Characterizations and In nite Divisibility of Certain 2016 Univariate … · 2017-05-08 · Oskouei et al. ; Generalized Log-Logistic Proportional Hazard (GLLPH) distribution of Khan

International Mathematical Forum, Vol. 12, 2017, no. 5, 195 - 228HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/imf.2017.612174

Characterizations and Infinite Divisibility of

Certain 2016 Univariate Continuous Distributions

G.G. Hamedani

Department of Mathematics, Statistics and Computer ScienceMarquette University

Milwaukee, WI 53201-1881, USA

F. Safavimanesh

Department of StatisticsShahid Beheshti University

Tehran, Iran

Copyright c© 2017 G.G. Hamedani and F. Safavimanesh. This article is distributed under

the Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

Twenty univariate continuous distributions appearing in 2016 will be dis-cussed and characterized. The present work is intended to complete, in someway, these twenty papers via establishing certain characterizations of theirdistributions in various directions. The infinite divisibility of some of thesedistributions will be determined as well.

1 Introduction

In designing a stochastic model for a particular modeling problem, an investigator willbe vitally interested to know if their model fits the requirements of a specific underlyingprobability distribution. To this end, the investigator will rely on the characterizations ofthe selected distribution. Generally speaking, the problem of characterizing a distributionis an important problem in various fields and has recently attracted the attention of manyresearchers. Consequently, various characterization results have been reported in the lit-erature. These characterizations have been established in many different directions. Thiswork deals with various characterizations of Odd Log-Logistic Generalized Half-Normal(OLLGHN) distribution of Cordeiro et al.; New Family of Additive Weibull-Generated

196 G.G. Hamedani and F. Safavimanesh

(NFAWG) distributions of Hassan and Hemeda; New Generalized Weibull (NGW) distri-bution of Mahmoud and Mandouh; Transmuted Burr Type XII (TBXII) distribution ofAl-Khazaleh; Odd generalized Exponential Modified Weibull (OGEMW) distribution ofAbdelall; Generalized Linear Failure Rate (GLFR) distribution of Kazemi et al.; AlphaPower Weibull (APW) distribution of Nassar et al.; Generalized Transmuted-G (GT-G)family of distributions of Nofal et al.; New Exponentiated Class (NEC) of distributionsof Rezaei et al.; Exponentiated Generalized Nadarajah-Haghighi (EGNH) distribution ofVedoVatto et al.; Generalized Gompertz-Power Series (GGPS) distribution of Tahmasebiand Jafari; Generalized Odd Half-Cauchy (GOHC) family of distributions of Cordeiro etal.; Generalized Exponentiated Modified Weibull (GEMW) distribution of Pu et al.; SkewGeneralized Inverse Weibull (SGIW) distribution of Mahdy and Ahmed ; Extended Arcsine(EAS) distribution of Cordeiro et al.; Weighted Half Exponential Power (WHEP) distri-bution of Ghitany et al. ; Beta Skew-Generalized Normal (BSGN) distribution of HassaniOskouei et al. ; Generalized Log-Logistic Proportional Hazard (GLLPH) distribution ofKhan and Khosa ; Exponentiated Marshall-Olkin G (EMO-G) family of distributions ofDias et al. and Extend Generalized Odd half-Cauchy-G (EGOHC-G) family of distribu-tions of Cordeiro et al.. These characterizations are presented in different directions: (i)based on a simple relationship between two truncated moments; (ii) in terms of the hazardfunction; (iii) in terms of the reverse (reversed) hazard function and (iv) based on theconditional expectation of certain functions of the random variable. Note that (i) canbe employed also when the cdf (cumulative distribution function) does not have a closedform. In defining the above nineteen distributions we shall try to use the same parameternotation as used by the original authors. Finally, we will discuss the infinite divisibility ofsome of these distributions.

The cdf and pdf (probability density function) of OLLGHN are given, respectively,by

F (x;α, θ, λ) =

2Φ[(

xθ

)λ]− 1α

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α , x ≥ 0, (1)

and

f (x;α, θ, λ) =α√

2π

(λx

) (xθ

)λe−

12(xθ )

2λ 2Φ[(

xθ

)λ]− 1α

2Φ[(

xθ

)λ]− 1α

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α2 , (2)

x > 0 , where α, θ, λ are positive parameters and Φ (x) is the cdf of the standard normalrandom variable.

The cdf and pdf of NFAWG are given, respectively, by

Characterizations and infinite divisibility ... 197

F (x; a, b, c, d) = 1− exp

−c[K (x)

K (x)

]d− a

[K (x)

K (x)

]b, x ∈ R, (3)

and

f (x; a, b, c, d) =k (x)(K (x)

)2cd

[K (x)

K (x)

]d−1

+ ab

[K (x)

K (x)

]b−1×

exp

−c[K (x)

K (x)

]d− a

[K (x)

K (x)

]b, (4)

x ∈ R , where a, b, c, d are positive parameters and K (x) is the baseline cdf withcorresponding pdf k (x) .

The cdf and pdf of NGW are given, respectively, by

F (x;λ, β, γ, α) =[1− e−λx−βxγ

]α, x ≥ 0, (5)

and

f (x;λ, β, γ, α) = α(λ+ βγxγ−1

)e−λx−βx

γ[1− e−λx−βxγ

]α−1, x > 0, (6)

where γ, α > 0, λ, β ≥ 0 are parameters such that λ+ β > 0.

Remark 1. The above NGW distribution is a special case of KGEW distribution

introduced by Kadilar (2014) whose cdf is F (x;α, β, γ, a, b) = 1−[1−

[1− e−(αx+βxγ

]a]b,

≥ 0. For b = 1, KGEW reduces to NGW. We mentioned here that KGEW does not requirethe condition λ + β > 0. The KGEW distribution has been characterized in Hamedani’supcoming Monograph.

The cdf and pdf of TBXII are given, respectively, by

F (x; c, k, s, λ) = 1−[1 +

(xs

)c]−k 1− λ+ λ

[1 +

(xs

)c]−k, x ≥ 0, (7)

and

f (x; c, k, s, λ) = ck

(1

x

)(xs

)c [1 +

(xs

)c]−(k+1)

1− λ+ 2λ[1 +

(xs

)c]−k, x > 0, (8)


where c, k, s all positive and |λ| ≤ 1 are parameters.

The cdf and pdf of OGEMW are given, respectively, by

F (x;α, β, γ, θ, λ) =

(1− e−λ

[eθx+γx

β−1])α

, x ≥ 0, (9)

and

f (x;α, β, γ, θ, λ) = λα(θ + γβxβ−1

)e−λ[eθx+γx

β−1](

1− e−λ[eθx+γx

β−1])α−1

, x > 0, (10)

where α, β, γ, θ, λ are all positive parameters.

The cdf and pdf of GLFR are given, respectively, by

F (x;α, β, a, b) =

(

1−(1−β(ax+ b2x2))

1/β)α, if β 6= 0(

1−e−(ax+ b2x

2))α

, if β=0, (11)

and

f (x;α, β, a, b) =

α(a+bx)(1−βz)

1β−1

(1−(1−βz)1/β)α−1

, if β 6= 0

α(a+bx)e−z(1−βz)1β−1

(1−e−z)α−1, if β=0, (12)

where α > 0, β ∈ R, a ≥ 0, b ≥ 0 (with a+ b > 0) are parameters and z = ax+ b2x

2.

Remarks 2. (A) The support of the above cdf F is as follows: i. (0,∞) if β ≤ 0

; ii.(

0, 1b

√a2 + 2b

β −ab

)if β > 0 and b 6= 0 ; iii.

(0, 1

αβ

)if β > 0 and b = 0. (B) For

β = 0 ,the above GLFR distribution is a special case of KGEW distribution introducedby Kadilar (2014), see Remark 1. For b = 1 and γ = 2 , KGEW reduces to GLFR, so weonly discuss the case β 6= 0 in the next section.

The cdf and pdf of APW are given, respectively, by

F (x;α, β, λ) =

1

1−α

(1−α1−e−λx

β), if α > 0 and α 6= 1(

1−e−λxβ), if α=1

, x ≥ 0, (13)

and


f (x;α, β, λ) =

α(a+bx)(1−βz)

1β−1

(1−(1−βz)1/β)α−1

, if β 6= 0

α(a+bx)e−z(1−βz)1β−1

(1−e−z)α−1, if β=0, x > 0, (14)

where α, β and λ are all positive parameters.

Remark 3. For α = 1 ,the above APW distribution is a Weibull distribution whichhas been characterized in Glanzel and Hamedani (2001). We only discuss the case α 6= 1in the next section.

The cdf and pdf of GT-G are given, respectively, by

F (x;λ, a, b, ϕ) = K (x;ϕ)a[(1 + λ)− λK (x;ϕ)b

], x ∈ R, (15)

and

f (x;λ, a, b, ϕ) = k (x;ϕ)K (x;ϕ)a−1[a (1 + λ)− λ (a+ b)K (x;ϕ)b

], x ∈ R, (16)

where λ (|λ| ≤ 1) , a > 0, b > 0 are parameters.

Remarks 4. (C) The Topp-Leone Generalized Family (TLGF) of Distributions byRezaei et al. (2016) has the following cdf , which in some way, is similar to (15),

F (x;α, θ) =

(G1 (x))θ[2− (G1 (x))θ

]α, x ∈ R,

where α and θ are positive parameters.

(D) Poisson-Odd Generalized Exponential Family (POGEF) of Distributions of Muham-mad, (2016) has the following cdf , which is the same as the above and must have beendone independently,

F (x;α, θ) =

(G1 (x))θ[2− (G1 (x))θ

]α, x ∈ R,

where α and θ are positive parameters.

(E) Both of the above distributions are mentioned and characterized in Hamedani’supcoming Monograph. We, however, characterize (15) in the next section since it is, insome way, a little different from the above cdf .

The cdf and pdf of NEC are given, respectively, by


F (x; a, b, θ, η) = 1−

1− 1− [1−K (x; η)]abθ, x ∈ R, (17)

and

f (x; a, b, θ, η) =abθk (x; η) [1−K (x; η)]a−1 1− [1−K (x; η)]ab−1

1− 1− [1−K (x; η)]ab1−θ , x ∈ R, (18)

where a, b, θ are all positive parameters and k (x; η) , K (x; η) are pdf and cdf of thebaseline distribution which depends on the parameter vector η.

Remark 5. The New Kumaraswamy Kumaraswamy (NKw-Kw) Family of Distribu-tions of Mahmoud et al. (2015) has the following cdf

F (x;α, β, a, b, η) = 1−

1−[1− (1− (K (x; η))α)β

]ab, x ∈ R,

which is a more general form of (17) . Rezaei et al.’s paper was submitted before theabove mentioned paper appeared and we have gathered that these papers were done in-dependently. The reason we mention the above paper, is that it has been characterized inHamedani’s upcoming Monograph, so we will not give a characterization of (17) here.

The cdf and pdf of EGNH are given, respectively, by

F (x; a, b, α, β) =

1−[e1−(1+ax)b

]αβ, x ≥ 0, (19)

and

f (x; a, b, α, β) =aαbβ (1 + ax)b−1

[e1−(1+ax)b

]α

1−[e1−(1+ax)b

]α1−β , x > 0, (20)

where a, b, α, β are all positive parameters.

The cdf and pdf of GGPS are given, respectively, by

F (x;α, β, γ, θ) =C (θtα)

C (θ), x ≥ 0, (21)

and


f (x;α, β, γ, θ) =θαβeγx

(e−βγ

(eγx−1)) [

1− e−βγ

(eγx−1)]α−1

C ′ (θtα)

C (θ), x > 0, (22)

where α, β, γ, θ are all positive parameters and C (θ) =∑∞

n=1 anθn , with an ≥ 0 ,

θ ∈ (0, s) (s can be ∞), t = 1− e−βγ

(eγx−1).

The cdf and pdf of GOHC are given, respectively, by

F (x;α, η) =2

πarctan

[(G (x; η))α

1− (G (x; η))α

], x ∈ R, (23)

and

f (x;α, η) =2αg (x; η) (G (x; η))α−1

π

(G (x; η))2α + [1− (G (x; η))α]2 , x ∈ R, (24)

where α > 0 is a parameter, η is a parameter vector of the baseline distribution G (x; η)with pdf , g (x; η).

The cdf and pdf of GEMW are given, respectively, by

F (x;α, β, δ, λ, k) = 1−γ

(δ,− log

[1− e−( xλ)

keβx]α)

Γ (δ), x ≥ 0, (25)

and

f (x;α, β, δ, λ, k) =1

Γ (δ)

(− log

[1− e−( xλ)

keβx]α)δ−1

(k + βx)

×(αλ

)(xλ

)k−1e−( xλ)

keβx+βx

[1− e−( xλ)

keβx]α−1

, (26)

x > 0, where α, δ, λ, k > 0 , β ≥ 0 are parameters and γ (δ, x) =∫ x

0 tδ−1e−tdt.

The cdf and pdf of SGIW are given, respectively, by

F (x;α, β, γ, λ) = exp(−γ (α/x)β

(1 + λ−β

)), x ≥ 0, (27)

and


f (x;α, β, γ, λ) = βγαβ(

1 + λ−β)x−(β+1) exp

(−γ (α/x)β

(1 + λ−β

)), x > 0, (28)

where α, β, γ, λ are all positive parameters.

Remarks 6. (F ) The Exponentiated Generalized Inverse Weibull (EGIW) Distribu-tion of Elbatal and Muhammed (2014) has the following cdf

F (x;α, β, θ, λ) =[1−

(1− exp

(− (λ/x)θ

))α]β, x ≥ 0.

(G) The SGIW distribution can be considered as a special case of EGIW distributionby taking β = 1 in the above cdf.

(H) The Generalized Inverse Generalized Weibull (GIGW) Distribution of Jain et al.(2014) is exactly the same as that of (EGIW) given above. We assume that they wereintroduced independently by the authors.

(I) The Extended Exponentiated Frechet (EEFr) distribution of Mansoor et al. (2016)is exactly the same as that of (EGIW) given above. We assume that the authors of (EEFr)were not aware of (EGIW) distribution of Elbatal and Muhammed (2014).

(J) The (EEFr) distribution is characterized in the upcoming Hamedani’s Monograph.Thus, SGIW will not be characterized here.

The cdf and pdf of EAS are given, respectively, by

F (x; a, b) =

1−

[1− 2

πarcsin

(√x)]ab

, x ∈ [0, 1) , (29)

and

f (x; a, b) =ab[1− 2

π arcsin (√x)]a−1

π (x− x2)1/2 1−[1− 2

π arcsin (√x)]a1−b , x ∈ (0, 1) , (30)

where a, b are positive parameters.

The cdf and pdf of WHEP are given, respectively, by

F (x;α, β, γ) = 1−(α+ 1)1/γ Γ

(1γ , βx

γ)− Γ

(1γ , (α+ 1)βxγ

)[(α+ 1)1/γ − 1

]Γ(

1γ

) , x ≥ 0, (31)


and

f (x;α, β, γ) = Ce−βxγ(

1− e−αβxγ), x > 0, (32)

where α, β, γ are positive parameters, Γ (a, z) =∫∞z ya−1e−ydy, a, z > 0 and C = [(α+1)β]1/γ

[(α+1)1/γ−1]Γ(

1γ

+1)

is the normalizing constant.

The cdf and pdf of BSGN are given, respectively, by

F (x; a, b, λ1, λ2) =1

B (a, b)

∫ Φ(x;λ1,λ2)

0ta−1 (1− t)b−1 dt, x ∈ R, (33)

and

f (x; a, b, λ1, λ2) =2

B (a, b)(Φ (x;λ1, λ2))a−1 (1− Φ (x;λ1, λ2))b−1 φ (x) Φ

(λ1x√

1 + λ2x2

), x ∈ R, (34)

where a, b > 0, λ1 ∈ R , λ2 ≥ 0 are parameters and Φ and φ are cdf and pdf of standardnormal.

The cdf and pdf of GLLPH are given, respectively, by

F (x; k, ρ, γ) = 1−[1 + γkxk

]−( ργ

)k, x ≥ 0, (35)

and

f (x; k, ρ, γ) =kρkxk−1

[1 + γkxk]−(ργ

)k+1, x > 0, (36)

where k, ρ, γ are all positive parameters.

The cdf and pdf of EMO-G are given, respectively, by

F (x;α, λ, p) =

1−

(G (x)

)λ1− p

(G (x)

)λα

, x ∈ R, (37)

and

f (x;α, λ, p) =αλ (1− p) g (x)

(G (x)

)λ−1[1−

(G (x)

)λ]α−1

[1− p

(G (x)

)λ]α+1 , x ∈ R, (38)


where α, λ > 0 , p < 1 are parameters and G (x) is the baseline cdf with correspondingpdf g (x).

The cdf and pdf of EGOHC-G are given, respectively, by

F (x;α, β, η) =2

πarctan

[G (x; η)α

1−G (x; η)β

], x ∈ R, (39)

and

f (x;α, β, η) =2g (x; η)G (x; η)α−1

[α+ (β − α)G (x; η)β

]π

G (x; η)2α +

[1−G (x; η)β

]2 , x ∈ R, (40)

where α, β > 0 are parameters and G (x; η) is the baseline cdf with corresponding pdfg (x; η) which may depend on a parameter vector η.

Remark 7. The EGOHC-G family of distributions is. in some way, a generalizationof a family by the same authors for α = β. The new family, however, has other interestingproperties discussed in the EGOHC-G paper.

2 Characterizations of distributions

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

2.1 Characterizations based on two truncated moments

This subsection deals with the characterizations of the above mentioned distributionsbased on the ratio of two truncated moments. Our first characterization employs a theoremof Glanzel (1987), see Theorem 1 below . The result, however, holds also when the intervalH is not closed.

Theorem 1. Let (Ω,F ,P) be a given probability space and let H = [d, e] be aninterval for some d < e (d = −∞, e =∞ might as well be allowed) . Let X : Ω→ H bea continuous random variable with the distribution function F and let q1 and q2 be tworeal functions defined on H such that

E [q2 (X) | X ≥ x] = E [q1 (X) | X ≥ x] ξ (x) , x ∈ H,

is defined with some real function η. Assume that q1, q2 ∈ C1 (H), ξ ∈ C2 (H) and Fis twice continuously differentiable and strictly monotone function on the set H. Finally,


assume that the equation ξq1 = q2 has no real solution in the interior of H. Then F isuniquely determined by the functions q1, q2 and ξ , particularly

F (x) =

∫ x

aC

∣∣∣∣ ξ′ (u)

ξ (u) q1 (u)− q2 (u)

∣∣∣∣ exp (−s (u)) du ,

where the function s is a solution of the differential equation s′ = ξ′ q1ξ q1 − q2

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 truncatedmoments is stable in the sense of weak convergence (see, Glanzel 1990), in particular, letus assume that there is a sequence Xn of random variables with distribution functionsFn such that the functions q1n , q2n and ξn (n ∈ N) satisfy the conditions of Theorem1 and let q1n → q1 , q2n → q2 for some continuously differentiable real functions q1 andq2 . Let, finally, X be a random variable with distribution F . Under the conditionthat q1n (X) and q2n (X) are uniformly integrable and the family Fn is relativelycompact, the sequence Xn converges to X in distribution if and only if ξn convergesto ξ , where

ξ (x) =E [q2 (X) | X ≥ x]

E [q1 (X) | X ≥ x].

This stability theorem makes sure that the convergence of distribution functions isreflected by corresponding convergence of the functions q1 , q2 and ξ , respectively. Itguarantees, for instance, the ’convergence’ of characterization of the Wald distribution tothat of the Levy-Smirnov distribution if α → ∞ , as was pointed out in Glanzel andHamedani (2001).

A further consequence of the stability property of Theorem 1 is the application of thistheorem to special tasks in statistical practice such as the estimation of the parameters ofdiscrete distributions. For such purpose, the functions q1, q2 and, specially, ξ shouldbe as simple as possible. Since the function triplet is not uniquely determined it isoften possible to choose ξ as a linear function. Therefore, it is worth analyzing somespecial cases which helps to find new characterizations reflecting the relationship betweenindividual continuous univariate distributions and appropriate in other areas of statistics.

In some cases, one can take q1 (x) ≡ 1, which reduces the condition of Theorem 1 toE [q2 (X) | X ≥ x] = ξ (x) , x ∈ H. We, however, believe that employing three functionsq1 , q2 and ξ will enhance the domain of applicability of Theorem 1.

Proposition 1. Let X : Ω → (0,∞) be a continuous random variable and let

q1 (x) =

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α2 4Φ[(

xθ

)λ]− 3

and

q2 (x) = q1 (x)

2Φ[(

xθ

)λ]− 1

2− 2Φ[(

xθ

)λ]for x > 0. Then, the random variable

X has pdf (2) if and only if the function ξ defined in Theorem 1 is of the form


ξ (x) =α

α+ 1

2Φ

[(xθ

)λ]− 1

2− 2Φ

[(xθ

)λ], x > 0.

Proof. Suppose the random variable X has pdf (2), then

(1− F (x))E [q1 (x) | X ≥ x] =

2Φ

[(xθ

)λ]− 1

α2− 2Φ

[(xθ

)λ]α, x > 0,

and

(1− F (x))E [q2 (x) | X ≥ x] =α

α+ 1

2Φ

[(xθ

)λ]− 1

α+12− 2Φ

[(xθ

)λ]α+1

, x > 0.

Further,

ξ (x) q1 (x)−q2 (x) = − 1

α+ 1q1 (x)

2Φ

[(xθ

)λ]− 1

2− 2Φ

[(xθ

)λ]< 0 for x > 0.

Conversely, if ξ is of the above form, then

s′ (x) =ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

2α2λθ−λxλφ((

xθ

)λ)3− 4Φ

[(xθ

)λ]2Φ[(

xθ

)λ]− 1

2− 2Φ[(

xθ

)λ] , x > 0,

and consequently

s (x) = −α ln

2Φ

[(xθ

)λ]− 1

2− 2Φ

[(xθ

)λ], x > 0,

where φ (x) is the pdf of the standard normal. Now, according to Theorem 1, X hasdensity (2) .

Corollary 1. Let X : Ω → (0,∞) be a continuous random variable and let q1 (x)be as in Proposition 1. The random variable X has pdf (2) if and only if there existfunctions q2 and ξ defined in Theorem 1 satisfying the following differential equation

ξ′ (x)h (x)

ξ (x)h (x)− g (x)=


xθ

)λ)3− 4Φ

[(xθ

)λ]2Φ[(

xθ

)λ]− 1

2− 2Φ[(

xθ

)λ] , x > 0.

The general solution of the above differential equation is


ξ (x) =

2Φ

[(xθ

)λ]− 1

2− 2Φ

[(xθ

)λ]−α×

−∫


xθ

)λ)3− 4Φ

[(xθ

)λ]×2Φ[(

xθ

)λ]− 1

2− 2Φ[(

xθ

)λ]α−1×

(q1 (x))−1 q2 (x) dx+D

,

where D is a constant. We like to point out that one set of functions satisfying the abovedifferential equation is given in Proposition 1 with D = 0. Clearly, there are other triplets(q1, q2, ξ) which satisfy conditions of Theorem1.

Remark 8. For q1 (x) =

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α2 2− 2Φ

[(xθ

)λ]1−α

and q2 (x) = q1 (x)

2− 2Φ[(

xθ

)λ], we have ξ (x) = 1

2

1 +

2Φ[(

xθ

)λ]− 1α

, for

x > 0. For q1 (x) =

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α2 2Φ[(

xθ

)λ]− 11−α

and

q2 (x) = q1 (x)

2Φ[(

xθ

)λ]− 1α

, we have ξ (x) = αα+1

2− 2Φ

[(xθ

)λ], for x > 0.

The following Proposition employs a special form of Theorem 1 mentioned above, inwhich q1 (x) is taken to be identically 1 and hence it depends on two functions q2 and ξ.

Proposition 2. Let X : Ω→ R be a continuous random variable and let q1 (x) ≡ 1

and q2 (x) = exp

−c[K(x)

K(x)

]d− a

[K(x)

K(x)

]bfor x ∈ R. Then, the random variable X has

pdf (4) if and only if the function ξ defined in Theorem 1 is of the form

ξ (x) =1

2exp

−c[K (x)

K (x)

]d− a

[K (x)

K (x)

]b, x ∈ R.

Proof. Suppose the random variable X has pdf (4), then

(1− F (x))E [q1 (x) | X ≥ x] = exp

−c[K (x)

K (x)

]d− a

[K (x)

K (x)

]b, x ∈ R,

and

(1− F (x))E [q2 (x) | X ≥ x] =1

2exp

−2c

[K (x)

K (x)

]d− 2a

[K (x)

K (x)

]b, x ∈ R.


Further,

ξ (x)− q2 (x) = −1

2exp

−c[K (x)

K (x)

]d− a

[K (x)

K (x)

]b< 0 for x ∈ R.

Conversely, if ξ is of the above form, then

s′ (x) =ξ′ (x)

ξ (x)− q2 (x)=

k (x)(K (x)

)2cd

[K (x)

K (x)

]d−1

+ ab

[K (x)

K (x)

]b−1, x ∈ R,

and consequently

s (x) =

−c[K (x)

K (x)

]d− a

[K (x)

K (x)

]b, x ∈ R.

Now, according to Theorem 1, X has density (4) .

Corollary 2. Let X : Ω→ R be a continuous random variable and let q1 (x) be asin Proposition 2. The random variable X has pdf (4) if and only if there exist functionsq2 and ξ defined in Theorem 1 satisfying the following differential equation

ξ′ (x)

ξ (x)− g (x)=

k (x)(K (x)

)2cd

[K (x)

K (x)

]d−1

+ ab

[K (x)

K (x)

]b−1, x ∈ R.


ξ (x) = exp

c

[K (x)

K (x)

]d+ a

[K (x)

K (x)

]b

−∫ k(x)

(K(x))2×

cd[K(x)

K(x)

]d−1+ ab

[K(x)

K(x)

]b−1×

exp

−c[K(x)

K(x)

]d− a

[K(x)

K(x)

]bq2 (x) +D

,

where D is a constant. We like to point out that one set of functions satisfying the abovedifferential equation is given in Proposition 2 with D = 0. Clearly, there are other triplets(q1, q2, ξ) which satisfy conditions of Theorem1.

A Proposition and a Corollary similar to that of proposition 1 and Corollary 1 will bestated (without proofs) for each one of the remaining distributions.


q1 (x) =

1− λ+ 2λ[1 +

(xs

)c]−k−1and q2 (x) = q1 (x)

[1 +

(xs

)c]−1for x > 0. Then,


the random variable X has pdf (8) if and only if the function ξ defined in Theorem 1 isof the form

ξ (x) =k

k + 1

[1 +

(xs

)c]−1, x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=kc(

1x

) (xs

)c−1

1 +(xs

)c , x > 0.


ξ (x) =[1 +

(xs

)c]k [ − ∫ kc ( 1x

) (xs

)c−1 [1 +

(xs

)c]−(k+1)×(q1 (x))−1 q2 (x) dx+D

].


q1 (x) =

(1− e−λ

[eθx+γx

β−1])1−α

and q2 (x) = q1 (x) e−λ[eθx+γx

β−1]

for x > 0. Then, the

random variable X has pdf (10) if and only if the function ξ defined in Theorem 1 is ofthe form

ξ (x) =1

2e−λ[eθx+γx

β−1], x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)= λ

(θ + γβxβ−1

)eθx+γxβ , x > 0.


ξ (x) = eλ[eθx+γx

β−1] [−∫λα(θ + γβxβ−1

)eθx+γxβe

−λ[eθx+γx

β−1]×

(q1 (x))−1 q2 (x) dx+D

].



q1 (x) =(

1−(1− β

(ax+ b

2x2))1/β)1−α

and q2 (x) = q1 (x)(1− β

(ax+ b

2x2))1/β

for

x > 0. Then, the random variable X has pdf (12) if and only if the function ξ defined inTheorem 1 is of the form

ξ (x) =1

2

(1− β

(ax+

b

2x2

))1/β

, x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

ax+ b

1− β(ax+ b

2x2) , x > 0.


ξ (x) =

(1− β

(ax+

b

2x2

))−1 [−∫

(a+ bx) (q1 (x))−1 q2 (x) dx+D

].


q1 (x) = αe−λxβ−1 and q2 (x) = q1 (x) e−λx

βfor x > 0. Then, the random variable X has

pdf (14) if and only if the function ξ defined in Theorem 1 is of the form

ξ (x) =1

2e−λx

β, x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)= λβxβ−1, x > 0.


ξ (x) = eλxβ

[−∫λβxβ−1e−λx

β(q1 (x))−1 q2 (x) dx+D

].


Proposition 7. Let X : Ω → R be a continuous random variable and let q1 (x) =[a (1 + λ)− λ (a+ b)K (x;ϕ)b

]−1and q2 (x) = q1 (x)K (x;ϕ)a for x ∈ R. Then, the ran-

dom variable X has pdf (16) if and only if the function ξ defined in Theorem 1 is of theform

ξ (x) =1

21 +K (x;ϕ)a , x ∈ R.

Corollary 7. Let X : Ω→ R be a continuous random variable and let q1 (x) be asin Proposition 7. The random variable X has pdf (16) if and only if there exist functionsq2 and ξ defined in Theorem 1 satisfying the following differential equation

ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=ak (x;ϕ)K (x;ϕ)a−1

1−K (x;ϕ)a, x ∈ R.


ξ (x) = [1−K (x;ϕ)a]−1

[−∫ak (x;ϕ)K (x;ϕ)a−1 (q1 (x))−1 q2 (x) dx+D

].


q1 (x) =

1−[e1−(1+ax)b

]α1−βand q2 (x) = q1 (x)

[e1−(1+ax)b

]αfor x > 0. Then, the


ξ (x) =α

α+ 1

[e1−(1+ax)b

]α, x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)= aαb (1 + ax)b−1 , x > 0.


ξ (x) =[e1−(1+ax)b

]−α [−∫aαb (1 + ax)b−1

[e1−(1+ax)b

]α(q1 (x))−1 q2 (x) dx+D

].



q1 (x) = (C ′ (θt))−1 and q2 (x) = q1 (x)[1− e−

βγ

(eγx−1)]α

for x > 0. Then, the random

variable X has pdf (22) if and only if the function ξ defined in Theorem 1 is of the form

ξ (x) =1

2

1 +

[1− e−

βγ

(eγx−1)]α

, x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=αβeγx

(e−βγ

(eγx−1)) [

1− e−βγ

(eγx−1)]α−1

1−[1− e−

βγ

(eγx−1)]α , x > 0.


ξ (x) =

1−[1− e−

βγ

(eγx−1)]α−1

−∫αβeγx

(e−βγ

(eγx−1))×[

1− e−βγ

(eγx−1)]α−1

(q1 (x))−1 q2 (x) dx+D

.

Proposition 10. Let X : Ω→ R be a continuous random variable and let q1 (x) =(G (x; η))2α + [1− (G (x; η))α]2

and q2 (x) = q1 (x) (G (x; η))α for x ∈ R. Then, the


ξ (x) =1

21 + (G (x; η))α , x ∈ R..

Corollary 10. Let X : Ω → R be a continuous random variable and let q1 (x) beas in Proposition 10. The random variable X has pdf (24) if and only if there existfunctions q2 and ξ defined in Theorem 1 satisfying the following differential equation

ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=αg (x; η) (G (x; η))α−1

1− (G (x; η))α, x ∈ R.



ξ (x) = 1− (G (x; η))α−1

[−∫αg (x; η) (G (x; η))α−1 (q1 (x))−1 q2 (x) dx+D

].


q1 (x) =

(− log

[1− e−( xλ)

keβx]α)δ−1

e−βx and q2 (x) = q1 (x)

[1− e−( xλ)

keβx]α

for x > 0.

Then, the random variable X has pdf (26) if and only if the function ξ defined in Theorem1 is of the form

ξ (x) =1

2

1 +

[1− e−( xλ)

keβx]α

, x > 0.

Corollary 11. Let X : Ω→ (0,∞) be a continuous random variable and let q1 (x)be as in Proposition 11. The random variable X has pdf (26) if and only if there existfunctions q2 and ξ defined in Theorem 1 satisfying the following differential equation

ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

(k + βx)(αλ

) (xλ

)k−1e−( xλ)

keβx[1− e−( xλ)

keβx]α−1

1−[1− e−( xλ)

keβx]α , x > 0.


ξ (x) =

1−

[1− e−( xλ)

keβx]α−1

−∫

(k + βx)(αλ

) (xλ

)k−1e−( xλ)

keβx+βx

×[1− e−( xλ)

keβx]α−1

(q1 (x))−1 q2 (x) dx+D

.Proposition 12. LetX : Ω→ (0, 1) be a continuous random variable and let q1 (x) =

1−[1− 2

π arcsin (√x)]a1−b

and q2 (x) = q1 (x)[1− 2

π arcsin (√x)]

for x ∈ (0, 1) . Then,the random variable X has pdf (30) if and only if the function ξ defined in Theorem 1is of the form

ξ (x) =a

a+ 1

[1− 2

πarcsin

(√x)], x ∈ (0, 1) .

Corollary 12. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x)be as in Proposition 12. The random variable X has pdf (30) if and only if there existfunctions q2 and ξ defined in Theorem 1 satisfying the following differential equation


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

a

π (x− x2)1/2 [1− 2π arcsin (

√x)] , x ∈ (0, 1) .


ξ (x) =

[1− 2

πarcsin

(√x)]−1 [

−∫

a

π

(x− x2

)−1/2(q1 (x))−1 q2 (x) dx+D

].


q1 (x) = xγ−1eβ(1−αβ)xγ and q2 (x) = q1 (x)(1− e−αβxγ

)2for x > 0. Then, the random


ξ (x) =1

2

1 +

(1− e−αβxγ

)2, x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

2αβγxγ−1e−αβxγ (

1− e−αβxγ)

1− (1− e−αβxγ )2 , x > 0.


ξ (x) =

1−

(1− e−αβxγ

)2−1 [

−∫

2αβγxγ−1e−αβxγ(

1− e−αβxγ)

(q1 (x))−1 q2 (x) dx+D

].

Proposition 14. Let X : Ω→ R be a continuous random variable and let q1 (x) =(1− Φ (x;λ1, λ2))1−b and q2 (x) = q1 (x) (Φ (x;λ1, λ2))a for x ∈ R. Then, the randomvariable X has pdf (34) if and only if the function ξ defined in Theorem 1 is of the form

ξ (x) =1

21 + (Φ (x;λ1, λ2))a , x ∈ R.



ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

αφ (x) (Φ (x;λ1, λ2))a−1 Φ

(λ1x√

1+λ2x2

)1− (Φ (x;λ1, λ2))a

, x ∈ R.


ξ (x) = 1− (Φ (x;λ1, λ2))a−1

[−∫αφ (x) (Φ (x;λ1, λ2))a−1 Φ

(λ1x√

1 + λ2x2

)(q1 (x))−1 q2 (x) dx+D

].

Remark 9. For λ2 = 0 , BSGN distribution reduces to BSN (Beta Skew-Normal)distribution of Mameli and Musio (2013).


q1 (x) = q2 (x)[1 + γkxk

]−1and q2 (x) =

[1 + γkxk

]( ργ

)k−1

for x > 0. Then, the randomvariable X has pdf (36) if and only if the function ξ defined in Theorem 1 is of the form

ξ (x) = 2[1 + γkxk

], x > 0.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

2kγkxk−1

1 + γkxk, x > 0.


ξ (x) =[1 + γkxk

]−1[−∫

2kγkxk−1 (q1 (x))−1 q2 (x) dx+D

].

Proposition 16. Let X : Ω→ R be a continuous random variable and let q1 (x) =1− p

(G (x)

)λα+1and q2 (x) = q1 (x)

[1−

(G (x)

)λ]αfor x ∈ R. Then, the random


ξ (x) =1

2

1 +

[1−

(G (x)

)λ]α, x ∈ R.



ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=αλg (x)

(G (x)

)λ−1[1−

(G (x)

)λ]α−1

1−[1−

(G (x)

)λ]α , x ∈ R.


ξ (x) =

1−[1−

(G (x)

)λ]α−1[−∫αλg (x)

(G (x)

)λ−1[1−

(G (x)

)λ]α−1×

(q1 (x))−1 q2 (x) dx+D

].

Proposition 17. Let X : Ω→ R be a continuous random variable and let q1 (x) =

G (x; η)β−αG (x; η)2α +

[1−G (x; η)β

]2

and q2 (x) = q1 (x)[α+ (β − α)G (x; η)β

]2

for x ∈ R. Then, the random variable X has pdf (40) if and only if the function ξdefined in Theorem 1 is of the form

ξ (x) =1

2

β2 +

[α+ (β − α)G (x; η)β

]2, x ∈ R.


ξ′ (x) q1 (x)

ξ (x) q1 (x)− q2 (x)=

2β (β − α) g (x; η)G (x; η)β−1[α+ (β − α)G (x; η)β

]β2 −

[α+ (β − α)G (x; η)β

]2 , x ∈ R.


ξ (x) =

β2 −

[α+ (β − α)G (x; η)β

]2−1

−∫

2β (β − α) g (x; η)G (x; η)β−1×[α+ (β − α)G (x; η)β

]×

(q1 (x))−1 q2 (x) dx+D

.


2.2 Characterization in terms of hazard function

The hazard function, hF , of a twice differentiable distribution function, F , satisfiesthe following first order differential equation

f ′(x)

f (x)=h′F (x)

hF (x)− hF (x).

It should be mentioned that for many univariate continuous distributions, the aboveequation is the only differential equation available in terms of the hazard function. Inthis subsection we present non-trivial characterizations of OLLGHN , NFAWG , TBXII ,GLFR , APW , EGNH , GEMW , EAS , WHEP , GLLPH and EMO-G distributionsin terms of the hazard function, which are not of the trivial form given above.

Proposition 18. Let X : Ω→ (0,∞) be a continuous random variable. The randomvariable X has pdf (2) if and only if its hazard function hF (x) satisfies the followingdifferential equation

h′F (x) + x−1hF (x)

= αλθ−λx−1 d

dx

xλ−1φ

([(xθ )

λ])

2Φ[(

xθ

)λ]− 1α−1

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α , x > 0.

Proof. It is clear that the above differential equation holds if X has pdf (2).Conversely, if the differential equation holds, then

d

dxxhF (x) = αλθ−λ

d

dx

xλ−1φ

([(xθ )

λ])

2Φ[(

xθ

)λ]− 1α−1

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α ,

or

hF (x) =αλθ−λx

λ−1φ([

(xθ )λ])

2Φ[(

xθ

)λ]− 1α−1

2Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α ,

which is the hazard function of the OLLGHN distribution.

Proposition 19. Let X : Ω → R be a continuous random variable. The randomvariable X has pdf (4) if and only if its hazard function hF (x) satisfies the followingdifferential equation


h′F (x)− k′ (x)

k (x)hF (x)

= k (x)d

dx

cd (K (x))d−1(K (x)

)d+1+ab (K (x))b−1(K (x)

)b+1

, x ∈ R

with the initial condition limx→−∞ hF (x) = 0 for b > 0 and d > 0.

Proof. It is clear that the above differential equation holds if X has pdf (4).Conversely, if the differential equation holds, then

d

dx

(k (x))−1 hF (x)

=

d

dx

cd (K (x))d−1(K (x)

)d+1+ab (K (x))b−1(K (x)

)b+1

,

or

hF (x) = k (x)

cd (K (x))d−1(K (x)

)d+1+ab (K (x))b−1(K (x)

)b+1

,

which is the hazard function of the NFAWG distribution.

Remark 10. For the special case of b = d, the differential equation has the followingsimpler form.

h′F (x)− k′ (x)

k (x)hF (x)

= b (a+ c) (k (x))2 (K (x))b−2

b− 1 + 2K (x)(K (x)

)b+2

.

A Proposition similar to that of Proposition 17 will be stated (without proofs) foreach one of the TBXII , GLFR , APW , EGNH , GEMW EAS , WHEP , GLLPH andEMO-G distributions.


h′F (x)− (c− 1)x−1hF (x)

= ck

(1

x

)(xs

)c d

dx

[1 +

(xs

)c]−1[

1 +λ[1 +

(xs

)c]−k1− λ+ λ

[1 +

(xs

)c]−k]

, x > 0,


with the initial condition limx→0+ hF (x) = 0 for c > 1.

Proposition 21. Let X : Ω→ (0,∞) be a continuous random variable. For α = 1,the random variable X has pdf (12) if and only if its hazard function hF (x) satisfiesthe following differential equation

h′F (x)− β (a+ bx)

1− β(ax+ b

2x2)hF (x) =

b

1− β(ax+ b

2x2) , x > 0,

with the initial condition hF (0) = a.


h′F (x)− (β − 1)x−1hF (x) = log (α)λβxβ−1 d

dx

e−λx

β

αe−λxβ − 1

, x > 0,

with the initial condition limx→0+ hF (x) = 0 for β > 1.

Proposition 23. Let X : Ω→ (0,∞) be a continuous random variable. For β = 1,the random variable X has pdf (20) if and only if its hazard function hF (x) satisfies thefollowing differential equation

h′F (x)− (b− 1) (1 + ax)−1 hF (x) = 0, x > 0,

with the initial condition hF (x) = aαb.


h′F (x) +λ−kxk−1 (k + βx) eβx + β

hF (x)

=(αλ

)e−( xλ)

keβx+βx

× d

dx

(k + βx)

(xλ

)k−1[1− e−( xλ)

keβx]α−1(

− log

[1− e−( xλ)

keβx]α)δ−1

γ

(δ, log

[1− e−( xλ)

keβx]α)

,


x > 0.

Proposition 25. Let X : Ω → (0, 1) be a continuous random variable. For b = 1,the random variable X has pdf (30) if and only if its hazard function hF (x) satisfies thefollowing differential equation

h′F (x) +1− 2x

x− x2hF (x) =

a

π (x− x2)1/2

d

dx

1[

1− 2π arcsin (

√x)] , x ∈ (0, 1) .


h′F (x) + βγxγ−1hF (x) = γ [(α+ 1)β]1/γ e−βxγ×

d

dx

1− e−αβxγ

(α+ 1)1/γ Γ(

1γ , βx

γ)− Γ

(1γ , (α+ 1)βxγ

) ,

x > 0.

Remark 11. For γ = 1 , we have the following differential equation

h′F (x)− αβhF (x) =α (α+ 1)β2

[(2 (α+ 1)− e−αβx

)e−αβx − (α+ 1)

](α+ 1− e−αβx)

2 , x > 0.


h′F (x)− kx−1hF (x) = −k2 (ργ)k x2k−1[1 + γkxk

]−2, x > 0.

Proposition 28. Let X : Ω → R be a continuous random variable. For α = 1, therandom variable X has pdf (38) if and only if its hazard function hF (x) satisfies thefollowing differential equation

h′F (x)− g′ (x)

g (x)hF (x) = λ (1− p) (g (x))2

p (1− λ)(G (x)

)λ − 1[1− p

(G (x)

)λ]2

, x > 0.


2.3 Characterization in terms of the reverse (or reversed) hazard function

The reverse hazard function, rF , of a twice differentiable distribution function, F , isdefined as

rF (x) =f (x)

F (x), x ∈ support of F.

In this subsection we present characterizations of OLLGHN , GLFR , APW , GT-G ,EGNH , GGPS , EAS , WHEP , BSGN and EMO-G distributions (without proofs) interms of the reverse hazard function.

Proposition 29. Let X : Ω → (0,∞) be a continuous random variable. Therandom variable X has pdf (2) if and only if its reverse hazard function rF (x) satisfiesthe following differential equation

r′F (x)−x−1hF (x) = α

√2

π

(λ

x

)d

dx

(xθ

)λ−1e−

12(xθ )

2λ 2− 2Φ

[(xθ

)λ]α−12Φ[(

xθ

)λ]− 1α

+

2− 2Φ[(

xθ

)λ]α , x > 0.

Proposition 30. Let X : Ω→ (0,∞) be a continuous random variable. For β < 0,the random variable X has pdf (12) if and only if its reverse hazard function rF (x)satisfies the following differential equation

r′F (x) +

((1− β) (a+ bx)

1− β(ax+ b

2x2)) rF (x) = α

(1− β

(ax+

b

2x2

)) 1β−1

×

d

dx

a+ bx

1−(1− β

(ax+ b

2x2))1/β

, x > 0.

Proposition 31. Let X : Ω→ (0,∞) be a continuous random variable. For α > 0,α 6= 1 , the random variable X has pdf (14) if and only if its reverse hazard functionrF (x) satisfies the following differential equation

r′F (x) + λβrF (x) = λβ log (α) e−λxβ d

dx

xβ−1α1−e−λxβ

1− α1−e−λxβ

, x > 0.

Proposition 32. Let X : Ω→ R be a continuous random variable. For b = 1 , therandom variable X has pdf (16) if and only if its reverse hazard function rF (x) satisfiesthe following differential equation


r′F (x)− k′ (x;ϕ)

k (x;ϕ)rF (x) = k (x;ϕ)

d

dx

a (1 + λ)− λ (a+ 1)K (x;ϕ)

K (x;ϕ) [1 + λ− λK (x;ϕ)]

, x > 0.


r′F (x)− a (b− 1) (1 + ax)−1 rF (x) = aαbβ (1 + ax)b−1 d

dx

[e1−(1+ax)b

]α1−

[e1−(1+ax)b

]α , x > 0.


r′F (x) + βeγxrF (x) = θαβe−βγ

(eγx−1) d

dx

eγx[1− e−

βγ

(eγx−1)]α−1

C ′ (θtα)

C (θtα)

, x > 0.

Proposition 35. Let X : Ω → (0, 1) be a continuous random variable. For a = 1,the random variable X has pdf (30) if and only if its reverse hazard function rF (x)satisfies the following differential equation

r′F (x) +1− 2x

x− x2rF (x) =

b

π (x− x2)1/2

d

dx

1[

1− 2π arcsin (

√x)] , x ∈ (0, 1) .

Proposition 36. Let X : Ω→ (0,∞) be a continuous random variable. For γ = 1,the random variable X has pdf (32) if and only if its reverse hazard function rF (x)satisfies the following differential equation

r′F (x) + βrF (x) = (α+ 1)βe−βxd

dx

1− e−αβx

α− [(α+ 1)− e−αβx] e−βx

, x > 0.

Proposition 37. Let X : Ω → R be a continuous random variable. For b = 1, therandom variable X has pdf (34) if and only if its reverse hazard function rF (x) satisfiesthe following differential equation


r′F (x) + xrF (x) = a

√2

πe−x

2/2 d

dx

Φ

(λ1x√

1+λ2x2

)Φ (x;λ1, λ2)

, x ∈ R.

Proposition 38. Let X : Ω → R be a continuous random variable. The randomvariable X has pdf (38) if and only if its reverse hazard function rF (x) satisfies thefollowing differential equation

r′F (x)− g′ (x)

g (x)rF (x) = αλ (1− p) g (x)

d

dx

(G (x)

)λ−1[1−

(G (x)

)λ] [1− p

(G (x)

)λ] , x ∈ R.

2.4. Characterization based on the conditional expectation of certain func-tion of the random variable

In this subsection we employ a single function ψ of X and characterize the distributionof X in terms of the truncated moment of ψ (X) . The following propositions have alreadyappeared in Hamedani’s previous work (2013), so we will just state them here which canbe used to characterize some of the above mentioned distributions.

Proposition H1. Let X : Ω → (a, b) be a continuous random variable with cdfF . Let ψ (x) be a differentiable function on (a, b) with limx→a+ ψ (x) = 1. Then forδ 6= 1 ,

E [ψ (X) | X > x] = δψ (x) , x ∈ (a, b) ,

if and only if

ψ (x) = (1− F (x))1δ−1 , x ∈ (a, b)

Proposition H2. Let X : Ω → (a, b) be a continuous random variable with cdfF . Let ψ1 (x) be a differentiable function on (a, b) with limx→b− ψ1 (x) = 1. Thenfor δ1 6= 1 ,

E [ψ1 (X) | X < x] = δ1ψ1 (x) , x ∈ (a, b)

implies

ψ1 (x) = (F (x))1δ1−1. x ∈ (a, b)


Remarks 12. (K) For (a, b) = (0,∞), ψ (x) = e−[K(x)

K(x)

]band δ = a

a+1 , Propo-sition H1 provides a characterization of NFAWG. (L) For (a, b) = (0,∞), ψ (x) =[1 +

(xs

)c]−1

1− λ+ λ[1 +

(xs

)c]−k1/kand δ = k

k+1 , Proposition H1 provides a char-

acterization of TBXII. (M) For (a, b) = (0,∞), ψ (x) = 1− e−λ[eθx+γx

β−1]

and δ = αα+1

, Proposition H2 provides a characterization of OGEMW. (N) For (a, b) = (0,∞),

ψ (x) = 1−(1− β

(ax+ b

2x2)) 1

β and δ = αα+1 , Proposition H2 provides a characterization

of GLFR. (O) For (a, b) = (0,∞), ψ (x) = 1−[e1−(1+ax)b

]αand δ = β

β+1 , Proposition H2

provides a characterization of EGNH. (P ) For (a, b) = (0, 1), ψ (x) = 2π arcsin (

√x) and

δ = bb+1 , proposition H2 provides a characterizations of EAS. (Q) For (a, b) = (0,∞),

ψ (x) =[1 + γkxk

]−1and δ = γk

γk+ρk, Proposition H1 provides a characterization of

GLLPH. (R) For (a, b) = R, ψ (x) =1−[1−(G(x))

λ]

1−p[1−(G(x))

λ] and δ = α

α+1 , Proposition H2

provides a characterization of EMO-G.

3 Infinite divisibility

Bondesson (1979) showed that all the members of the following families

f (x) = C xβ−1 (1 + c xα)−γ , x > 0 , 0 < α ≤ 1 , (41)

f (x) = C xβ−1 exp −c xα , x > 0 , 0 < |α| ≤ 1 , (42)

f (x) = C xβ−1 exp−(c1x+ c2x

−1)

, x > 0,−∞ < β <∞, (43)

f (x) = C x−1 exp− (log x− µ)2 /

(2σ2)

, x > 0, (44)

where the natural restrictions are put on the unspecified parameters, are infinitely divisible.The last one is the lognormal density.

Remark 13. Bondesson (1992, Theorem 6.2.4) pointed out that multiplying den-sities (41)− (44) by C1 (δ + x)−ν for δ > 0 and ν > 0 , will result in densities whichare also infinitely divisible.

We list below the distributions from subsection 2.1 whose densities can be expressedas one of the forms (41) or (42) mentioned above.


NGW: For α = 1, λ = 0 and 0 < γ ≤ 1, the pdf of NGW is of the form (42) .

TBXII: For λ = 0 and 0 < c ≤ 1, the pdf of TBXII is of the form (41) .

ENGH: For β = 1 and 0 < b ≤ 1, the pdf of ENGH is of the form (42) .

SGIW: For 0 < β ≤ 1, the pdf of SGIW is of the form (42) .

GLLPH: For γ > ρ, the pdf of GLLPH is of the form (41) .

4 Concluding Remarks

In designing a stochastic model for a particular modeling problem, an investigator willbe vitally interested to know if their model fits the requirements of a specific underlyingprobability distribution. To this end, the investigator will vitally depend on the charac-terizations of the selected distribution. A good number of 2016 introduced distributionswhich have important applications in many different fields have been mentioned in thiswork. Certain characterizations of these distributions have been established. We hopethat these results will be of interest to the investigators who may believe their models havedistributions mentioned here and are looking for justifying the validity of their models.It is known that determining a distribution is infinitely divisible or not via the existingrepresentations is not easy. We have used Bondesson’s classifications to show that someof the distributions taken up in this work are infinitely divisible. This could be helpful tosome researchers.

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