Page 23 - Contributed Paper Session (CPS) - Volume 5

CPS649 A-Hadi N. Ahmed et al.
The corresponding variance and index of dispersion (ID) are

(1 − )(4 + 2) Variance(X) 2(2 + 1)
Variance(X) = and ID(X) = = .
2
(1 + ) 2 E(X) (1 + )(2 + )

The moment generating function is
2
̅
(2 − )

() = ( ) = (1 + )(1 − ) , < log(1 − ) and ∈ (0,1).

̅ 2

The th descending factorial moment of is given (for = 0,1,2, . . . ) by

2
2
[(1 − ) + 1 + − 2 ]!
′ [] = (1 + ) ,

′
′
where [] = [( + 1) … ( + − 1)]. Clearly, for = 0, we obtain
[0]
2
= (1 + − 2 )/ (1 + ) and the mean of follows as ′ [1] = ().

5. Estimation
In this section, we estimate the parameter of the NDL distribution by the
maximum likelihood method and the method of moments. Both maximum
likelihood estimator (MLE) and moment estimator (ME) are the same and are
available in closed forms. 1 +
Let , . . . , be a random sample of size from the NDL distribution, then the
1

log-likelihood function is given by

ℓ(|) ∝ 2 () − (1 + ) + ̅ (1 − ).

The maximum likelihood estimator of follows by solving ℓ(|) = 0,
then we have
1 1
̂
= √1 + 8/(1 + ̅) − .
2 2

Remark 4: It can be shown that the method of moments yields the same
estimator derived using the MLE method.

6. Two applications
In this section, we use two real datasets to illustrate the importance and
superiority of the NDL distribution over the existing models namely discrete
Lindley (DL) (Bakouch et al., 2014), discrete Burr (DB) and discrete Pareto (DP)
(Krishna and Pundir, 2009) distributions.

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