Page 22 - Contributed Paper Session (CPS) - Volume 5

CPS649 A-Hadi N. Ahmed et al.
Proof.
We have
2
() (1 + )(1 − )
() = = 1 2 1 .
2
() (1 + )(1 − )

2
2
1

Clearly, one can see that ( + 1) ≤ () ∀ > .
2
1
Theorem 2 shows that the NDL distribution is ordered according to the
strongest stochastic order (v).

Corollary 2: Based on the chain of stochastic orders in the definition (4),
≤ ℎ , ≤ ℎ , ≤ ≤ .

Definition 6: The discrete random variable is said to be smaller than in
weak likelihood ratio ordering (denoted by ≤ ) if (+1) ≤ (0) ∀ ≥
(0)
(+1)

0 (see, Khider et al., 2002).

Theorem 3: Let ~( ) and ~( ). Then, is said to be smaller than
1
2
in weak likelihood ratio ordering, denoted by ≤ , for all > .
1
2

Proof.
According to Definition 2.5 of Khider et al. (2002), we can prove that

( + 1) (0) .

( + 1) ≤ (0)

Then, we obtain

2
(1 + )[(1 − ) +1 − (1 − ) +1 ] ≤ 0, ∀ < . Hence ≤ .
1
2
2
1
2
(1 + )(1 − ) +1 1 2
2
2
1

The mean residual life (MRL) function of the NDL distribution is defined by
1 − (1 − )(2 − )
() = ( = | ≥ ) = () + ,
2 (1 + + )
where ()is the hrf of the NDL distribution.

4. Moments
The first four raw moments of the NDL distribution are, respectively, given by

3
2
(1 − )(2 + ) + − 8 + 6
′
′
= () = (1 + ) , = (1 + ) ,
2
1
2

2
3
(1 − )( + 2 − 24 + 24)
′
= (1 + )
3
3
and
2
3
4
(1 − )(− − 2 + 78 − 192 + 120)
′
̅
= (1 + ) , ℎ = (1 − ).
4
4

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