Several characterizations of a New Modified Weibull distribution, introduced by Doostmoradi et al. (2014), are presented. These characterizations are based on: (i) truncated moment of a function of the random variable; (ii) the hazard function; (iii) a single function of the random variable; (iv) truncated moment of certain function of the 1st order statistic

Hamedani, Gholamhossein

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Marquette Universitye-Publications@MarquetteMathematics, Statistics and Computer ScienceFaculty Research and PublicationsMathematics, Statistics and Computer Science,Department of1-1-2014Characterizations of New Modified WeibullDistributionGholamhossein HamedaniMarquette University, gholamhoss.hamedani@marquette.eduPublished version. Journal of Statistical Research of Iran, Vol. 11, No. 2 (2014): 223-229. Publisherlink. © 2014 Statistical Research and Training Center. Used with permission.c Statistical Research and Training CenterJ. Statist. Res. Iran 11 (2014): 223–229 ٢٢٩–٢٢٣ ﺺﺻ ،١٣٩٣ نﺎﺘﺴﻣز و ﺰﯿﯾﺎﭘ ،٢ ی هرﺎﻤﺷ ،١١ ی هرودCharacterizations of New ModiﬁedWeibull DistributionG. G. HamedaniMarquette UniversityAbstract. Several characterizations of a New Modiﬁed Weibull distribution,introduced by Doostmoradi et al. (2014), are presented. These character-izations are based on: (i) truncated moment of a function of the randomvariable; (ii) the hazard function; (iii) a single function of the random vari-able; (iv) truncated moment of certain function of the 1st order statistic.Keywords. Characterization; hazard function; modiﬁed Weibull distribu-tion; truncated moment.MSC 2010: 62E10.1 IntroductionCharacterizations of distributions are important to many researchers in theapplied ﬁelds. An investigator will be vitally interested to know if their modelﬁts the requirements of a particular distribution. To this end, one will dependon the characterizations of this distribution which provide conditions underwhich the underlying distribution is indeed that particular distribution. Vari-ous characterizations of distributions have been established in many diﬀerentdirections in the literature. In this short note, several characterizations ofa New Modiﬁed Weibull (NMW) distribution, introduced by Doostmoradiet al. (2014), are presented in four directions. These characterizations arebased on: (i) truncated moment of a function of the random variable; (ii) thehazard function; (iii) a single function of the random variable; (iv) truncatedmoment of certain function of the 1st order statistic. Our characterizationsof NMW distribution (i) will employ a version of an interesting result due224 Characterizations of New Modiﬁed Weibull Distributionto Glanzel (1987) (Theorem 1 of Section 2 below). The advantage of thesetype of characterizations is that, cumulative distribution function, F , neednot have a closed form and are given in terms of an integral whose integranddepends on the solution of a ﬁrst order diﬀerential equation, which can serveas a bridge between probability and diﬀerential equation.The cdf (cumulative distribution function) F (x) and pdf (probabilitydensity function) f (x) of NMW are given byF (x) = F (x;; ; ; ) = 1  expn ex + e xo; x > 0; (1)andf (x) = f (x;; ; ; )=x 1ex+ x 1e xexpn ex + e xo; x > 0; (2)where  > 0;  > 0;  > 0 and  > 0 are parameters. For further propertiesand the domain of applicability of NMW distribution, we refer the interestedreader to Doostmoradi et al. (2014).The presentation of the content of this work is as follows: In Section 2, wepresent our characterization results based on truncated moment of a functionof the random variable. Section 3 is devoted to characterization of NMW interms of the hazard function. Section 4 deals with characterization of NMWin terms of the conditional expectation of a function of the random variable.Finally, in Section 5, we present a characterization of NMW in terms of the1st order statistic.2 Characterizations Based on Truncated MomentIn this section we present characterizations of NMW distribution in terms oftruncated moment. As mentioned in the Introduction, our characterizationresults presented here will employ a version of an interesting result due toGlanzel (1987) (Theorem 1, below). The advantage of the characterizationsgiven here is that, cdf F need not have a closed form and are given interms of an integral whose integrand depends on the solution of a ﬁrst orderdiﬀerential equation, which can serve as a bridge between probability anddiﬀerential equation.c 2014, SRTC IranG. G. Hamedani 225Theorem 1. Let (;F ; P) be a given probability space and let H = [a; b] bean interval for some a < b (a =  1; b =1 might as well be allowed) : LetX :  ! H be a continuous random variable with the distribution functionF and let g be a real function deﬁned on H such thatE fg (X) j X > xg =  (x) ; x 2 H;is deﬁned with some real function . Assume that g 2 C1 (H),  2 C2 (H)and F is twice continuously diﬀerentiable and strictly monotone function onthe set H. Finally, assume that the equation  = g has no real solution inthe interior of H. Then F is uniquely determined byF (x) =Z xaC 0 (u) (u)  g (u) exp f s (u)g du;where the function s is a solution of the diﬀerential equation s0 = 0 g andC is a constant, chosen to make RH dF = 1.Proposition 1. Let X :  ! (0;1) be a continuous random variable andlet g (x) = e ex+e xfor x 2 (0;1) : Then, pdf of X is (2) if and only ifthe function  deﬁned in Theorem 1 has the form (x) =12e ex+e x; x > 0: (3)Proof. Let X have pdf (2), thenf1  F (x)gE [g (X) j X > x] = 12e 2(ex e x ); x > 0;and (x)  g (x) =  12e ex+e x< 0 for x > 0:Conversely, if  (x) is given by (3), thens0 (x) =0 (x) (x)  g (x) =x 1ex+ x 1e x;from which we obtains (x) = ex   e x ; x > 0:Now, in view of Theorem 1, X has cdf (1) and pdf (2).J. Statist. Res. Iran 11 (2014): 223–229226 Characterizations of New Modiﬁed Weibull DistributionCorollary 1. Let X : ! (0;1) be a continuous random variable. Then,pdf of X is (2) if and only if there exist functions g and  deﬁned in Theorem1 satisfying the following diﬀerential equations0 (x) =0 (x) (x)  g (x) =x 1ex+ x 1e xx > 0: (4)Remarks 1. (a) The general solution of the diﬀerential equation (4) is (x) = eex e x( Zg (x)x 1ex+ x 1e x e ex+e xdx+D);for x > 0 , where D is a constant. One set of functions (g; ) satisfying theabove equation is given in Proposition 1 for D = 0:(b) Clearly there are other pairs of functions (g; ) satisfying the condi-tions of Theorem 1. We presented one such pair in Proposition 1.3 Characterization Based on Hazard FunctionFor the sake of completeness, we state the following deﬁnition.Deﬁnition 1. Let F be an absolutely continuous distribution with thecorresponding pdf f . The hazard function corresponding to F is denoted byhF and is deﬁned byhF (x) =f (x)1  F (x) ; x 2 Supp F;where Supp F is the support of F .It is obvious that the hazard function of a twice diﬀerentiable distributionfunction satisﬁes the ﬁrst order diﬀerential equationc 2014, SRTC IranG. G. Hamedani 227h0F (x)hF (x)  hF (x) = q (x)where q (x) is an appropriate integrable function. Although this diﬀerentialequation has an obvious form sinceh0F (x)hF (x)  hF (x) = f0 (x)f (x); (5)for many univariate continuous distributions (5) seems to be the only diﬀer-ential equation in terms of the hazard function. The goal of the characteriza-tion based on hazard function is to establish a diﬀerential equation in termsof hazard function, which has as simple form as possible and is not of thetrivial form (5). Here, we present a characterization of the of NMW modelbased on a nontrivial diﬀerential equation in terms of the hazard function.Proposition 2. LetX : ! (0;1) be a continuous random variable. Then,X has pdf (2) if and only if its hazard function hF satisﬁes the diﬀerentialequationh0F (x) + x 1hF (x) = 2x 2 (1 + x) ex+ 2x 21  xe x; x > 0: (6)Proof. If X has pdf (2), then clearly (6) holds. Now, if (6) holds, then aftermultiplying both sides of (6) by x, we arrive atddxfxhF (x)g = ddxnxex+ xe xo;from which we havehF (x) = x 1nxex+ xe xo;orhF (x) =f (x)1  F (x) = x 1ex+ x 1e x: (7)Integrating both sides of (7) from 0 to x, we arrive at  ln f1  F (x)g = ex   e x :J. Statist. Res. Iran 11 (2014): 223–229228 Characterizations of New Modiﬁed Weibull Distributionfrom which we obtainF (x) = 1  e ex+e x; x > 0:4 Characterizations Based on Single Function ofthe Random VariableIn this section we employ a single function  of X and state characterizationresults in terms of  (X) : The following proposition has already appearedin a Technical Report (Hamedani, 2013), so we will just state it here for thesake of completeness.Proposition 3. Let X :  ! (a; b) be a continuous random variable withcdf F . Let  (x) be a diﬀerentiable function on (a; b) with limx!a  (x) = 1.Then for  6= 1,E [ (X) jX > x] =  (x) ; x 2 (a; b) ;if and only if (x) = f1  F (x)g 1 1 ; x 2 (a; b) :Remark 2. Taking, e.g.,  (x) = e ( 1  )(ex e x ), Proposition 3 gives acharacterization of NMW distribution.5 Characterizations Based on Truncated Momentof Certain Functions of the 1st Order StatisticLet X1:n 6 X2:n 6 : : : 6 Xn:n be n order statistics from a continuous cdf F:We present here a characterization result base on some functions of the 1storder statistic. Our characterization will be a consequence of the followingproposition, which is an extended version of the one appeared in our previouswork (Hamedani, 2010).c 2014, SRTC IranG. G. Hamedani 229Proposition 4. Let X :  ! (a; b) be a continuous random variable withcdf F . Let  (x) and q (x) be two diﬀerentiable functions on (a; b) such thatlim x!b  (x) [1  F (x)]n = 0 andR baq0(t)[q(t)  (t)]dt =1. ThenE f (X1:n) j X1:n > tg = q (t) ; t > a;impliesF (x) = 1  exp Z xaq0 (t)n f (t)  q (t)gdt; a 6 x < b:Remark 3. Take, e.g.,  (x) = en(eex e x ), q (x) = 12 (x) and (a; b) =(0;1) : Clearly there are other suitable functions  (x) and q (x).ReferencesDoostmardi, A., Zadkarami, M.R. and Roshani Sheykhabad, A. (2014). A New ModiﬁedWeibull Distribution and Its Applications. Journal of Statistical Research of Iran, 11, 97-118.Glanzel, W. (1987). A Characterization Theorem Based on Truncated Moments and Its Ap-plication to Some Distribution Families, Mathematical Statistics and Probability Theory (BadTatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht, 75-84.Hamedani, G.G. (2010). Characterizations of Continuous Univariate Distributions Based onthe Truncated Moments of Functions of Order Statistics, Studia Sci. Math. Hungar., 47,462-484.Hamedani, G.G. (2013). On Certain Generalized Gamma Convolution Distributions II, Tech-nical Report, No. 484, MSCS, Marquette University.G. G. HamedaniDepartment of Mathematics, Statisticsand Computer Science,Marquette University,Milwaukee, WI 53201-1881 USA.email: gholamhoss.hamedani@marquette.eduJ. Statist. Res. Iran 11 (2014): 223–229

Characterizations of New Modified Weibull Distribution

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Characterizations of New Modified Weibull Distribution

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