution. Further, the performances of these estimators are studied through Monte Carlo simulation.

Maximum Likelihood Estimation
-----------------------------

The most popular and efficient method of classical estimation of the parameter (s) is maximum likelihood estimation. The estimators obtained by this method passes several optimum properties. The maximum likelihood estimation theory required formulation of the likelihood function. Thus, let us suppose that $X_1,X_2,\cdots,X_n$ are the iid random sample of size $n$ taken from PMaD $(\alpha,\beta)$. The likelihood function is written as; $$\begin{split}
L(\alpha,\theta)&=\prod_{i=1}^{n}\dfrac{4}{\sqrt{\pi}} \alpha^{\frac{3}{2}}\beta x_i^{3\beta-1}e^{-\alpha x_i^{2\beta}}=\dfrac{4^{n}}{\pi^{n/2}} \alpha^{\frac{3n}{2}}\beta^{n} e^{-\alpha\sum_{i=1}^{n} x_i^{2\beta}}\left( \prod_{i=1}^{n}x_i^{3\beta-1}\right) 
\end{split}$$ Log-likelihood function is written as; $$\ln L(\alpha,\theta)=l=n\ln 4-\frac{n}{2}\ln \pi+\dfrac{3n}{2}\ln \alpha+n\ln\beta
-\alpha\sum_{i=1}^{n}x_i^{2\beta}+(3\beta-1)\sum_{i=1}^{n}\ln x_i$$ for MLEs of $\alpha$ and $\beta$, $$\dfrac{\partial l}{\partial\alpha}=0~~~\&~~~\dfrac{\partial 