*3.2. Method of Moments Estimation*

Let *X*1, *X*2, ... , *Xn* be a random sample from the BNDL distribution with parameter *p*. The moment estimate (ME) of the parameter *p* can be obtained by solving the following equation.

$$\frac{(p+2)(1-p)}{p+1} = \frac{1}{n} \sum\_{i=1}^{n} X\_i.$$

#### *3.3. Method of Proportions Estimation*

Let *X*1, *X*2, ... , *Xn* be a random sample from the BNDL distribution with parameter *p*. For *i* = 1, 2, . . . , *n*, we define the indicator functions

$$I(X\_i) = \begin{cases} 1 & \text{if } X\_i = 0 \\ 0 & \text{if } X\_i > 0 \end{cases}$$

Therefore, the proportion of 0s in the sample Π = <sup>1</sup> *<sup>n</sup>* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *I*(*Xi*). The proportion estimate (PE) of the parameter *p* can be obtained by solving the following equation with respect to *p*

$$
\Pi = \frac{1 + 2p - p^2}{(p+1)(2-p)^2}.
$$
