Characterizations and Infinite Divisibility of Certain Recently Introduced Distributions IV

Certain characterizations of recently proposed univariate continuous distributions are presented in different directions. This work contains a good number of reintroduced distributions and may serve as a source of preventing the reinvention and/or duplication of the existing distributions in the future.

hazard function; (iii) in terms of the reverse (reversed) hazard function and (iv) based on the conditional expectation of certain function of the random variable.Note that (i) can be employed also when the cd f (cumulative distribution function) does not have a closed form.We would also like to mention that due to the nature of some of these 41 distributions, our chracterizations may be the only possible ones.In defining the above distributions we shall try to employ the same parameter notation as used by the original authors.We follow the same order as listed above.
1) The cd f and pd f (probability density function) of EGLFR are given, respectively, by and where α > 0, β ∈ R, a ≥ 0 and b ≥ 0 (with a + b > 0) are parameters, z = ) and i.If β ≤ 0, then the support of the cd f is x ≥ 0.
ii.If β > 0, then the support of the cd f is 0 iii.If β > 0 and b = 0, then the support of the cd f is 0 ≤ x ≤ 1 aβ .
2) The cd f and pd f of McG are given, respectively, by where a, b, c, θ, γ are all positive parameters.
4) The cd f and pd f of LE are given, respectively, by and f (x; a, s, k) = ska k e sx (e sx + a − 1) k+1 , x > 0, (8) where a, s, k are positive parameters.Remark 1.For α = 1 and G 1 (x) = e sx −1 e sx +a−1 , x ≥ 0, the cd f of NF (A New Family) of distributions of Alizadeh et al (2015) reduces to (7).The NF distribution has been characterized in the upcoming Research Monograph by Hamedani and Maadooliat (2017).
6) The cd f and pd f of Go-G are given, respectively, by and where θ, γ are positive parameters and g (x; η) and G (x; η) are pd f and cd f of the baseline distribution which depends on the vector parameter η.
7) The cd f and pd f of TIHL-G are given, respectively, by and where λ is a positive parameters and g (x; η) and G (x; η) are pd f and cd f of the baseline distribution which depends on the vector parameter η.

Special TIHL-G distributions
The following distributions are listed as sub-models of the TIHL-G distribution: The type I half-logistic normal; The type I half-logistic gamma and The type I half-logistic Fréchet.Remark 3.For α = 1, the cd f of EHL-G (The Exponentiated Half-Logistic Family) of distributions of Cordeiro et al (2014) reduces to (13).The EHL-G distribution has been characterized in the upcoming Research Monograph by Hamedani and Maadooliat (2017).
8) The cd f and pd f of TGW are given, respectively, by x ≥ 0 and where θ, δ, η are positive parameters.
10) The cd f and pd f of TIIHL-G are given, respectively, by and where λ is a positive parameter and G (x; ζ) , g (x; ζ) are cd f and pd f of the baseline distribution which depends on the parameter vector ζ.
12) The cd f and pd f of Po-G are given, respectively, by and where θ is a positive parameter and g (x; η) and G (x; η) are pd f and cd f of the baseline distribution which depends on the parameter vector η.
14) The cd f and pd f of EGGP are given, respectively, by and x ∈ R , where a, b, λ are positive parameters and g (x; ψ) and G (x; ψ) are pd f and cd f of the baseline distribution which depends on the parameter vector ψ.
15) The cd f and pd f of BIR are given, respectively, by where a, b, θ are positive parameters.
16) The cd f and pd f of BIR 1 (a = b = 1 2 ) are given, respectively, by where θ is a positive parameter.
17) The cd f and pd f of BWP are given, respectively, by and where α, β, a, b, λ are positive parameters and c = 18) The cd f and pd f of GMW are given, respectively, by where α, β, a, λ are positive parameters.
19) The cd f and pd f of TD are given, respectively, by where α, θ, γ are positive parameters and g (x; η), G (x; η) are pd f and cd f of the baseline distribution with parameter vector η.

21)
The cd f and pd f of NEPL are given, respectively, by and where α, β, λ are positive parameters.

22)
The cd f and pd f of PWR (WLOG for µ = 0, σ = 1) are given, respectively, by where α is a positive parameter.
23) The cd f and pd f of OLLSN (WLOG for µ = 0, σ = 1) are given, respectively, by and where λ is a positive parameter, ϕ (x) , Φ (x) are pd f and cd f of the standard normal distribution, du and Φ S N (x; λ) = 1 − Φ S N (x; λ) .
26) The pd f of LKGG (WLOG for µ = 0, σ = 1) is given, respectively, by x ∈ R, where λ, φ positive , q ∈ R are parameters and , as before, The cd f and pd f of ERK are given, respectively, by and 0 < x < 1, where β, λ are positive parameters.
28) The cd f and pd f of EL are given, respectively, by where α, θ, λ are positive parameters.Remark 5. A generalization of the EL distribution was proposed by Mead (2015) which is characterized in the upcoming Research Monograph by Hamedani and Maadooliat (2017).
29) The cd f and pd f of AEE are given, respectively, by and where α, β are positive parameters.
31) The cd f and pd f of EIWG are given, respectively, by and where α, γ positive and q (0 < q < 1) are parameters.Remark 6.The EIWG distribution is a special case of BGIWG (The Beta Generalized Inverse Weibull Geometric) distribution of Elbatal et al. (2017).We believe that Chung et al. were not aware of Elbatal et al.'s paper as these two papers were published within a couple of months of each other.The BGIWG distribution was characterized in Hamedani (2017) paper listed in the references here.
32) The cd f and pd f of NEE are given, respectively, by and where α, β are positive parameters.
34) The cd f and pd f of GW are given, respectively, by and where α, β, ν are positive parameters.Remark 7. Generalizations of GW distributions were presented by Bidram et al. (2015) and by Elbatal et al. (2016).These distributions are characterized in the upcoming Research Monograph by Hamedani and Maadooliat (2017).
35) The cd f and pd f of ENHE are given, respectively, by x > 0, where α, β, λ are all positive parameters.
36) The cd f and pd f of GW are given, respectively, by and where α, θ, c are all positive parameters.
37) The cd f and pd f of TW-G are given, respectively, by and where α, β are positive parameters.Remark 8.The TW-G distribution is not new.It was first introduced by Gomes et al. (2015), which is also characterized in the upcoming Research Monograph by Hamedani and Maadooliat (2017).
38) The cd f and pd f of GAW-G are given, respectively, by and where a, b, c, d are positive parameters, G (x; η) and g (x; η) are cd f and pd f of the baseline distribution which depends on the parameter vector η .
40) The pd f of GWW is given by where α, β, λ, υ are positive parameters and C = ) is the normalizing constant and B (a, b) is the beta function.
Remark 9.The Generalized Weighted Exponential (GWE) distribution of Kharazmi et al. ( 2017) is a special case of GWW distribution.
41) The cd f and pd f of GB are given, respectively, by and where θ, λ are positive parameters.

Characterizations of Distributions
We present our characterizations (i) − (iv) in four subsections.

Characterizations Based on Two Truncated Moments
This subsection deals with the characterizations of distributions listed in Section 1 based on the ratio of two truncated moments.Our first characterization employs a theorem due to Glänzel (1987), see Theorem 1 of Appendix A .The result, however, holds also when the interval H is not closed, since the condition of the Theorem is on the interior of H.
Remark 1.1.For β = 0 , the distribution (1) has the simple form which has been characterized in our previous work.We will concentrate on the following three cases: I) β < 0 ; II) β > 0 and b 0 and III) β > 0 and b = 0.
)) −1 for x > 0. Then for β < 0, the random variable X has pd f (2) if and only if the function ξ defined in Theorem 1 is of the form )) , x > 0.
Proof.Suppose the random variable X has pd f (2), then and Further, Conversely, if ξ is of the above form, then and consequently Now, according to Theorem 1, X has density (2) .Corollary 1.1.Let X : Ω → (0, ∞) be a continuous random variable and let q 2 (x) be as in Proposition 1.1.For β < 0, the random variable X has pd f (2) if and only if there exist functions q 1 and ξ defined in Theorem 1 satisfying the following differential equation Remark 1.2.The general solution of the differential equation in Corollary 1.1 is where D is a constant.We like to point out that one set of functions satisfying the above differential equation is given in Proposition 1.1 with D = 0. Clearly, there are other triplets (q 1 , q 2 , ξ) which satisfy conditions of Theorem1.
) be a continuous random variable and let q 1 (x) and q 2 (x) be as in Then for β > 0 and b 0, the random variable X has pd f (2) if and only if the function ξ defined in Theorem 1 is of the form Proof.Suppose the random variable X has pd f (2), then Similarly, Further, Conversely, if ξ is of the above form, then and consequently ) be a continuous random variable and let q 2 (x) be as in Proposition 1.2.For β > 0 and b 0, the random variable X has pd f (2) if and only if there exist functions q 1 and ξ defined in Theorem 1 satisfying the following differential equation Remark 1.3.The general solution of the differential equation in Corollary 1.2 is where D is a constant.CASE III : This case is similar to CASE II.
----A Proposition, a Corollary and a Remark similar to Proposition 1.1, Corollary 1.1 and Remark 1.2 will be stated, without proofs, for each of the remaining distributions listed in Section 1.
Proposition 1.3.Let X : Ω → (0, ∞) be a continuous random variable and let )] ac for x > 0.Then, the random variable X has pd f (4) if and only if the function ξ defined in Theorem 1 is of the form Corollary 1.3.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.3.The random variable X has pd f (4) if and only if there exist functions q 2 and ξ defined in Theorem 1 satisfying the following differential equation Remark 1.4.The general solution of the differential equation in Corollary 1.3 is where D is a constant.Proposition 1.4.Let X : Ω → (0, ∞) be a continuous random variable and let and q 2 (x) = q 1 (x) Then, the random variable X has pd f (6) if and only if the function ξ defined in Theorem 1 is of the form Corollary 1.4.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.4.The random variable X has pd f (6) if and only if there exist functions q 2 and ξ defined in Theorem 1 satisfying the following differential equation Remark 1.5.The general solution of the differential equation in Corollary 1.4 is where D is a constant.
Proposition 1.5.Let X : Ω → R be a continuous random variable and let q 1 (x) ≡ 1 and Then, the random variable X has pd f (12) if and only if the function ξ defined in Theorem 1 is of the form Corollary 1.5.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.5.The random variable X has pd f (12) if and only if there exist functions q 2 and ξ defined in Theorem 1 satisfying the following differential equation Remark 1.6.The general solution of the differential equation in Corollary 1.5 is where D is a constant.
Proposition 1.6.Let X : Ω → (0, ∞) be a continuous random variable and let The random variable X has pd f (16) if and only if the function ξ defined in Theorem1 has the form Corollary 1.6.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.6.The random variable X has pd f (16) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.7.The general solution of the differential equation in Corollary 1.6 is where D is a constant.Remark 1.8.Proposition 1.6, Corollary 1.6 and Remark 1.7 were mentioned incorrectly in Nofal et al. (2017).Proposition 1.7.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = The random variable X has pd f (18) if and only if the function ξ defined in Theorem1 has the form Corollary 1.7.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.7.The random variable X has pd f (18) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.9.The general solution of the differential equation in Corollary 1.7 is where D is a constant.Proposition 1.8.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = The random variable X has pd f (20) if and only if the function ξ defined in Theorem1 has the Corollary 1.8.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.8.The random variable X has pd f (20) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation ξ Remark 1.10.The general solution of the differential equation in Corollary 1.8 is where D is a constant.
Proposition 1.9.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = for x > 0. The random variable X has pd f (22) if and only if the function ξ defined in Theorem1 has the form Corollary 1.9.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.9.The random variable X has pd f (22) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.11.The general solution of the differential equation in Corollary 1.9 is where D is a constant.Proposition 1.10.Let X : Ω → R be a continuous random variable and let q 1 (x) ≡ 1 and q 2 (x) = exp [ θG (x; η) ] for x ∈ R. The random variable X has pd f (24) if and only if the function ξ defined in Theorem1 has the form Corollary 1.10.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.10.The random variable X has pd f (24) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.12.The general solution of the differential equation in Corollary 1.10 is where D is a constant.
Proposition 1.11.Let X : Ω → (0, ∞) be a continuous random variable and let The random variable X has pd f (26) if and only if the function ξ defined in Theorem1 has the form Corollary 1.11.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.11.The random variable X has pd f (26) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.13.The general solution of the differential equation in Corollary 1.11 is where D is a constant.Proposition 1.12.Let X : Ω → R be a continuous random variable and let q 1 (x) = exp The random variable X has pd f (28) if and only if the function ξ defined in Theorem1 has the form Corollary 1.12.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.12.The random variable X has pd f (28) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.14.The general solution of the differential equation in Corollary 1.12 is where D is a constant.Proposition 1.13.Let X : Ω → (0, ∞) be a continuous random variable and let for x > 0. The random variable X has pd f (30) if and only if the function ξ defined in Theorem1 has the form Corollary 1.13.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.13.The random variable X has pd f (30) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation ξ Remark 1.15.The general solution of the differential equation in Corollary 1.13 is Disaconstant.Proposition 1.14.LetX:Ω → (0, ∞) be a continuous random variable and let q ) for x > 0. The random variable X has pd f (32) if and only if the function ξ defined in Theorem1 has the form ξ Corollary 1.14.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.14.The random variable X has pd f (32) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation ) , x > 0.
Remark 1.16.The general solution of the differential equation in Corollary 1.14 is where D is a constant.Proposition 1.15.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = ( e λ − e λe −βx α ) 1−a and q 2 (x) = q 1 (x) ( e λe −βx α − 1 ) for x > 0. The random variable X has pd f (34) if and only if the function ξ defined in Theorem1 has the form Corollary 1.15.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.15.The random variable X has pd f (34) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.17.The general solution of the differential equation in Corollary 1.15 is where D is a constant.Proposition 1.16.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = ( αx β e λx ) 1−a and q 2 (x) = q 1 (x) e λx−αx β e λx for x > 0. The random variable X has pd f (36) if and only if the function ξ defined in Theorem1 has the form ξ (x) = 1 2 e −αx β e λx , x > 0.
Remark 1.18.The general solution of the differential equation in Corollary 1.16 is where D is a constant.
Proposition 1.17.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = and q 2 (x) = q 1 (x) The random variable X has pd f (38) if and only if the function ξ defined in Theorem1 has the form ξ Corollary 1.17.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.17.The random variable X has pd f (38) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.19.The general solution of the differential equation in Corollary 1.17 is where D is a constant.
Proposition 1.18.Let X : Ω → R be a continuous random variable and let for x ∈ R. The random variable X has pd f (40) if and only if the function ξ defined in Theorem1 has the form ξ , x ∈ R.
Corollary 1.18.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.18.The random variable X has pd f (40) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.20.The general solution of the differential equation in Corollary 1.18 is where D is a constant.Proposition 1.19.Let X : Ω → (0, ∞) be a continuous random variable and let The random variable X has pd f (42) if and only if the function ξ defined in Theorem1 has the form Corollary 1.19.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.19.The random variable X has pd f (42) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.21.The general solution of the differential equation in Corollary 1.19 is where D is a constant.Proposition 1.20.Let X : Ω → R be a continuous random variable and let q 1 (x) = e −αe −e x and q 2 (x) = q 1 (x) e −e x for x ∈ R. The random variable X has pd f (44) if and only if the function ξ defined in Theorem1 has the form Corollary 1.20.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.20.The random variable X has pd f (44) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.22.The general solution of the differential equation in Corollary 1.20 is where D is a constant.
Proposition 1.21.Let X : Ω → R be a continuous random variable and let ] α−1 and q 2 (x) = q 1 (x) Φ (λx) for x ∈ R. The random variable X has pd f (46) if and only if the function ξ defined in Theorem1 has the form Corollary 1.21.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.21.The random variable X has pd f (46) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation ξ Remark 1.23.The general solution of the differential equation in Corollary 1.21 is where D is a constant.Proposition 1.22.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = and q 2 (x) = q 1 (x) exp ) τ ) for x > 0. The random variable X has pd f (48) if and only if the function ξ defined in Theorem1 has the form Corollary 1.22.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.22.The random variable X has pd f (48) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.24.The general solution of the differential equation in Corollary 1.22 is where D is a constant.
Proposition 1.23.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) and q 2 (x) = q 1 (x) )] 2 for x > 0. The random variable X has pd f (50) if and only if the function ξ defined in Theorem1 has the form Corollary 1.23.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.23.The random variable X has pd f (50) if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.25.The general solution of the differential equation in Corollary 1.23 is where D is a constant.Proposition 1.24.Let X : Ω → R be a continuous random variable and let Corollary 1.27.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.27.The random variable X has pd f (57) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation ξ ′ (x) q 1 (x) ξ (x) q 1 (x) − q 2 (x) = α, x > 0.
Remark 1.29.The general solution of the differential equation in Corollary 1.27 is where D is a constant.
Proposition 1.28.Let X : Ω → (0, ∞) be a continuous random variable and let } −1 and q 2 (x) = q 1 (x) e −λx for x > 0. The random variable X has pd f (59) , if and only if the function ξ defined in Theorem1 has the form Corollary 1.28.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.28.The random variable X has pd f (59) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.30.The general solution of the differential equation in Corollary 1.27 is where D is a constant.Proposition 1.29.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = (1 + βx) −1 and q 2 (x) = q 1 (x) e −αx for x > 0. The random variable X has pd f (63) , if and only if the function ξ defined in Theorem1 has the form ξ (x) = 1 2 e −αx , x > 0.
Corollary 1.29.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.29.The random variable X has pd f (63) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.31.The general solution of the differential equation in Corollary 1.29 is where D is a constant.Proposition 1.30.Let X : Ω → (0, ∞) be a continuous random variable and let The random variable X has pd f (65) , if and only if the function ξ defined in Theorem1 has the form Corollary 1.30.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.30.The random variable X has pd f (65) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.32.The general solution of the differential equation in Corollary 1.30 is where D is a constant.Proposition 1.31.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = ( 1 − e 1−(1+αx) β ) 1−λ and q 2 (x) = q 1 (x) e 1−(1+αx) β for x > 0. The random variable X has pd f (69) , if and only if the function ξ defined in Theorem1 has the form ξ Corollary 1.31.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.31.The random variable X has pd f (69) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.33.The general solution of the differential equation in Corollary 1.31 is where D is a constant.Proposition 1.32.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = ( 1 − e −x c + αe −x c ) θ+1 and q 2 (x) = q 1 (x) e 1−(1+αx) β for x > 0. The random variable X has pd f (71) , if and only if the function ξ defined in Theorem1 has the form Corollary 1.32.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.32.The random variable X has pd f (71) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.34.The general solution of the differential equation in Corollary 1.32 is where D is a constant.Proposition 1.33.Let X : Ω → R be a continuous random variable and let The random variable X has pd f (75) , if and only if the function ξ defined in Theorem1 has the form Corollary 1.33.Let X : Ω → R be a continuous random variable and let q 1 (x) be as in Proposition 1.33.The random variable X has pd f (75) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.35.The general solution of the differential equation in Corollary 1.33 is where D is a constant.Proposition 1.34.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = and q 2 (x) = q 1 (x) e −λx for x > 0. The random variable X has pd f (77) , if and only if the function ξ defined in Theorem1 has the form ξ (x) = 1 2 e −λx , x > 0.
Corollary 1.34.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.34.The random variable X has pd f (77) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.36.The general solution of the differential equation in Corollary 1.34 is where D is a constant.Proposition 1.35.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = ( 1 − e −λ(αx) β ) −υ and q 2 (x) = q 1 (x) e −λx β for x > 0. The random variable X has pd f (78) , if and only if the function ξ defined in Theorem1 has the form ξ (x) = 1 2 e −λx β , x > 0.
Corollary 1.35.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.35.The random variable X has pd f (78) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.37.The general solution of the differential equation in Corollary 1.35 is where D is a constant.Proposition 1.36.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) = e (x/θ) λ and q 2 (x) = q 1 (x) ( 1 − e −(x/θ) λ ) 2 for x > 0. The random variable X has pd f (80) , if and only if the function ξ defined in Theo-rem1 has the form ξ Corollary 1.36.Let X : Ω → (0, ∞) be a continuous random variable and let q 1 (x) be as in Proposition 1.36.The random variable X has pd f (80) , if and only if there exist functions q 2 and ξ defined in Theorem1 satisfying the following differential equation Remark 1.38.The general solution of the differential equation in Corollary 1.36 is where D is a constant.

Characterization in Terms of Hazard Function
The hazard function, h F , of a twice differentiable distribution function, F, satisfies the following first order differential equation It should be mentioned that for many univariate continuous distributions, the above equation is the only differential equation available in terms of the hazard function.In this subsection we present non-trivial characterizations of EGLFR (for α = 1) , McG (for a = b = c = 1) , Go-G , EGGP (for b = 1) , BIR (for a = 1) , BWP (for a = 1) , EGG (for α = 1), NEPL (for α = 1), EGGG, ETQL (for λ = 0, β = 1), ERK (for λ = 1), AEE, NEE , ETW (for ν = 1), ENHE (for λ = 1), GW (for θ = 1), GAW-G (for b = 0 or d = 0), GB distributions in terms of the hazard function, which are not of the above trivial form.Proposition 2.1.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (2) (for β < 0 and α = 1) if and only if its hazard function h F (x) satisfies the following differential equation Proof.If X has pd f (2) for β < 0 and α = 1, then clearly the above differential equation holds.If the differential equation holds, then from which we arrive at the hazard function corresponding to the pd f (2) .Remark 2.1.Similar Propositions can be stated for the cases II and III.
A Proposition similar to that of Proposition 2.1 will be stated (without proof) for each one of the distributions listed in subsection 2.1.Proposition 2.2.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (4) (for a = b = c = 1) if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.3.Let X : Ω → R be a continuous random variable.The random variable X has pd f (12) if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.4.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (20) if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.5.Let X : Ω → R be a continuous random variable.The random variable X has pd f (28) ,for b = 1, if and only if its hazard function h F (x) satisfies the following differential equation x ∈ R. Proposition 2.6.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (30) ,for a = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.7.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (34) ,for a = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.8.Let X : Ω → R be a continuous random variable.The random variable X has pd f (40) ,for α = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.9.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (42) ,for α = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.10.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (48), if and only if its hazard function h F (x) satisfies the following differential equation Remark 2.2.For k = 1, the above differential equation has the following simpler form Proposition 2.11.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X, has pd f (50), for λ = 0, β = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.12.Let X : Ω → (0, 1) be a continuous random variable.The random variable X, has pd f (53), for λ = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.13.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X, has pd f (57) if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.14.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X, has pd f (63), if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.15.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X, has pd f (65), for ν = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.16.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X, has pd f (69), for λ = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.17.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X, has pd f (71), for θ = 1, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.18.Let X : Ω → R be a continuous random variable.The random variable X, has pd f (73), for d = 0, if and only if its hazard function h F (x) satisfies the following differential equation Proposition 2.19.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (80) if and only if its hazard function h F (x) satisfies the following differential equation

Characterization in Terms of the Reverse (or reversed) Hazard Function
The reverse hazard function, r F , of a twice differentiable distribution function, F , is defined as In this subsection we present characterizations of EGLFR, McG (for b = 1) , Kw-TEMW (for b = 1 , λ = 0) , ALLGE , Po-G , EGGP (for a = 1) , BIR (for b = 1) , BWP (for b = 1), GMW (for a = 1), TW (for λ = 0), PWR (for µ = 0, σ = 1), OLLSN (for µ = 0, σ = 1), EGGG (for λ = 1), ETQL (for λ = 0), ERK, ETW, ENHD, GWW (for α = 1) distributions (without proofs) in terms of the reverse hazard function.Proposition 3.1.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (2) (for β < 0) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.2.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (4) (for b = 1) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.3.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (6) (for b = 1, λ = 0) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.4.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (18) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.5.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (20) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.6.Let X : Ω → R be a continuous random variable.The random variable X has pd f (24) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.7.Let X : Ω → R be a continuous random variable.The random variable X has pd f (28) if and only if its reverse hazard function r F (x) satisfies the following differential equation x ∈ R. Proposition 3.8.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (30) ,for b = 1, if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.9.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (34) ,for b = 1, if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.11.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (38) ,for λ = 0, if and only if its reverse hazard function r F (x) satisfies the following differential equation r ′ F (x) + (θ + 1) x −1 r F (x) = α 2 βθ 2 x −2(θ+1) (1 + αx −θ ) −2 , x > 0.
Proposition 3.12.Let X : Ω → R be a continuous random variable.The random variable X has pd f (44) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.13.Let X : Ω → R be a continuous random variable.The random variable X has pd f (46) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proposition 3.14.Let X : Ω → (0, ∞) be a continuous random variable.The random variable X has pd f (48) ,for λ = 1, if and only if its reverse hazard function r F (x) satisfies the following differential equation is defined with some real function η.Assume that q 1 , q 2 ∈ C 1 (H), ξ ∈ C 2 (H) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation ξq 1 = q 2 has no real solution in the interior of H. Then F is uniquely determined by the functions q 1 , q 2 and ξ , particularly ξ ′ (u) ξ (u) q 1 (u) − q 2 (u) exp (−s (u)) du , where the function s is a solution of the differential equation s ′ = ξ ′ q 1 ξq 1 −q 2 and C is the normalization constant, such that ∫ H dF = 1.We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see, Glänzel [2]), in particular, let us assume that there is a sequence {X n } of random variables with distribution functions {F n } such that the functions q 1n , q 2n and ξ n (n ∈ N) satisfy the conditions of Theorem 1 and let q 1n → q 1 , q 2n → q 2 for some continuously differentiable real functions q 1 and q 2 .Let, finally, X be a random variable with distribution F .Under the condition that q 1n (X) and q 2n (X) are uniformly integrable and the family {F n } is relatively compact, the sequence X n converges to X in distribution if and only if ξ n converges to ξ , where This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions q 1 , q 2 and ξ , respectively.It guarantees, for instance, the 'convergence' of characterization of the Wald distribution to that of the Lévy-Smirnov distribution if α → ∞.
A further consequence of the stability property of Theorem 1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions.For such purpose, the functions q 1 , q 2 and, specially, ξ should be as simple as possible.Since the function triplet is not uniquely determined it is often possible to choose ξ as a linear function.Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics.