Characterizations of Discrete Weibull Distributions

Seven versions of the discrete Weibull distribution are characterized via conditional expectation of function of the random variable as well as based on the hazard or reverse hazard function.

where > 0 and ∈ (0,1) , ∈ ℝ are parameters. The reason for using (1.1.3) and (1.1.3) * is that for all of the distributions, except one, we will be using the hazard function.
( ) The cdf , pmf, hf and rhf of DWII distribution are given, respectively, by where ≤ 1,0 < ≤ 1 are parameters. Remark 1.1. The authors consider two case ≤ 1 and > 1. The case ≤ 1 is considered here since in this case, ∈ ℕ which seems compatible with the other six distributions taken up in this work. The formulas given above are slightly different from those of the authors.

Characterizations Based on Conditional Expectation
In this section we present our characterizations of all the distributions listed in the Introduction in terms of the truncated moments of certain functions of the random variables. The choice of each function depends on the form of the pmf. We devote a sub-section to each one of the seven distributions. Most of the proofs follow the same scheme, we will give all of them for the sake of completeness. Nakagawa and Osaki (1975) proposed the first analogue of two-parameter continuous Weibull distribution. Proof. If has pmf (1.1.2) , then for ∈ ℕ * , the left-hand side of (2.1.1) will be

Discrete Weibull (DWI) Distribution
. From the above equality, we have which, in view of (1.1.3), implies that has pmf (1.1.2).

Discrete Weibull (DWII) Distribution
As mentioned by Almalki and Nadarajah (2014), Stein and Dattero (1984) introduced a discrete Weibull distribution called type II discrete Weibull distribution by taking the lifetimes as the integer part of the continuous Weibull distribution.
has pmf (1.1.5) , then for ∈ ℕ, the left-hand side of (2.2.1) will be Similarly, for < 0 and ∈ ℤ , we obtain the right hand side of (2.2.1).
). From the above equality, after some manipulations, we have has mpf (1.1.5).

Discrete Weibull (DWIII) Distribution
Again as pointed Almalki and Nadarajah (2014), the third discrete version of the Weibull distribution was introduced by Padgett and Spurrier (1985). It exhibit increasing, decreasing and constant hazard function.
Proof. We take −1 ≤ < 0 or > 0 to make sure that the infinite series is convergent. If has pmf (1.1.8) , then for ∈ ℕ * , the left-hand side of (2.3.1) will be From the above equality, we have which, in view of (1.1.9), implies that has pmf (1.1.8).

Discrete Modified Weibull (DMW) Distribution
This distribution was proposed by Nooghabi et al. (2011) which is the discrete analogue of the modified Weibull distribution of Lai et al. (2003). Proof. If has pmf (1.1.14) , then for ∈ ℕ * , the left-hand side of (2.5.1) will be From the above equality, after some manipulations, we have +2 − 1, which in view of (1.1.15), implies that has mpf (1.1.14). Bebbington et al. (2012) proposed the four-parameter discrete additive Weibull distribution.

Characterizations of distributions based on hazard function
This section consists of 6 sub-sections devoted to 6 of the 7 distributions listed in the Introduction. The characterizations presented here are in terms of the hazard function. Most of the proofs follow the same scheme and some can be omitted. We, however, give all the proofs and the reader may choose to skip the proofs. Due to the fact that these 6 distributions form a subset of those mentioned in Section 2, we will not repeat the statements given at the beginning of the corresponding sections for them in this section.

Discrete Weibull (DWI) Distribution
Proposition 3.1.1. Let : Ω → ℕ * be a random variable. The pmf of is (1.1.2) if and only if its hazard function satisfies the difference equation with the initial condition ℎ (0) = 1− .