Various characterizations of the distributions of the ratio of two independent gamma and exponential random variables as well as that of two independent Weibull random variables are presented. These characterizations are based, on a simple relationship between two truncated moments ; on hazard function ; and on functions of order statistics

Hamedani, Gholamhossein

English

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Marquette Universitye-Publications@MarquetteMathematics, Statistics and Computer ScienceFaculty Research and PublicationsMathematics, Statistics and Computer Science,Department of8-1-2013Characterizations of Distributions of Ratio ofCertain Independent Random VariablesGholamhossein HamedaniMarquette University, gholamhoss.hamedani@marquette.eduPublished version. Journal of Applied Mathematics, Statistics and Information, Vol. 9, No. 1 (August2013): 15-25. DOI. © 2013 Walter de Gruyter. Used with permission.Journal of Applied Mathematics, Statistics and Informatics (JAMSI), 9 (2013), No. 1CHARACTERIZATIONS OF DISTRIBUTIONSOF RATIOS OF CERTAIN INDEPENDENTRANDOM VARIABLESG.G. HAMEDANIAbstractVarious characterizations of the distributions of the ratio of two independent gamma and exponen-tial random variables as well as that of two independent Weibull random variables are presented.These characterizations are based, on a simple relationship between two truncated moments ; onhazard function ; and on functions of order statistics.Mathematics Subject Classiﬁcation 2000: 46F101. INTRODUCTIONAs pointed out by Nadarajah and Kotz (2006), the distribution of U / V forindependent random variables U and V have been studied by several authorswhen these random variables come from the same family of distributions (see theirpertinent references). Recently, Shakil and Ahsanullah (2011) also pointed out thatthe distribution of the ratio of independent random variables arises in many ﬁeldsof studies such as biology, economics, engineering, genetics and order statistics, toname a few. For the detailed explanation of the importance of the distribution ofthe ratio of independent random variables, we refer the interested reader to Shakiland Ahsanullah (2011) where they consider the distributional properties of recordvalues of the ratio of independent Rayleigh random variables. Nadarajah (2010)studied the distributional properties as well as estimation of the ratio of independentWeibull random variables. For the detailed discussion, domain of applicability andpractical examples we refer the interested reader to Nadarajah (2010). A moreinteresting case, however, is when the random variables have diﬀerent distributions.Nadarajah and Kotz (2006) considered the case when the independent randomvariables U and V have diﬀerent but similar distributions (gamma and Weibull)and obtained the exact distribution of the ratio U / V . The goal of the presentwork is to establish various characterizations of certain sub-family of distributionsof the ratios given in Nadarajah and Kotz (2006) and Nadarajah (2010).The problem of characterizing a distribution is an important problem which hasrecently attracted the attention of many researchers. Thus, various characteriza-tions have been established in many diﬀerent directions. In Section 2, we presentcharacterizations of the distributions of the ratios of two independent gamma andexponential random variables as well as two Weibull random variables in diﬀerentDOI 10.2478/jamsi-2013-0002©University of SS. Cyril and Methodius in TrnavaBrought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM16G.G. HAMEDANIdirections. Our results in subsections 2.1-2.3, will be based: on two truncated mo-ments ; on hazard function ; and on truncated moments of certain functions of orderstatistics, respectively.Let U and V have a gamma and an exponential distributions respectively.The pdf ′s (probability density functions) f U and fV are given respectively byfU (u) =μαΓ (α)uα − 1 e− μ u , u > 0andfV (v) = λ e− λ v , v > 0where α , μ and λ are positive parameters.The pdf f and cdf (cumulative distribution function) F of the distribution ofthe ratio, X = UV , of two independent gamma and exponential random variablesU and V (given above) are respectively (see Nadarajah and Kotz (2006))f (x) = f (x;α, γ) = α γ xα − 1 (x+ γ)− (α+1) , x > 0 , (1.1)andF (x) = xα (x+ γ)− α, x ≥ 0 , (1.2)where γ = λμ reducing the number of parameters which is desirable as far ascharacterization results are concerned. We denote the random variable X withcdf (1.2) by RGE.Let U and V have Weibull distributions with pdf ′s given respectively byfU (u) = β λ uβ − 1 e− λ uβ, u > 0andfV (v) = β μ vβ − 1 e− μ vβ, y > 0where β , λ and μ are positive parameters.Brought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM17CHARACTERIZ. OF DISTR. OF RATIOS OF CERTAIN INDEPEND. RANDOM VARIABLESThe pdf f , cdf F and hazard function ξF of the ratio, X = UV , of twoindependent Weibull random variables U and V (given above) are respectively(see Nadarajah and Kotz (2006))f (x) = f (x;β, λ, μ) = βλμ xβ − 1(λxβ + μ)−2, x > 0 , (1.3)F (x) = λxβ(λxβ + μ)− 1, x ≥ 0 , (1.4)andξF (x) = βλ xβ − 1 (λxβ + μ)−1 , x > 0 . (1.5)We denote the random variable X with cdf (1.4) by RWW.2. Characterization ResultsAs pointed out in the introduction, the distribution of the ratio of independentrandom variables has applications in many ﬁelds of study. So, an investigator will bevitally interested to know if their model ﬁts the requirements of RGE or RWWdistributions. To this end, the investigator relies on characterizations of thesedistributions, which provide conditions under which the underlying distributionis indeed that of RGE or of RWW . In this section we will present severalcharacterizations of these distributions.Throughout this section we assume, where necessary, that the distribution func-tion F is twice diﬀerentiable on its support.2.1. Characterization based on two truncated momentsIn this subsection we present characterizations of RGE and RWW distributionsin terms of truncated moments. We like to mention here the works of Galambos andKotz (1978), Kotz and Shanbahag (1980), Glänzel (1987, 1988, 1990), Glänzel et al.(1984, 1994), Glänzel and Hamedani (2001) and Hamedani (1993, 2002, 2006) in thisdirection. Our characterization results presented here will employ an interestingresult due to Glänzel (1987) (Theorem G below).Theorem G. Let (Ω,F ,P) be a given probability space and let H = [a, b]be an interval for some a < b (a = −∞ , b = ∞ might as well be allowed) . LetX : Ω → H be a continuous random variable with the distribution function Fand let g and h be two real functions deﬁned on H such thatE [g (X) | X ≥ x] = E [h (X) | X ≥ x] η (x) , x ∈ H ,Brought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM18G.G. HAMEDANIis deﬁned with some real function η . Assume that g , h ∈ C1 (H) , η ∈ C2 (H)and F is twice continuously diﬀerentiable and strictly monotone function on theset H . Finally, assume that the equation hη = g has no real solution in theinterior of H . Then F is uniquely determined by the functions g , h and η ,particularlyF (x) =∫ xaC∣∣∣∣ η′ (u)η (u)h (u)− g (u)∣∣∣∣ exp (−s (u)) du ,where the function s is a solution of the diﬀerential equation s′ = η′ hη h − g andC is a constant, chosen to make∫HdF = 1 .Remarks 2.1.1. (a) In Theorem G, the interval H need not be closed. (b)The goal is to have the function η as simple as possible. For a more detaileddiscussion on the choice of η , we refer the reader to Glänzel and Hamedani (2001)and Hamedani (1993, 2002, 2006).Proposition 2.1.2. Let X : Ω → (0,∞) be a continuous random variableand let h (x) = x−(α−1) and g (x) = x−2α (x+ γ)α+1 for x ∈ (0,∞) . The pdfof X is (1.1) if and only if the function η deﬁned in Theorem G has the formη (x) =(1 + γx−1)α, x > 0 .Proof. Let X have pdf (1.1) , then(1− F (x)) E [h (X) | X ≥ x] = γ (x+ γ)− α , x > 0 ,and(1− F (x)) E [g (X) | X ≥ x] = γx− α, x > 0 ,and ﬁnallyη (x)h (x)− g (x) = −γx−2 α (x+ γ)α < 0 , for x > 0.Conversely, if η is given as above, thens′ (x) =η′ (x) h (x)η (x) h (x)− g (x) = α (x+ γ)−1, x > 0 ,and henceBrought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM19CHARACTERIZ. OF DISTR. OF RATIOS OF CERTAIN INDEPEND. RANDOM VARIABLESs (x) = ln((x+ γ)α).Now, in view of Theorem G (with C = γ ) , X has cdf (1.2) and pdf (1.1) .Corollary 2.1.3. Let X : Ω → (0,∞) be a continuous random variable andlet h (x) = x−(α−1) for x ∈ (0,∞) . The pdf of X is (1.1) if and only if thereexist functions g and η deﬁned in Theorem G satisfying the diﬀerential equationη′ (x) x−(α−1)η (x) x−(α−1) − g (x) = α (x+ γ)−1, x > 0.Remark 2.1.4. The general solution of the diﬀerential equation given in Co-rollary 2.1.3 isη (x) = (x+ γ)α[−∫g (x) α xα − 1 (x+ γ)−(α+1) dx+D],for x > 0 , where D is a constant. One set of appropriate functions is given inProposition 2.1.2 with D = 0.Proposition 2.1.5. Let X : Ω → (0,∞) be a continuous random variable andlet h (x) = x− (β+1)(λxβ + μ)2 and g (x) = x−2β (λxβ + μ)2 for x ∈ (0,∞) .The pdf of X is (1.3) with β 	= 1 if and only if the function η deﬁned inTheorem G has the formη (x) =1βx1−β , x > 0 .Proof. Let X have pdf (1.3) , then(1− F (x)) E [h (X) | X ≥ x] = βλμx−1 , x > 0 ,and(1− F (x)) E [g (X) | X ≥ x] = λμx− β , x > 0 ,and ﬁnally for β 	= 1η (x)h (x)− g (x) = (1− β)βx−2 β(λxβ + μ)2 	= 0 , for x > 0.Brought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM20G.G. HAMEDANIConversely, if η is given as above, thens′ (x) =η′ (x) h (x)η (x) h (x)− g (x) = x−1, x > 0 .Integrating both sides of the above equation from x0 > 0 to x we haves (x) = ln(xx0),for which, without loss of generality, we take s (x0) = 0. Now, in view of TheoremG (with C = x0 = βλμ ) , X has cdf (1.4) and pdf (1.3) .Corollary 2.1.6. Let X : Ω → (0,∞) be a continuous random variable andlet h (x) = x−(β+1)(λxβ + μ)2 for x ∈ (0,∞) . The pdf of X is (1.3) ifand only if there exist functions g and η deﬁned in Theorem G satisfying thediﬀerential equationη′ (x) x−(β+1)(λxβ + μ)2η (x) x−(β+1) (λxβ + μ)2 − g (x)=1x, x > 0.Remark 2.1.7. The general solution of the diﬀerential equation given in Co-rollary 2.1.6 isη (x) = x[−∫g (x) xβ−1(λxβ + μ)−2dx+D],for x > 0 , where D is a constant. One set of appropriate functions is given inProposition 2.1.5 with D = 0.Remark 2.1.8. Clearly there are other triplet functions (h, g, η) satisfyingconditions of Proposition 2.1.2 or those of Proposition 2.1.5.2.2. Characterization based on hazard functionFor the sake of completeness, we state the following simple fact.Let F be an absolutely continuous distribution with the corresponding pdf f. The hazard function corresponding to F isξF (x) =f (x)1− F (x) , x ∈ Supp F , (2.2.1)where Supp F is the support of F .Brought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM21CHARACTERIZ. OF DISTR. OF RATIOS OF CERTAIN INDEPEND. RANDOM VARIABLESIt is obvious that the hazard function of a twice diﬀerentiable distribution func-tion satisﬁes the ﬁrst order diﬀerential equationξ′F (x)ξF (x)− ξF (x) = k (x) , (2.2.2)where k (x) is an appropriate integrable function. Although this diﬀerential equ-ation has an obvious form sincef ′ (x)f (x)=ξ′F (x)ξF (x)− ξF (x)α ,for many univariate continuous distributions (2.2.2) seems to be the only diﬀe-rential equation in terms of the hazard function. The goal here is to establish adiﬀerential equation which has as simple form as possible and is not of the trivialform (2.2.2). For some general families of distributions this may not be possible.Here is our characterization result for RWW distribution.Proposition 2.2.1. Let X : Ω → (0,∞) be a continuous random variable.The pdf of X is (1.3) if and only if its hazard function ξF satisﬁes thediﬀerential equationξ′F (x)− (β − 1) x−1ξF (x) + ξ2F (x) = 0, x > 0. (2.2.3)Proof. If X has pdf (1.3) , then obviously (2.2.3) holds. If ξF satisﬁes(2.2.3) , thenddx(xβ−1 ( ξF (x))−1)= xβ−1 ,or after integrating both sides with respect to x from x0 > 0 to x and somegrouping of terms, we arrive atξF (x) =f (x)1− F (x) =βxβ−1xβ + C,where C is a constant. Upon another integration with respect to x from 0 tox , we obtain (1.4) with C = μλ .Remark 2.2.2. For characterizations of other well-known continuous distri-butions based on hazard function, we refer the reader to Hamedani (2004) andHamedani and Ahsanullah (2005) .2.3. Characterization based on truncated moments of certain functionsof order statisticsBrought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM22G.G. HAMEDANILet X1:n ≤ X2:n ≤ ... ≤ Xn:n be n order statistics from a continuous cdf F. Wepresent here characterization results base on some functions of these order statistics.We refer the reader to Ahsanullah and Hamedani (2007) , Hamedani et al. (2008)and Hamedani (2010) , among others , for characterizations of other well-knowncontinuous distributions in this direction. The proof of the following proposition issimilar to that of Theorem 2.2 of Hamedani (2010) in which k (x) = xγ for someγ > 0. We give a brief proof, however, for the sake of completeness.Proposition 2.3.1. Let X : Ω → (0,∞) be a continuous random va-riable with cdf F and k (x) be a diﬀerentiable function such that lim x→∞k (x) (1− F (x))n = 0 . Let g (x, n) be a real-valued function which is diﬀeren-tiable with respect to x on [0,∞) and ∫∞0k′(x)g(x,n)dx = ∞. ThenE [k (X1:n) | X1:n > t] = k (t) + g (t, n) , t > 0 , (2.3.1)implies thatF (x) = 1−(g (0, n)g (x, n)) 1ne−∫ x0k′(t)n g(t,n)dt , x ≥ 0.Proof. Condition (2.3.1) and the assumption lim x→∞ k (x) (1− F (x))n = 0imply that∫ ∞tk′ (x) (1− F (x))n dx = g (t, n) (1− F (t))n . (2.3.2)Diﬀerentiating (2.3.2) with respect to t, we have−k′ (t) (1− F (t))n=[∂∂tg (t, n) (1− F (t))− n g (t, n) f (t)](1− F (t))n−1 ,from whichf (t)1− F (t) =∂∂tg (t, n)n g (t, n)+k′ (t)n g (t, n). (2.3.3)Integrating (2.3.3) with respect to t , from 0 to x , results inBrought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM23CHARACTERIZ. OF DISTR. OF RATIOS OF CERTAIN INDEPEND. RANDOM VARIABLESF (x) = 1−(g (0, n)g (x, n)) 1ne−∫ x0k′(t)n g(t,n)dt , x ≥ 0.Remarks 2.3.2. (i) For k (x) = xβ and g (x, n) = 1(n−1) λ(λxβ + μ), Propo-sition 2.3.1 gives a characterization of (1.4). (ii) For α = 1 , h (x) = (x+ γ)n−1and g (x, n) = (n− 1) (x+ γ)n−1 , Proposition 2.3.1 gives a characterization of(1.2) .Let Xj , j = 1, 2, ..., n be n i.i.d. random variables with cdf F andcorresponding pdf f and let X1:n ≤ X2:n ≤ ... ≤ Xn:n be their correspondingorder statistics. Let X∗1:n−i+1 be the 1st order statistic from a sample of sizen − i + 1 of random variables with cdf Ft (x) = F (x)−F (t)1−F (t) , x ≥ t (t is ﬁxed)and corresponding pdf ft (x) =f(x)1−F (t) , x ≥ t . Then(Xi:n | Xi−1:n = t) d= X∗1:n−i+1 ( d= means equal in distribution) ,that isfXi:n | Xi−1:n (x|t) = fX∗1:n−i+1 (x) = (n− i+ 1) (1− Ft (x))n−i f (x)1− F (t) , x ≥ t .Now we can state the following characterizations of REE (RGE with α = 1)and RWW distributions in yet somewhat diﬀerent direction. The proofs aresimilar to that of Propositions 2.3.1 and hence will be omitted.Corollary 2.3.3. Let X : Ω → (0,∞) be a continuous random variable withcdf F . Then(i)E[(Xi:n + γ)n−i | Xi−1:n = t]= (t+ γ)n−i + (n− i) (t+ γ)n−i , t > 0 ,for n > i and some γ > 0 if and only if X has cdf (1.2) ;(ii)E[Xβi:n | Xi−1:n = t]= tβ +1(n− i)λ(λtβ + μ), t > 0 ,for n > i and some β > 0 , λ > 0 , μ > 0 if and only if X has cdf (1.4) .Brought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM24G.G. HAMEDANIReferencesAhsanullah, M. and Hamedani, G.G., Certain characterizations of the powerfunction and beta distributions based on order statistics, JSTA, 6 (2007) , 220−225.Galambos, J. and Kotz, S., Characterizations of probability distributions. Auniﬁed approach with an emphasis on exponential and related models, LectureNotes in Mathematics, 675, Springer, Berlin, 1978.Glänzel, W., A characterization of the normal distribution, Studia Sci. Math.Hungar. 23 (1988) , 89− 91.Glänzel, W., A characterization theorem based on truncated moments and itsapplication to some distribution families, Mathematical Statistics and ProbabilityTheory (Bad Tatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht, 1987, 75− 84.Glänzel, W., Some consequences of a characterization theorem based on truncatedmoments, Statistics 21 (1990) , 613− 618.Glänzel, W. and Hamedani, G.G., Characterizations of univariate continuousdistributions, Studia Sci. Math. 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Hungar. 43 (2006) , 361− 385.Hamedani, G.G., Characterizations of univariate continuous distributions basedon hazard function, J. of Applied Statistical Science, 13 (2004) , 169− 183.Hamedani, G.G., Characterizations of continuous univariate distributions basedon the truncated moments of functions of order statistics, Studia Sci. Math. Hungar.47 (2010) , 462− 484.Hamedani, G.G. and Ahsanullah, M., Characterizations of univariate continuousdistributions based on hazard function II, J. of Statistical Theory and Applications,4 (2005) , 218− 238.Hamedani, G.G., Ahsanullah, M. and Sheng, R., Characterizations of certaincontinuous univariate distributions based on truncated moment of the ﬁrst orderstatistic, Aligrah J. of Statistics, 28 (2008) , 75− 81.Kotz, S. and Shanbhag, D.N., Some new approaches to probability distributions,Adv. in Appl. Probab. 12 (1980) , 903− 921.Nadarajah, S., Distribution properties and estimation of the ratio of independentWeibull random variables, Adv. Stat. Anal. 94 (2010) , 231− 246.Nadarajah, S. and Kotz, S., On the product and ratio of gamma and Weibullrandom variables, Econometric Theory, 22 (2006) , 338− 344.Shakil, M. and Ahsanullah, M., Record values of the ratio of Rayleigh randomBrought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM25CHARACTERIZ. OF DISTR. OF RATIOS OF CERTAIN INDEPEND. RANDOM VARIABLESvariables, Pak. J. Statist. 27 (2011) , 307− 325.G.G. HamedaniDepartment of Mathematics, Statistics and Computer ScienceMarquette UniversityMilwaukee, WI 53201-1881Received May 2013Brought to you by | Marquette University Raynor Memorial LibrariesAuthenticated | 134.48.158.157Download Date | 2/5/14 5:29 PM

Characterizations of Distributions of Ratio of Certain Independent Random Variables

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Characterizations of Distributions of Ratio of Certain Independent Random Variables

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