Characterizations and Infinite Divisibility of Certain Univariate Continuous Distributions II

Transcription

1 International Mathematical Forum, Vol. 12, 2017, no. 12, HIKARI Ltd, Characterizations Infinite Divisibility of Certain Univariate Continuous Distributions II G.G. Hamedani Department of Mathematics, Statistics Computer Science Marquette University, Milwaukee, WI , USA Copyright c 2017 G.G. Hamedani. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract Twenty univariate continuous distributions appearing in 2016 were discussed characterized in Hamedani Safavimanesh 2017). These characterizations were intended to complete, in some way, the cited papers. The present work is a continuation of their 2017) paper dealing with other interesting univariate continuous distributions, some of which will appear in These characterizations are also intended to complete, in the same way, the new cited papers. The present work deals with certain characterizations of these new distributions established in various directions. The infinite divisibility of some of these distributions will be determined as well. 1 Introduction As mentioned in our 2017) paper, in designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution. To this end, the investigator will rely on the characterizations of the selected distribution. Generally speaking, the problem of characterizing a distribution is an important problem in various fields has recently attracted the attention of many researchers. Consequently, various characterization results have been reported in the literature. These characterizations have been established in many different directions. This work is the continuation of Hamedani Safavimanesh 2017) which deals with various characterizations of Generalized Inverse Lindley GIL) distribution of Asgharzadeh et al. ; Exponentiated Kumaraswamy-Power

2 566 G.G. Hamedani Function EKPF) distribution of Bursa Kadilar ; Exponentiated Power Lindley Poisson EPLP) distribution of Pararai et al. ; Complementary Geometric Transmuted-G CGT-G) family of distributions of Afify et al. ; Lindley Family LF) of distributions of Cakmakyapan Ozel ; Odd Burr Lindley OBL) distribution of Altun et al. ; Exponentiated Weibull Rayleigh EWR) distribution of Elgarhy ; Transmuted Weibull TW) distribution of Khan et al. ; Generalized Odd Half-Cauchy GOHC) family of distributions of Cordeiro et al. ; Quasi X Gamma QXG) distribution of Sen Chra ; Generalization of Alpha Skew Normal GASN) of Sharafi et al. ; New Exponential Class NEC) of distributions of Rezaei et al. ; Geometric Power Half-Normal GPHN) distribution of Gómez Bolfarine ; -Logarithmic Transformation Exponential ALTE) distribution of KaraKaya et al. ; Transmuted Two- Parameter Lindley TTPL) distribution of Kemaloglu Yilmaz ; Modified Burr III G MBIIIG) family of distributions of Arifa et al. ; Alpha-Power Transformed Weibull APTW) distribution of Dey Sharma ; Geometric Power Lindley Poisson GPLP) distribution of Mansour et al.; Double Weighted Weibull DWW) distribution of Saghir Saleem; Maxwell Length Biased MLB) distribution of Saghir et al.; Odd Lindley-G OL-G) family of distributions of Gomes-Silva et al.; Generalized Transmuted Lindley GTL) distribution of Mansour Mohamed ; New Wider NW) family of distributions of Cordeiro et al.; Transmuted Exponentiated Gumbel TEG) distribution of Deka et al.; Inverted Kumaraswamy IK) distribution of Al-Fattah et al.; Beta Generalized Inverse Weibull Geometric BGIWG) distribution of Elbatal et al.; Reflected Generalized Topp- Leone Power Series RGTLPS) distribution of Condino Domma; Exponentiated Poisson-Exponential EPE) distribution of Louzada et al.; Transmuted Exponentiated Additive Weibull TEAW) distribution of Nofal et al.; Transmuted Kumaraswamy Exponentiated Inverse Rayleigh TKEIR) distribution of Badr; Kumaraswamy Odd Burr G KOBG) family of distributions of Nasir et al.; Extension of Inverse Lindley EIL) distribution of Sharma Khelwal; Marshall-Olkin Extended Generalized Gompertz MOEGG) distribution of Benkhelifa; Exponentiated Generalized Stardized Half-Logistics EGSHL) distribution of Cordeiro et al.; Extension of the Kumaraswamy EKw) distribution of Carrasco Cordeiro; Gamma generalized Pareto GGP) distribution of de Andrade et al.. These characterizations are presented in different directions: i) based on a simple relationship between two truncated moments; ii) in terms of the hazard function; iii) in terms of the reverse reversed) hazard function; iv) based on the conditional expectation of certain functions of the rom variable v) based on a functional equation. Note that i) can be employed also when the cdf cumulative distribution function) does not have a closed form. In defining the above distributions we shall try to employ the same parameter notation as used by the original authors. Finally, we will discuss the infinite divisibility of some of these distributions. The cdf pdf probability density function) of GIL are given, respectively, by F x;, λ) = 1 + λ + λx e λx, x 0, 1) 1 + λ f x;, λ) = λ2 1 + x ) x 1 e λx, x > 0, 2) 1 + λ

3 Characterizations infinite divisibility of where, λ are positive parameters. The cdf pdf of EKPF are given, respectively, by F x;, β, θ, a, b) = 1 1 x ) ) aβ b θ 1, 0 x, 3) f x;, β, θ, a, b) = θabβxaβ 1 aβ x ) ) aβ b 1 1 x ) ) aβ b θ 1 1, x 0, ) 4) where, β, θ, a, b are all positive parameters. The cdf pdf of EPLP are given, respectively, by ω} exp θ βx β+1 e βx) 1 F x;, β, ω, θ) = e θ, x 0, 5) 1 β 2 ωθ f x;, β, ω, θ) = β + 1) e θ 1) 1 + x ) x 1 e βx ) ω βx β + 1 e βx ) ω } exp θ βx β + 1 e βx, x > 0. 6) where, β, ω, θ are positive parameters. The cdf pdf of CGT-G are given, respectively, by F x; θ, λ, ϕ) = θg x; ϕ) 1 + λ λg x; ϕ), x R, 7) 1 1 θ) G x; ϕ) 1 + λ λg x; ϕ) θg x; ϕ) 1 + λ 2λG x; ϕ) f x; θ, λ, ϕ) = 2, x R, 8) 1 1 θ) G x; ϕ) 1 + λ λg x; ϕ)} where θ 0, 1), λ 1 are parameters ϕ is the parameter of the baseline distribution G x; ϕ) with pdf g x; ϕ). The cdf pdf of LF are given, respectively, by F x; θ, ξ) = 1 1 θ ) } ) θ log G x; ξ) G x; ξ), x R, 9) θ + 1

4 568 G.G. Hamedani f x; θ, ξ) = θ2 θ + 1 g x; ξ) 1 log G x; ξ) ) G x; ξ) ) θ 1, x R, 10) where θ > 0 is a parameter g x; ξ) is pdf of the baseline distribution, G x; ξ), with parameter vector ξ. The cdf pdf of OBL are given, respectively, by λx 1+λ ) a λx 1+λ e λx ) a e λx λx 1+λ F x; a, b, λ) = 1 b ) a, x 0, 11) e λx ) a 1 abλ 2 1+λ x) e λx 1 + λx 1+λ e λx f x; a, b, λ) = ) a ) a } b λx 1+λ e λx λx 1+λ e λx 1 + λx ) ab 1 e λx, x > 0, 12) 1 + λ where a, b, λ are positive parameters. The cdf pdf of EWR are given, respectively, by F x;, β, λ, a) = ) β a 1 exp e λx2 1), x 0, 13) ) β 1 ) β f x;, β, λ, a) = 2aβλ e λx2 1 exp λx 2 e λx2 1 ) β a 1 1 exp e λx2 1), x > 0, 14) where, β, λ, a are positive parameters. Special cases of EWR: a) Exponentiated Generalized Weibull Gompertz EGWG) of El-Damcese et al. 2015): The cdf pdf of EGWG are given, respectively, by F x; a, b, c, d, θ) = θ 1 exp ax b e cxd 1)}, x 0,

5 Characterizations infinite divisibility of f x) = abθx b 1 e axb e cxd 1 )+cx 1 d + cdb ) xd e cxd θ 1 1 exp ax b e cxd 1)}, x > 0, where a, b, c, d θ are all positive parameters. Note 1: For b = 0 d = 2, EGWG is a special case of EWR. A special case of EGWG for θ = 1, was taken up by El-Bassiouny et al. 2015) characterized based on the upper record values. Another special case of EGWG for θ = d = 1 b = 0 has appeared in a paper by Maiti Pramanik 2015). b) Transmuted Generalized Gompertz TGG) distribution of Khan et al. 2016): The cdf pdf of TGG are given, respectively, by F x;, β, ξ, λ) = 1 exp ) } β e ξx 1 ξ 1 + λ λ 1 exp e ξx 1) } } β, x 0, ξ f x;, β, ξ, λ) = βe ξx exp 1 + λ 2λ e ξx 1) } 1 exp ) } β 1 e ξx 1 ξ ξ 1 exp e ξx 1) } } β, x > 0, ξ where, β, ξ > 0, λ 1 are parameters. Note 2: For λ = 0, TGG is a special case of EWR. c) Burr Type X Weibull BrX-W) distribution of Rasekhi et al. 2016): The cdf pdf of BrX-W are given, respectively, by F x; θ, β) = } 2 θ 1 exp e xβ 1), x 0, f x; θ, β) = 2θβx β 1 1 e xβ) ) } 2 exp 2x β e xβ 1 } 2 θ 1 1 exp e xβ 1), x > 0,

6 570 G.G. Hamedani where θ, β are positive parameters. Note 3: For β = 2, BrX-W is a special case of EWR. The cdf pdf of TW are given, respectively, by ) ) F x;, β, λ) = 1 e x β 1 + λe x β, x 0, 15) f x;, β, λ) = β ) x 1 ) ) e x β 1 λ + 2λe x β, x > 0, 16) β where, β both positive λ 1 are parameters. The cdf pdf of GOHC are given, respectively, by F x; ) = 2 G x; ξ) π arctan 1 G x; ξ), x R, 17) f x; ) = 2g x; ξ) G x; ξ) 1 x R, 18) π G x; ξ) G x; ξ) 2}, where is a positive parameter G x; ξ) is a baseline cdf, which may depend on the vector parameter ξ, with the corresponding pdf g x; ξ). The cdf pdf of QXG are given, respectively, by ) θx + θ 2 2 F x;, θ) = 1 x2 e θx, x 0, 19) 1 + f x;, θ) = θ ) + θ x2 e θx, x > 0, 20) where, θ are positive parameters. The cdf pdf of GASN are given, respectively, by F x;, λ) = 1 C, λ) bδφ ) Φ x; λ) x ) ϕ x; λ) x 1 + λ 2 ) λ 2 W x 1 + λ 2 )}, 21) x R f x;, λ) = 1 x)2 + 1 ϕ x) Φ λx), x R, 22) C, λ)

7 Characterizations infinite divisibility of where, λ are positive parameters, C, λ) = 1 bδ + 2 2, b = 2 π, δ = λ, ϕ.), 1+λ 2 Φ.) are pdf cdf of stard normal distribution, ϕ x; λ), Φ x; λ) pdf cdf of skew normal distribution W.) = ϕ.) Φ.). The cdf pdf of NEC are given, respectively, by F x; a, b, θ, ξ) = G x; ξ)) a } b} θ, x R, 23) f x; a, b, θ, ξ) = abθg x; ξ) 1 G x; ξ)) a G x; ξ)) a } b G x; ξ)) a } b} 1 θ, x R, 24) where a, b, θ are positive parameters, G x; ξ), g x; ξ) are cdf pdf of the baseline distribution which depend on the parameter vector ξ. Note 4. The NEC is a special case of the New Kumaraswamy Kumaraswamy NKw- Kw) family of distributions of Mahmoud et al. 2015), which has cdf given by F x; a, b, θ,, ξ) = G x; ξ)) ) a } b} θ, x R. For = 1, cdf of NKw-Kw reduces to 23). We believe that these two classes of distributions were obtained independently. We also like to mention that NKw-Kw has been characterized in upcoming monograph by Hamedani Maadooliat. The cdf pdf of GPHN are given, respectively, by F x;, θ, ) = 1 1 θ) 1 2Φ ) ) x 1 1 θ 1 2Φ ) ) x, x 0, 25) 1 f x;, θ, ) = 2 1 θ) ϕ ) x 2Φ x ) ) 1 1 θ 1 θ 1 2Φ ) ) x } 2, x > 0, 26) 1 where > 0, > 0, θ 0, 1) are parameters, Φ x), ϕ x) are cdf pdf of the stard normal distribution. The cdf pdf of ALTE are given, respectively, by F x;, β) = log e x/β), x 0, 27) log 1 + ) f x;, β) = e x/β β log 1 + ) e x/β), x > 0, 28)

8 572 G.G. Hamedani where 1, ) 0}, β > 0 are parameters. The cdf pdf of TTPL are given, respectively, by x 0, F x;, θ, λ) = 1 + λ) 1 λ 1 θ + + θx θ + θ 2 f x;, θ, λ) = 1 + λ) θ + θ + + θx 2λ 1 ) θ + + θx e θx θ + ) 2 e θx, 29) θ x) e θx x > 0, where > 0, θ > 0 λ λ 1) are parameters. The cdf pdf of MBIIIG are given, respectively, by ) e θx, 30) ) G x; ξ) β γ F x;, β, γ) = 1 + γ, x R, 31) G x; ξ) f x;, β, γ) = β x; ξ) G x; ξ)) β+1) ) G x; ξ) β γ ) β 1) γ, x R, 32) G x; ξ) G x; ξ) where, β, γ are positive parameters G x; ξ), g x; ξ) are cdf pdf of the baseline distribution. ) β Note 5. The term can be replaced with β, Gx;ξ)) which may be easier to Gx;ξ) Gx;ξ) deal with. That being said, we will use the authors form nonetheless. The cdf pdf of APTW are given, respectively, by F x;, β, λ) = 1 e βxλ ) 1, e βxλ, =1, x 0, 33) f x;, β, λ) = log )βλx λ 1 e βxλ 1 e βxλ ) 1, 1 1 βλx λ 1 e βxλ, =1, x > 0, 34)

9 Characterizations infinite divisibility of where, β, λ are positive parameters. Note 6. i) For = 1, APTW reduces to a Weibull distribution which has been characterized in our previous work. We consider the case 1 in the present work. ii) For λ = 1, APTW reduces to Alpha Power Exponential APE) distribution of Mahdavi Kundu 2017). We believe that APTW APE were introduced independently. The cdf pdf of GPLP are given, respectively, by exp 1 e λ γ λ θ+1+θx β θ+1 1 exp λ F x; θ, λ, β, γ) = ) e θxβ) e λ ) ), x 0, 35) e θxβ θ+1+θx β θ+1 f x; θ, λ, β, γ) = exp θ + 1) θx β λ λθ 2 1 γ) βx β 1 1 e λ) 1 + x β) ) )} 1 e λ γ 1 exp λ θ+1+θx β 2 θ+1 e θxβ θ θx β θ + 1 ) e θxβ ), 36) x > 0, where θ, λ, β all positive γ 0 < γ < 1) are parameters. The cdf pdf of DWW are given, respectively, by F x; c, λ) = C x 0 λt λ e tλ 1 e cλ t λ) dt, x 0, 37) f x; c, λ) = Cx λ e xλ 1 e cλ x λ), x > 0, 38) where c, λ are positive parameters C = 1+c λ ) 1+ λ 1 Γ1+ λ) 1 1+c λ ) 1+ λ 1 1 The cdf pdf of MLB are given, respectively, by ) F x; ) = 1 e x2 x , x 0, 39) f x; ) = x3 x e 2 2, x > 0, 40) where > 0 is a parameter. The cdf pdf of PL-G are given, respectively, by ).

10 574 G.G. Hamedani F x; a, η) = 1 a + K x; η) 1 + a) K x; η) exp a K x; η) K x; η) }, x R, 41) } f x; a, η) = a2 k x; η) 1 + a) K x; η) 3 exp K x; η) a, x R, 42) K x; η) where a > 0 is a parameter, K x; η) K x; η) = 1 K x; η)) k x; η) are cdf pdf of the baseline distribution which depend on the parameter vector η. The cdf pdf of GTL are given, respectively, by F x;, θ, γ, λ) = 1 + λ) 1 θ θx γ e θx θ + 1 λ 1 θ θx e θx, 43) θ + 1 f x;, θ, γ, λ) = θ2 1 + x) e θx θ λ) γ λ γ 1 1 θ+1+θx θ+1 e θx 1 1 θ+1+θx θ+1 e θx, 44) x > 0, where, θ, γ all positive 1 < λ < 0 or both, θ positive, 2 γ < λ < 1 are parameters. The cdf pdf of NW are given, respectively, by 1 K x; η) λ F x;, λ, p, η) = 1 1 pk x; η) λ, x R, 45) ) f x;, λ, p, η) = λ 1 p) k x; η) K x; η) λ 1 1 K x; η) λ 1 1 pk x; η) λ +1, x R, 46) where, λ positive, p 0, 1) are parameters, K x; η) k x; η) are cdf pdf of the baseline distribution which depend on the parameter vector η. The cdf pdf of TEG are given, respectively, by

11 Characterizations infinite divisibility of F x;, µ,, λ) = 1 1 exp exp x µ 1 λ + λ 1 exp exp ))} ))}, 47) x µ f x;, µ,, λ) = 1 exp exp exp exp x µ ))} exp 1 λ + 2λ 1 exp exp x µ )) x µ x µ ))} 1 x R, where, positive, µ R λ 1 are parameters. The cdf pdf of IK are given, respectively, by ))}, 48) F x;, β) = x) ) β, x 0, 49) f x;, β) = β 1 + x) 1) x) ) β 1, x > 0, 50) where, β are positive parameters. Note 7. IK is a special case of BBXII of Paranaiba et al.2011), which was characterized in Hamedani 2016). The cdf pdf of BGIWG are given, respectively, by F x;, θ, γ, p) = e γx) θ x 0, 51) 1 p 1 e γx) θ, f x;, θ, γ, p) = 1 p) θγ x) θ 1 e γx) θ 1 p 1 e γx) θ} 2, x > 0, 52) where, θ, γ positive p 0, 1) are parameters. The cdf pdf of RGTLPS are given, respectively, by F x;, θ, ν) = 1 A θ 1 G x)), 0 x 1, 53) A θ) f x;, θ, ν) = θg x) A θ 1 G x)), 0 < x < 1, 54) A θ)

12 576 G.G. Hamedani where 0, 2, θ > 0, ν > 0 are parameters, G x) = G x;, ν) = 1 1 x) ν 1) 1 x) ν, 0 x 1 is a cdf with corresponding pdf g x) A θ) = z=1 a zθ z finite for a z 0. The cdf pdf of EPE are given, respectively, by ) e θ e λ x e θ F x; θ, λ, ) = 1 e θ, x 0, 55) f x; θ, λ, ) = ) 1 θλe λx θe λ x e θe λx e θ 1 e θ ), x > 0, 56) where θ, λ, are positive parameters. The cdf pdf of TEAW are given, respectively, by F x;, β, γ, θ, δ, λ) = 1 e xθ γx β) δ 1 + λ λ 1 e xθ γx β) δ, x 0, 57) f x;, β, γ, θ, δ, λ) = δ θx θ 1 + γβx β 1) e xθ γx β 1 e xθ γx β) δ λ 2λ 1 e xθ γx β) δ, 58) x > 0, where, β, γ, θ, δ > 0, with 0 < θ < β or 0 < β < θ) λ 1 are parameters. Note 8. For γ = 0, TEAW reduces to TExGW of Yousof et al. entitled A new four-parameter Weibull model for lifetime data. The cdf of TKEIR is given by ) F, θ, λ, a, b) = 1 1 e θ a ) b x 2 ) 1 + λ λ 1 e θ a ) } b x 2, x 0, where, θ, a, b all positive λ 1 are parameters. Note 9. It is easy to see that the number of parameters can be reduced to three as follows ) F x; β, λ, b) = 1 1 e β b x 2 ) 1 + λ λ 1 e β b } x 2, x 0, 59)

13 Characterizations infinite divisibility of where β, b positive λ 1 are parameters. So, there is no need to have two unnecessary extra parameters, which could create identifiability problem. The corresponding pdf is f x; β, λ, b) = 2bβx 3 e βx 2 1 e βx 2) b λ 2λ 1 1 e βx 2) } b, x > 0. 60) The cdf pdf of KOBG are given, respectively, by F x; a, b, c, k) = 1 1 B c,k x)} a ) b, x R, 61) f x; a, b, c, k) = abb c,k x) B c,k x)} a 1 1 B c,k x)} a ) b 1, x R, 62) c } k where a, b, c, k are all positive parameters, B c,k x) = Rx;η)), bc,k x) = d dx B c,k x), R x; η) r x; η) are cdf pdf of the baseline distribution which depend on the parameter η. The pdf of EIL is given by f x;, θ) = θ 1 + x 1)) 1 + θ) Γ ) x +1 e θ/x, x > 0, where > 1 θ > 0 are parameters. Note 10. It would be more appropriate to parametrized the above pdf by taking β = 1 > 0. With this new parameter, the pdf will be given by f x; β, θ) = θ β βx) β 1 + θ) Γ β) x β+2 e θ/x, x > 0, 63) where β > 0 θ > 0 are parameters. The cdf pdf of MOEGG are given, respectively, by 1 e β λe λx 1) ) F x;, β, λ, θ) = θ + θ 1 e β λe λx 1) ) x 0, 64) f x;, β, λ, θ) = βθe λx e β λe λx 1) θ + θ 1 e β λe λx 1) ) 1 θ + θ 1 e β λe λx 1) ) 2, x > 0, 65)

14 578 G.G. Hamedani where, β, λ, θ 0 < θ < 1) are all positive parameters. The cdf pdf of EGSHL are given, respectively, by 1 + e x ) a 2 a e ax b F x; a, b) = 1 + e x ) a, x 0, 66) f x; a, b) = ab2a e ax 1 + e x ) a 2 a e ax b e x ) ab+1, x > 0, 67) where a, b are positive parameters. The cdf pdf of EKw are given, respectively, by F x;, β, γ, η, λ) = λ B γ, η) 1 1 x ) β 0 u γλ 1 1 u λ) η 1, 0 x 1, 68) f x;, β, γ, η, λ) = λβ B γ, η) x 1 1 x ) β x ) β γλ x ) β } λ η 1, 69) where, β, γ, η, λ are all positive parameters. The cdf pdf of GGP are given, respectively, by γ a, 1 ηx log1+ η ) / Γa), if η 0 F x; a, η, ) = γa, x ), 70) / Γa), if η=0 f x; a, η, ) = 1 ηx η a 1 log1+ Γa) ) a 1 1+ ηx ) η 1 1, if η 0, 71) 1 a Γa) xa 1 exp x/), if η=0 where a, are positive η R are parameters. The range of x is x > 0 for η 0 0 < x < /η for η < 0. Note 11. WLOG, we take the case η > 0, x > 0. 2 Characterizations of distributions We present our characterizations i) v) in five subsections. 2.1 Characterizations based on two truncated moments This subsection deals with the characterizations of the above mentioned distributions based on the ratio of two truncated moments. Our first characterization employs a theorem

15 Characterizations infinite divisibility of of 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. Proposition 1.1. Let X : Ω 0, ) be a continuous rom variable let q 1 x) = 1 + x ) 1 q 2 x) = q 1 x) e λx for x > 0. Then, the rom variable X has pdf 2) if only if the function ξ defined in Theorem 1 is of the form ξ x) = e λx ), x > 0. Proof. Further, Suppose the rom variable X has pdf 2), then 1 F x)) E q 1 x) X x = λ 1 e λx ), x > 0, 1 + λ 1 F x)) E q 2 x) X x = λ 1 e 2λx ), x > λ) ξ x) q 1 x) q 2 x) = 1 2 q 1 x) 1 e λx ) > 0 for x > 0. Conversely, if ξ is of the above form, then consequently s x) = ξ x) q 1 x) ξ x) q 1 x) q 2 x) = λx 1 e λx 1 e λx, x > 0, s x) = log 1 e λx )}, x > 0. Now, according to Theorem 1, X has density 2). Corollary 1.1. Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition 1.1. The rom variable X has pdf 2) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) h x) ξ x) h x) g x) = λx 1 e λx 1 e λx, x > 0. The general solution of the differential equation in Corollary 1.1 is ξ x) = 1 e λx ) 1 λx 1 e λx q 1 x)) 1 q 2 x) dx + D, 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 = 1 2. Clearly, there are other triplets q 1, q 2, ξ) which satisfy conditions of Theorem1. The following Proposition may employ a special form of Theorem 1, in which q 1 x) is taken to be identically 1 hence it depends on two functions q 2 ξ see the last paragraph of Appendix A).

16 580 G.G. Hamedani Proposition 1.2. q 1 x) 1 q 2 x) = Let X : Ω 0, ) be a continuous rom variable let 1 1 ) x aβ b θ ) for x 0, ). Then, the rom variable X has pdf 4) if only if the function ξ defined in Theorem 1 is of the form ξ x) = 1 x ) ) aβ b θ , x 0, ). Proof. Suppose the rom variable X has pdf 4), then 1 F x)) E q 1 x) X x = 1 1 x ) ) aβ b θ 1, x 0, ), 1 F x)) E q 2 x) X x = x ) ) aβ b 2θ 1, x 0, ). Further, ξ x) q 2 x) = x ) ) aβ b θ 1 > 0, x 0, ). Conversely, if ξ is of the above form, then s x) = ξ x) ξ x) q 2 x) = θβb ) x aβ 1 1 x 1 ) ) aβ b x 1 x ) aβ ) b θ 1 ) aβ ) b θ, 0, ), consequently s x) = log 1 1 x ) ) aβ b θ 1. Now, according to Theorem 1, X has density 4). Corollary 1.2. Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition 1.2. The rom variable X has pdf 4) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation

17 Characterizations infinite divisibility of ξ x) ξ x) g x) = θβb ) x aβ 1 1 x 1 ) ) aβ b x 1 x ) aβ ) b θ 1 ) aβ ) b θ, 0, ). The general solution of the differential equation in Corollary 1.2 is ξ x) = 1 1 x ) ) aβ b θ 1 1 x aβ 1 x ) ) aβ b 1 x ) ) aβ b θ 1 θβb q 2 x) + D, ) where D is a constant. We like to point out that one set of functions satisfying the above differential equation is given in Proposition 1.2 with D = 1 2. Clearly, there are other triplets q 1, q 2, ξ) which satisfy conditions of Theorem1. A Proposition a Corollary similar to that of Proposition 1.1 or Proposition 1.2) Corollary 1.1 or Corollary 1.2) will be stated without proofs) for each one of the remaining distributions. Proposition 1.3. θ 1 q 1 x) = exp Let X : Ω 0, ) be a continuous rom variable let q 2 x) = q 1 x) βx 1 + βx β+1 e βx) ω} β+1 e βx) ω for x > 0. Then, the rom variable X has pdf 6) if only if the function ξ defined in Theorem 1 is of the form ξ x) = 1 ) ω } βx 2 β + 1 e βx, x > 0. Corollary 1.3. Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition 1.3. The rom variable X has pdf 6) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = β 2 ωθ β x ) x 1 e βx βx β+1 e βx) ω βx β+1 e βx ) ω, x > 0. The general solution of the differential equation in Corollary 1.3 is ξ x) = βx β + 1 e βx ) ω } 1 β 2 ωθ β x ) x 1 e βx βx β+1 e βx) ω 1 q 1 x)) 1 q 2 x) dx + D.

18 582 G.G. Hamedani Proposition 1.4. Let X : Ω 0, ) be a continuous rom variable let q 1 x) = 1 1 θ) G x; ϕ) 1 + λ λg x; ϕ)} 2 q 2 x) = q 1 x) 1 + λ λg x; ϕ) 2 for x > 0. Then, the rom variable X has pdf 8) if only if the function ξ defined in Theorem 1 is of the form ξ x) = λ λg x; ϕ) λ) 2}, x > 0. 2 Corollary 1.4. Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition 1.4. The rom variable X has pdf 8) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = 4g x; ϕ) 1 + λ 2λG x; ϕ) 1 + λ λg x; ϕ) 2 1 λ) 2, x > 0. The general solution of the differential equation in Corollary 1.4 is ξ x) = 1 + λ λg x; ϕ) λ) 2} 1 4g x; ϕ) 1 + λ 2λG x; ϕ) q 1 x)) 1 q 2 x) dx + D Proposition 1.5. Let X : Ω R be a continuous rom variable let q 1 x) = 1 log G x; ξ) ) 1 q2 x) = q 1 x) G x; ξ) for x R. Then, the rom variable X has pdf 10) if only if the function ξ defined in Theorem 1 is of the form ξ x) = θ G x; ξ), x R. θ + 1 Corollary 1.5. Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition 1.5. The rom variable X has pdf 10) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) θg x; ξ) = ξ x) q 1 x) q 2 x) G x; ξ), x R. The general solution of the differential equation in Corollary 1.5 is ξ x) = G x; ξ) ) 1 θg x; ξ) q 1 x)) 1 q 2 x) dx + D. Proposition 1.6. Let X : Ω 0, ) be a continuous rom variable ) a ) a } b+1 ) 1 ab let q 1 x) = λx 1+λ e λx λx 1+λ e λx 1 + λx 1+λ e λx ) afor q 2 x) = q 1 x) 1 + λx 1+λ e λx x > 0. Then, the rom variable X has pdf 12) if only if the function ξ defined in Theorem 1 is of the form ξ x) = λx 1 + λ ) e λx a }, x > 0. Corollary 1.6. Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition 1.6. The rom variable X has pdf 12) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation.

19 Characterizations infinite divisibility of ξ x) q 1 x) ξ x) q 1 x) q 2 x) = ) a 1 aλ 2 1+λ x) 1 + λx 1+λ e λx ) a, x > λx 1+λ e λx The general solution of the differential equation in Corollary 1.6 is ξ x) = λx 1 + λ aλ x) λ ) e λx a } λx ) a 1 e λx q 1 x)) 1 q 2 x) dx + D 1 + λ Proposition 1.7. Let X : Ω 0, ) be a continuous rom variable let ) β 1 a ) ) β q 1 x) = 1 exp e λx2 1) q 2 x) = q 1 x) exp e λx2 1 for x > 0. The rom variable X has pdf 14) if only if the function η defined in Theorem 1 has the form ξ x) = 1 ) ) β 2 exp e λx2 1, x > 0. Corollary 1.7. Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition 1.7. The pdf of X is 14) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ) β 1 ξ x) q 1 x) q 2 x) = βλxeλx2 e λx2 1, x > 0. The general solution of the differential equation in Corollary 1.7 is ) ) β ξ x) = exp e λx2 1 1) β 1 βλxeλx2 eλx2 ) ) β exp e λx2 1 q 1 x)) 1. q 2 x) + D Proposition 1.8. Let X : Ω 0, ) be a continuous rom variable let ) ) q 1 x) = 1 λ + 2λe x β q 2 x) = q 1 x) e x β for x > 0. The rom variable X has pdf 16) if only if the function ξ defined in Theorem 1 has the form ξ x) = 1 2 e x β ), x > 0. Corollary 1.8. Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition 1.8. The pdf of X is 16) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = β ) x 1, x > 0. β.

20 584 G.G. Hamedani The general solution of the differential equation in Corollary 1.8 is ξ x) = e x β ) β x β ) 1 ) e x β q 1 x)) 1 q 2 x) + D. Proposition 1.9. Let X : Ω R be a continuous rom variable let q 1 x) = G x; ξ) G x; ξ) 2} q 2 x) = q 1 x) G x; ξ) for x R. The rom variable X has pdf 18) if only if the function η defined in Theorem 1 has the form ξ x) = G x; ξ) }, x R. Corollary 1.9. Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition 1.9. The pdf of X is 18) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) g x; ξ) G x; ξ) 1 = ξ x) q 1 x) q 2 x) 1 G x; ξ), x R. The general solution of the differential equation in Corollary 1.9 is ξ x) = 1 G x; ξ) 1 g x; ξ) G x; ξ) 1 q 1 x)) 1 q 2 x) + D. Proposition Let X : Ω 0, ) be a continuous rom variable let ) 1 q 1 x) = + θ2 2 x2 q2 x) = q 1 x) e θx for x > 0. The rom variable X has pdf 20) if only if the function ξ defined in Theorem 1 has the form ξ x) = 1 2 e θx, x > 0. Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The pdf of X is 20) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) = θ, x > 0. ξ x) q 1 x) q 2 x) The general solution of the differential equation in Corollary 1.10 is ξ x) = e θx θe θx q 1 x)) 1 q 2 x) + D. Proposition Let X : Ω R be a continuous rom variable let 1 q 1 x) = 1 x) Φ λx)} q2 x) = q 1 x) Φ x) for x R. The rom variable X has pdf 22) if only if the function ξ defined in Theorem 1 has the form ξ x) = Φ x)), x R. 2

21 Characterizations infinite divisibility of Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The pdf of X is 22) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = ϕ x) 1 Φ x), x R. The general solution of the differential equation in Corollary 1.11 is ξ x) = 1 Φ x)) 1 ϕ x) q 1 x)) 1 q 2 x) + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = 1 θ 1 2Φ ) ) x } 2 1 q2 x) = q 1 x) 2Φ ) ) x 1 for x > 0. The rom variable X has pdf 26) if only if the function ξ defined in Theorem 1 has the form ξ x) = 1 2 x ) ) 1 + 2Φ 1, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The pdf of X is 26) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = 2ϕ x ) 2Φ x ) ) Φ x ) 1 ), x > 0. The general solution of the differential equation in Corollary 1.12 is ξ x) = x ) ) 1 1 2Φ 1 x ) x ) 1 2ϕ 2Φ 1) q1 x)) 1 q 2 x) + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = e x/β) q 2 x) = q 1 x) e x/β for x > 0. The rom variable X has pdf 28) if only if the function ξ defined in Theorem 1 has the form ξ x) = 1 2 e x/β, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The pdf of X is 28) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = 1 β, x > 0. The general solution of the differential equation in Corollary 1.13 is 1 ξ x) = e x/β β e x/β q 1 x)) 1 q 2 x) + D.

22 586 G.G. Hamedani Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = θ λ) 1 + x) 2λ θ + ) e θx + 2λ θ + + θx) } 1 q2 x) = q 1 x) e θx for x > 0. The rom variable X has pdf 30) if only if the function ξ defined in Theorem 1 has the form ξ x) = 1 2 e θx, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The pdf of X is 30) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) = θ, x > 0. ξ x) q 1 x) q 2 x) The general solution of the differential equation in Corollary 1.14 is ξ x) = e θx θe θx q 1 x)) 1 q 2 x) + D. Proposition Let X : Ω R be a continuous rom variable let ) β γ +1 ) 1 q 1 x) = 1 + γ Gx;ξ) q Gx;ξ) 2 x) = q 1 x) Gx;ξ) for x R. The rom Gx;ξ) variable X has pdf 32) if only if the function ξ defined in Theorem 1 has the form ξ x) = β β + 1 ) G x; ξ) 1, x R. G x; ξ) Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The pdf of X is 32) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = βg x; ξ) G x; ξ) G x; ξ), x R. The general solution of the differential equation in Corollary 1.15 is ) G x; ξ) β βg x; ξ) G x; ξ)) β+1) ξ x) = ) G x; ξ) β 1) q 1 x)) 1 q 2 x) + D. G x; ξ) Proposition Let X : Ω 0, ) be a continuous rom variable let 1 e ) q 1 x) = βxλ q 2 x) = q 1 x) e βxλ for x > 0. The rom variable X has pdf 34), for 1, if only if the function ξ defined in Theorem 1 has the form ξ x) = 1 2 e βxλ, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The pdf of X is 34),for 1, if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation

23 Characterizations infinite divisibility of ξ x) q 1 x) ξ x) q 1 x) q 2 x) = βλxλ 1, x > 0. The general solution of the differential equation in Corollary 1.16 is Proposition e λ γ q 1 x) = ξ x) = e βxλ 1 exp βλx λ 1 e βxλ q 1 x)) 1 q 2 x) + D. Let X : Ω 0, ) be a continuous rom variable let ) λ θ+1+θx β e θxβ)} 2 q2 x) = q 1 x) exp λ θ+1 for x > 0. Then, the rom variable X has pdf 36) if only if the function ξ defined in Theorem 1 is of the form ξ x) = exp λ θ θx β θ + 1 ) e θxβ )}, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 36) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) λθ2βxβ 1 = ξ x) q 1 x) q 2 x) ξ x) = 1 + x β ) ) exp θx β λ θ+1+θx β θ+1 e θxβ) ) ), x > 0. 1 exp λ e θxβ θ+1+θx β θ+1 The general solution of the differential equation in Corollary 1.17 is 1 exp λ θ θx β θ + 1 λθ 2 βx β x β) exp Proposition ) e θxβ )} 1 θx β λ θ θx β θ + 1 θ+1+θx β θ+1 ) ) e θxβ q 1 x)) 1 q 2 x) dx + D. Let X : Ω 0, ) be a continuous rom variable let q 1 x) = x 1 1 e cλ x λ) q 2 x) = q 1 x) e xλ for x > 0. Then, the rom variable X has pdf 38) if only if the function ξ defined in Theorem 1 is of the form ξ x) = 1 2 e xλ, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let h x) be as in Proposition The rom variable X has pdf 38) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = λxλ 1, x > 0. The general solution of the differential equation in Corollary 1.18 is ) e θxβ)

24 588 G.G. Hamedani ξ x) = e xλ λx λ 1 e xλ q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = x 2 q 2 x) = q 1 x) e x2 /2 2 for x > 0. Then, the rom variable X has pdf 40) if only if the function ξ defined in Theorem 1 is of the form ξ x) = 1 2 e x2 /2 2, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let h x) be as in Proposition The rom variable X has pdf 40) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = x 2, x > 0. The general solution of the differential equation in Corollary 1.19 is ξ x) = e x2 /2 2 x /2 2 2 e x2 q 1 x)) 1 q 2 x) dx + D. Proposition Let} X : Ω R be a continuous rom variable let q 1 x) 1 q 2 x) = exp a Kx;η) for x R. Then, the rom variable X has pdf 42) if Kx;η) only if the function ξ defined in Theorem 1 is of the form ξ x) = 1 } 2 exp K x; η) a, x R. K x; η) Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 42) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) k x; η) = a ξ x) q 1 x) q 2 x) K x; η) 2 x R. The general solution of the differential equation in Corollary 1.20 is } K x; η) ξ x) = exp a K x; η) } k x; η) a K x; η) 2 exp K x; η) a q 1 x)) 1 q 2 x) dx + D. K x; η) Proposition Let X : Ω 0, ) be a continuous rom variable γ 1 ) 1 let q 1 x) = 1 + λ) γ 1 θ+1+θx θ+1 e θx λ 1 θ+1+θx θ+1 e 1 θx q 2 x) = q 1 x) e θx for x > 0. Then, the rom variable X has pdf 44) if only if the function ξ defined in Theorem 1 is of the form ξ x) = 1 2 e θx, x > 0.

25 Characterizations infinite divisibility of Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 44) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) = θ, x > 0. ξ x) q 1 x) q 2 x) The general solution of the differential equation in Corollary 1.21 is ξ x) = e θx θe θx q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω R be a continuous rom variable let q 1 x) = 1 pkx;η)λ +1 1 Kx;η) λ q 2 x) = q 1 x) K x; η) λ for x R. Then, the rom variable X has pdf 46) if only if the function ξ defined in Theorem 1 is of the form ξ x) = K x; η) λ, x R. Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 46) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) λk x; η) K x; η)λ 1 = ξ x) q 1 x) q 2 x) 1 K x; η) λ x R. The general solution of the differential equation in Corollary 1.22 is ξ x) = 1 K x; η) λ 1 λk x; η) K x; η) λ 1 q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω R be a continuous rom variable let q 1 x) = 1 λ + 2λ 1 exp exp x µ ))} 1 q2 x) = q 1 x) 1 exp exp x µ ))} for x R. Then, the rom variable X has pdf 48) if only if the function ξ defined in Theorem 1 is of the form ξ x) = 1 exp exp x µ ))}, x R. + 1 Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 48) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = exp x µ ) exp exp x µ )) )) x R. 1 exp exp x µ The general solution of the differential equation in Corollary 1.23 is

26 590 G.G. Hamedani ξ x) = 1 exp exp x µ exp x µ ) exp ))} 1 exp x µ )) q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = 1 p 1 e γx) θ} 2 q2 x) = q 1 x) e γx) θ for x > 0. Then, the rom variable X has pdf 52) if only if the function ξ defined in Theorem 1 is of the form ξ x) = e γx) θ}, x > 0. 2 Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 52) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = θγ x) θ 1 e γx) x > 0. 1 e γx) θ The general solution of the differential equation in Corollary 1.24 is θ ξ x) = 1 e γx) θ} 1 θγ x) θ 1 e γx) θ q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω 0, 1) be a continuous rom variable let q 1 x) = A θ 1 G x))} 1 q 2 x) = q 1 x) G x) for x 0, 1). Then, the rom variable X has pdf 54) if only if the function ξ defined in Theorem 1 is of the form ξ x) = G x)}, x 0, 1). 2 Corollary Let X : Ω 0, 1) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 54) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = g x) 1 G x) x 0, 1). The general solution of the differential equation in Corollary 1.25 is ξ x) = 1 G x)} 1 g x) q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω 0, ) be a continuous rom variable let ) 1 q 1 x) = e θe λx e θ q2 x) = q 1 x) e θe λx for x > 0. Then, the rom variable X has pdf 56) if only if the function ξ defined in Theorem 1 is of the form

27 Characterizations infinite divisibility of ξ x) = e θe λx}, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 56) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ λx x) q 1 x) ξ x) q 1 x) q 2 x) = θλe λx θe x > 0. 1 e θe λx The general solution of the differential equation in Corollary 1.26 is ξ x) = e θe λx e θ) 1 θλe λx θe λx q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = 1 + λ 2λ 1 e xθ γx β) δ 1 q 2 x) = q 1 x) 1 e xθ γx β) δ for x > 0. Then, the rom variable X has pdf 58) if only if the function ξ defined in Theorem 1 is of the form ξ x) = e xθ γx β) } δ, x > 0. 2 Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 58) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) δ θx θ 1 + γβx β 1) e xθ γx β 1 e xθ γx β) δ 1 ξ x) q 1 x) q 2 x) = 1 1 e xθ γx β) x > 0. δ The general solution of the differential equation in Corollary 1.27 is ξ x) = q 1 x) = 1 Proposition λ 2λ 1 e xθ γx β) } δ 1 1 δ θx θ 1 + γβx β 1) e xθ γx β 1 e xθ γx β) δ 1 q1 x)) 1. q 2 x) dx + D Let X : Ω 0, ) be a continuous rom variable let 1 } e βx 2) b 1 q 2 x) = q 1 x) 1 e βx 2) for x > 0. Then, the rom variable X has pdf 60) if only if the function ξ defined in Theorem 1 is of the form ξ x) = b 1 e βx 2), x > 0. b + 1

28 592 G.G. Hamedani Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 60) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the following differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = 2bβx 3 e βx 2 1 e βx 2 x > 0. The general solution of the differential equation in Corollary 1.28 is ξ x) = 1 e βx 2) 1 2bβx 3 e βx 2 q 1 x)) 1 q 2 x) dx + D. Proposition Let X : Ω R be a continuous rom variable let q 1 x) = 1 B c,k x)) a 1 b q 2 x) = q 1 x) B c,k x)) a for x R. The rom variable X has pdf 62) if only if the function ξ defined in Theorem 1 has the form ξ x) = B c,k x)) a, x R. Corollary Let X : Ω R be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 62) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = ab c,k x) B c,k x)) a 1 1 B c,k x)) a x R. The general solution of the differential equation in Corollary 1.29 is ξ x) = 1 B c,k x)) a 1 ab c,k x) B c,k x)) a 1 q 1 x)) 1 q 2 x) + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = 1 + βx) 1 e θ/x q 2 x) = x q 1 x) for x > 0. The rom variable X has pdf 63) if only if the function ξ defined in Theorem 1 has the form ξ x) = βx β + 1, x > 0. Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 63) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = βx 1, x > 0. The general solution of the differential equation in Corollary 1.30 is ξ x) = x β βx β 1 q 1 x)) 1 q 2 x) + D.

29 Characterizations infinite divisibility of Proposition θ + θ q 1 x) = Let X : Ω 0, ) be a continuous rom variable let 1 e β λe λx 1) ) for x > 0. The 1 e β λe λx 1) ) 2 q2 x) = q 1 x) rom variable X has pdf 65) if only if the function ξ defined in Theorem 1 has the form ξ x) = e β λe λx 1) ) }, x > 0. 2 Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 65) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) βeλx e β λe λx 1) 1 e β λe λx 1) ) 1 ξ x) q 1 x) q 2 x) = 1 1 e β λe λx 1) ), x > 0. The general solution of the differential equation in Corollary 1.31 is ξ x) = 1 1 e β λe λx 1) ) } 1 βe λx e β λe λx 1) 1 e β λe λx 1) ) 1 q1 x)) 1 q 2 x) + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = e a 1)x 1 + e x ) a 2 a e ax b 1 q2 x) = q 1 x) 1 + e x ) ab for x > 0. The rom variable X has pdf 67) if only if the function ξ defined in Theorem 1 has the form ξ x) = e x) } ab, x > 0. 2 Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 67) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = abe x 1 + e x ) ab e x ) ab, x > 0. The general solution of the differential equation in Corollary 1.32 is ξ x) = e x) } ab 1 abe x 1 + e x) ab 1 q1 x)) 1 q 2 x) + D. Proposition Let X : Ω 0, 1) be a continuous rom variable let q 1 x) = x ) β } λ 1 η q 2 x) = q 1 x) 1 1 x ) β γλ for 0 < x < 1. The rom variable X has pdf 69) if only if the function ξ defined in Theorem 1 has the form

30 594 G.G. Hamedani ξ x) = x ) β } γλ, 0 < x < 1. Corollary Let X : Ω 0, 1) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 69) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) γλβx 1 1 x β 1 ) 1 1 x ) β γλ 1 ξ x) q 1 x) q 2 x) = x ) β, 0 < x < 1. γλ The general solution of the differential equation in Corollary 1.33 is ξ x) = x ) β } γλ 1 γλβx 1 1 x ) β x ) β γλ 1 q1 x)) 1 q 2 x) + D. Proposition Let X : Ω 0, ) be a continuous rom variable let q 1 x) = log 1 + ηx ) 1 a q2 x) = q 1 x) 1 + ηx ) 1 for x > 0. The rom variable X has pdf 71) if only if the function ξ defined in Theorem 1 has the form ξ x) = ηx ) 1, x > η Corollary Let X : Ω 0, ) be a continuous rom variable let q 1 x) be as in Proposition The rom variable X has pdf 71) if only if there exist functions q 2 ξ defined in Theorem 1 satisfying the differential equation ξ x) q 1 x) ξ x) q 1 x) q 2 x) = ηx ) 1, x > 0. The general solution of the differential equation in Corollary 1.34 is ξ x) = 1 + ηx ) 1/η ηx ) 1 η 1 q 1 x)) 1 q 2 x) + D. 2.2 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 f x) f x) = h F x) h F x) h F x). 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 EKPF for θ = 1), LF, EWR

31 Characterizations infinite divisibility of for a = 1) TW for λ = 1), QXC, GPHN, APTW for 1), MLB, PL-G, NW, RGTLPS KOBG distributions in terms of the hazard function, which are not of the trivial form given above. Proposition 2.1. Let X : Ω 0, ) be a continuous rom variable. The rom variable X has pdf 4) for θ = 1) if only if its hazard function h F x) satisfies the following differential equation h F x) aβ 1) x 1 h F x) = bβ 2 x ) 2aβ 1) x ) aβ 2 1, x 0, ). Proof. It is clear that the above differential equation holds if X has pdf 4) for θ = 1. Conversely, if the differential equation holds, then d } x aβ 1) h F x) = bβ d x ) } aβ 1 1, dx dx or h F x) = bβ ) x aβ 1 1 ) x aβ, x 0, ), which is the hazard function of the EKPF distribution. A Proposition similar to that of Proposition 2.1 will be stated without proof) for each one of the above mentioned distributions. Proposition 2.2. Let X : Ω R be a continuous rom variable. The rom variable X has pdf 10) if only if its hazard function h F x) satisfies the following differential equation h F x) g x; ξ) g x; ξ) h F x) = θ2 d g x; ξ) θ + 1 dx 1 θ θ+1 1 log G x; ξ) ) ) } log G x; ξ) G x; ξ), x R. Proposition 2.3. Let X : Ω 0, ) be a continuous rom variable. The pdf of X for a = 1, is 14) if only if its hazard function h F x) satisfies the differential equation ) β 2 ) h F x) x 1 h F x) = 4βλ 2 x 2 e λx2 e λx2 1 2e λx2 1, x > 0. Proposition 2.4. Let X : Ω 0, ) be a continuous rom variable. The pdf of X for λ = 1, is 16) if only if its hazard function h F x) satisfies the differential equation h F x) 1) β 2 x 1 h F x) = 0, x > 0. Proposition 2.5. Let X : Ω 0, ) be a continuous rom variable. The pdf of X is 20) if only if its hazard function h F x) satisfies the differential equation

32 596 G.G. Hamedani h F x) 2θx + h θ2 2 x2 F x) = θ θx) + θ2 2 x2 ) θx + θ2 2 x2 ) 2, x > 0. Proposition 2.6. Let X : Ω 0, ) be a continuous rom variable. The pdf of X is 26) if only if its hazard function h F x) satisfies the differential equation d ϕ x dx h F x) 2 1) ϕ ) x 2Φ x ) 1 2Φ x ) ) θ 1 2Φ x ) ) ) 1 ) h F x) = 2 1 }, x > 0. x ) 1 2Φ 1) Note that for = 1, we have, clearly a much simpler differential equation. Proposition 2.7. Let X : Ω 0, ) be a continuous rom variable. The pdf of X, for 1, is 34) if only if its hazard function h F x) satisfies the differential equation h F x) + βλx λ 1 h F x) = log ) βλe βxλ d dx x λ 1 1 e βxλ 1 e βxλ ), x > 0. Note 8. For = 2, the above differential equation has the following simple form: h F x) + βλx λ 1 h F x) = log 2) βλ λ 1) x λ 2 e βxλ, x > 0. Proposition 2.8. Let X : Ω 0, ) be a continuous rom variable. Then, X has pdf 40) if only if its hazard function h F x) satisfies the differential equation h F x) 3x 1 h F x) = 2x4 2 x ) 2, x > 0, with the initial condition h F 0) = 0. Proposition 2.9. Let X : Ω R be a continuous rom variable. Then X has pdf 42) if only if its hazard function h F x) satisfies the differential equation h F x) k x) k x) h F x) = a2 k x; η) 2 ) 2a + 3K x; η) K x; η) 3 a + K x; η) ) 2, with the initial condition h F 0) = ak 0; η). Proposition Let X : Ω R be a continuous rom variable. Then X has pdf 46) if only if its hazard function h F x) satisfies the differential equation

33 Characterizations infinite divisibility of h F x) k x) k x) h F x) = λk x; η) 2 K x; η) λ p) λ 1) K x; η) λ + p 1 2λ) K x; η) 2λ 1 K x; η) λ 2 1 pk x; η) λ 2 = λk x; η) d dx K x; η) λ 1 1 K x; η) λ 1 pk x; η) λ. Proposition Let X : Ω 0, 1) be a continuous rom variable. Then, X has pdf 54) if only if its hazard function h F x) satisfies the differential equation h F x) g x) g x) h A θ 1 G x)) F x) = θg x) A θ 1 G x)) A ) } θ 1 G x)) 2, x 0, 1). A θ 1 G x)) Proposition Let X : Ω R be a continuous rom variable. Then, X has pdf 62) if only if its hazard function h F x) satisfies the differential equation h F x) b c,k x) b c,k x) h F x) = abb2 c,k x) B c,k x)) a 1 a 1 + B c,k x)) a 1 B c,k x)) a 2, x R. 2.3 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 r F x) = f x), x support of F. F x) In this subsection we present characterizations of GIL, EKPF, CGT-G, OBL, EWR, TW for λ = 1), GPHN, ALTE, MBIIIG, APTW, GTL, NW, BGIWG, EPE, TEAW, KOBG, MOEGG, EGSHL EKw for η = 1) distributions without proofs) in terms of the reverse hazard function. Proposition 3.1. Let X : Ω 0, ) be a continuous rom variable. The rom variable X has pdf 2) if only if its reverse hazard function r F x) satisfies the following differential equation r F x) + + 1) x 1 h F x) = λ 2 x 1 d } 1 + x dx 1 + λ + λx, x > 0. Proposition 3.2. Let X : Ω 0, ) be a continuous rom variable. The rom variable X has pdf 4) if only if its reverse hazard function r F x) satisfies the following differential equation

34 598 G.G. Hamedani bβ ) x aβ 1 r F x) aβ 1) x 1 r F x) = 1 1 x 1 x ) aβ ) b 1 ) aβ ) b 2, x 0, ). Proposition 3.3. Let X : Ω 0, ) be a continuous rom variable. The rom variable X has pdf 8) if only if its reverse hazard function r F x) satisfies the following differential equation r F x) g x; ϕ) g x; ϕ) r F x) = g x; ϕ) d dx 1 + λ 2λG x; ϕ) G x; ϕ) 1 + λ λg x; ϕ) 1 1 θ) G x; ϕ) 1 + λ λg x; ϕ)} for x > 0. Proposition 3.4. Let X : Ω 0, ) be a continuous rom variable. For b = 1, the rom variable X has pdf 12) if only if its reverse hazard function r F x) satisfies the following differential equation }, r F x) + λr F x) = aλ2 1 + λ e λx d dx λx 1+λ 1 + x) 1 + λx ) a e λx λx 1+λ 1+λ e λx ) a 1 ) e λx a } λx 1+λ ) e λx, for x > 0. Proposition 3.5. Let X : Ω 0, ) be a continuous rom variable. The rom variable X has pdf 14) if only if its reverse hazard function r F x) satisfies the following differential equation r F x) x 1 r F x) = 2aβλx d dx ) } β λx 2 e λx2 1 1 exp e λx2 1 ) } β ) β 1 e λx2 1 exp, x > 0. Proposition 3.6. Let X : Ω 0, ) be a continuous rom variable. The pdf of X for λ = 1, is 16) if only if its hazard function r F x) satisfies the differential equation r F x) 1) β 2 x 1 r F x) = 0, x > 0. Proposition 3.7. Let X : Ω 0, ) be a continuous rom variable. The pdf of X is 26) if only if its hazard function r F x) satisfies the differential equation