On Marshall-Olkin Extended Weibull Distribution

Transcription

1 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 O Marshall-Olki Exteded Weibull Distributio Haa Haj Ahmad Departmet of Mathematics, Uiversity of Hail Hail, KSA haaahm@yahoo.com Omar M. Bdair Departmet of Physics ad Basic Scieces Faculty of Egieerig Techology, Al-Balqa Applied Uiversity Amma 34, Jorda bdairmb@yahoo.com M. Ahsaullah Departmet of Maagemet Scieces, Rider Uiversity Lawreceville, NJ 8648, USA ahsa@rider.edu Received 4 May 26 Accepted 5 December 26 I this paper, we itroduce a ew form of distributios called Marshall-Olki Exteded Weibull distributio MOEW. We study some of its structural properties ad the shape of its hazard rate fuctio. We provide some methods of poit estimatio for the ukow parameters ad discuss its asymptotic properties. Fially, umerical methods ad simulatios are used to compare betwee differet methods of estimatio, the we study the behavior of parameters uder differet sample sizes ad differet values of parameters. Keywords: Marshall-Olki distributio; Hazard rate fuctio; Maximum likelihood estimators; Method of momet estimators; Least squares estimators; Percetiles estimators; Simulatios; Characterizatio. 2 Mathematics Subject Classificatio: 62H, 62F, 78M5, 37M5, 62E.. Itroductio: Statistical distributios are commoly used to describe real world pheomea. For this feature, the theory of statistical distributio is widely studied ad ew distributios are developed. Scietists are iterested i developig more flexible statistical distributios, therefore, may geeralized classes of distributios have bee developed ad applied to describe various pheomea. A commo advatage of these geeralized distributios is that they have more parameters. [6] proposed a importat Correspodig author. Copyright 27, the Authors. Published by Atlatis Press. This is a ope access article uder the CC BY-NC licese

2 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 method of addig a ew parameter to a existig distributio. Thus a ew family of distributios called Marshall-Olki MO) has bee defied. This ew family of distributios icludes the origial distributio as a special case ad it gives more flexibility to model. [2] itroduced a detailed study about the physical iterpretatio of the MO family. Weibull distributio is a well kow distributio that has may applicatios i survival aalysis ad reliability theory. Its importace appears sice Weibull distributio cosidered as a geeralizatio of may other distributios such as: Expoetial, Rayleigh ad geeralized extreme value distributios. Weibull distributio has may ice properties, amog others, its hazard rates ca be decreasig, costat or icreasig. This is oe of the attractive properties that make the Weibull distributio so applicable. Weibull distributio has bee extesively used over the past decades for modelig data i reliability, egieerig ad biological studies. The eed for exteded form of the Weibull distributio is importat i may areas. Some exteded forms of Weibull distributio ad its applicatios were referred to [24], [3], [2] ad [23]. The extesio idea is simply based o addig more parameters to a well-defied distributio provides more flexible ew classes of distributios. Let Fx) = Fx) deote the survivor fuctio of a cotiuous radom variable X ad f x) = dfx) dx be the probability desity fuctio associated with the cumulative distributio fuctio cdf) Fx), the the MO exteded distributio has the survival fuctio Gx) = Fx) Fx) + Fx), < x <, >,.) where =. It is clear that Eq..) provides a method to obtai a ew distributio from a existig oe. Whe =, Gx) = Fx), it is readily see that Fx) is a special case of Gx). The probability desity fuctio pdf) correspodig to Eq..) takes the form gx) = f x) Fx)) 2, < x <, >. Some special cases of Eq..) were recetly studied i literature. [5] cosidered Pareto distributio, [8] cosidered gamma distributio while [6] took lomax distributio. [] studied the ifereces of Marshall-Olki expoetial distributio ad [9] studied the momets of order statistics based o Marshall-Olki expoetial distributio. [8] ad [7] studied a ew geeralizatio of the geometric ad ormal distributios usig the MO idea. [4] cosidered the MO exteded Weibull distributio with shape ad scale parameters, they studied some of its mathematical properties as well as estimatio of the model parameters. [7] discussed the Marshall-Olki exteded Weibull family of distributios, some structural properties icludig momets ad order statistics were discussed. [2] cosidered Marshall-Olki expoetial distributio ad he itroduced a compariso betwee differet methods of methods of poit estimatio of the distributio parameters. I this paper we itroduce MO exteded Weibull distributio MOEW). Some structural properties of the desity fuctio ad hazard rate fuctio are studied. Our goal is to itroduce differet methods of poit estimatio for the ukow parameters. Numerical methods are used to make compariso betwee these methods, also we study the behavior of estimated parameters for differet sample sizes ad for differet parameters) values. Basically we compare, the maximum likelihood estimators, momet estimators, estimators based o percetiles ad the least square estimators by usig simulatio techiques. 2

3 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 The orgaizatio of this paper is as follows. I sectio 2, we itroduce MOEW desity fuctio ad show that it ca be expressed as a liear combiatio of Weibull desity fuctio. The hazard rate fuctio for the MOEW is give i sectio 3. Sectio 4 describes the methods of poit estimatio. Some characterizatio results are preseted i sectio 5, while umerical results ad simulatio are provided i sectio The probability desity fuctio Cosider Weibull distributio with probability desity fuctio pdf) give by f x;,,µ) = x µ ) exp[ x µ ) ], x µ, where > is the shape parameter, > is a scale parameter ad µ R is the locatio parameter. The cumulative distributio fuctio cdf) is give by Fx;,,µ) = exp[ x µ ) ], ad the survival fuctio is Fx;,,µ) = exp[ x µ ) ]. Usig Eq..), the survival fuctio of Marshall-Olki exteded Weibull distributio MOEW) is give by the correspodig pdf of MOEW is Gx;,,,µ) = x µ exp[ ) ] exp[ x µ ) ], 2.) gx;,,,µ) = x µ ) exp[ x µ ) ] exp[ x µ ) ]) 2, 2.2) where >, >, >, µ R, ad x µ. For < <, usig biomial expasio of the deomiator we ca rewrite it as: exp[ x µ ) ]) 2 = j= j + ) j exp[ j x µ ) ], 2.3) where Γ.) is the gamma fuctio. Applyig the expasio 2.3) i 2.2), yields gx;,,µ) = = j= j exp[ x µ ) ] x µ ) η j f WE x;,,, µ), 2.4) j= where f WE x;,, µ) is the pdf of Weibull distributio with parameters, ad µ, where =, η j+) / j = j. 3

4 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 For >, we ca use similar argumet as i Eq. 2.3), ad after some algebraic maipulatios we ca obtai gx;,,,µ) = v j f WE x;,, µ), 2.5) j= where v j = ) j j+) k= j k + ) ) k j )k. Note that j= η j = j= v j =. Hece the MOEW desity fuctio ca be expressed as a ifiite liear combiatio of Weibull desity fuctio. Therefore expressio 2.4) ad 2.5) ca be used to fid may mathematical properties such as the momets of MOEW distributio. 3. The Hazard Rate Fuctio Hazard rate fuctio or failure rate is importat i survival aalysis as well as reliability theory. The hazard rate fuctio for MOEW is of the form hx;,,,µ) = x µ ) exp[ x µ ) ], for x µ. I order to determie the shape of hx;,,,µ) it is quite eough to determie the shape of log hx;,,,µ), the first derivative of log hx;,,,µ) is d loghx;,,,µ) dx = sx) x µ) exp[ x µ ) ]), where sx) = exp[ x µ ) ]) ) exp[ x µ ) ] x µ ). The possible shapes of the hazard rate fuctio are: - If < <, the sx) is egative for x µ, ad hece the hazard fuctio is a decreasig fuctio with hµ) = ad h) =. 2- If >, < <, the sx) has two roots x ad x. The hazard fuctio is icreasig o µ,x ), decreasig o x,x ) ad icreasig o x,) with hµ) = ad h) =. 3- If >, >, the sx) is positive. The hazard rate is icreasig with hµ) = ad h) =. 4. Poit Estimatio I this sectio we itroduce four differet methods of poit estimatio for the parameters of MOEW distributio. The asymptotic properties are discussed for the maximum likelihood ad method of momet estimators. I sectio 6, umerical techiques will be used to obtai the estimated values of parameters, the a compariso is doe betwee these estimated values to decide which method is the best. 4.. Maximum Likelihood Estimatio The maximum likelihood estimatio MLE) is widely used i iferetial statistics as it has may ice properties such as ivariace, cosistecy ad ormal approximatio property. It depeds basically o maximizig the likelihood fuctio of MOEW distributio. Let X,X 2,...,X be a radom 4

5 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 sample from MOEW, the the log-likelihood fuctio for the vector of parameters γ =,,,µ) ca be expressed as lγ) = log ) x i µ ) + ) log x i µ ) 2 log exp[ x i µ ) ]). 4.) From the above log-likelihood equatio we compute the derivatives with respect to the parameter vector γ. Sice x µ, the the MLE of the parameter µ is assumed to be x ), where x ) is the first order statistic. So, we eed to solve the followig three ormal equatios after equatig them to zero: f,,) = lγ) = 2 exp[ x i µ ) ] exp[ x i µ ) ] f 2,,) = lγ) = log x i µ ) x ) i µ ) f 3,,) = lγ) = + x i µ ) exp[ x i µ exp[ x i µ ) ] x i µ ) exp x i µ ) ) ) ] x i µ ) log x i µ ) exp[ x i µ ) ] The solutio for the above ormal equatios is ot a easy task as they have ot a explicit solutio. The MLE s ca be obtaied umerically usig Newto-Raphso method to solve the above ormal equatio. May umerical methods were used i the literature to solve such system of oliear equatios. I this paper, we use the Newto-Raphso method which is oe of the mostly used methods. This method depeds o the followig iterated equatio i+) i) i+) = i) i+) i) f f f f 2 f 2 f 2 f 3 f 3 f 3 γ) f γ) f 2 γ). f 3 γ) The above iterated iterated equatio depeds o the choice of the iitial poit ), ), ) ). We use Mathematica package to produce the ew values iterated values) for, ad based o their past or iitial) values util they coverge to their MLE s results. The ormal approximatio of the MLE of vector parameter γ ca be used to costruct approximate cofidece itervals ad for testig hypotheses o the parameters,, ad µ. From the asymptotic property of the MLE we have that γ γ) d N 4,K γ)), where Kγ) is the uit expected iformatio matrix ad Kγ) = lim I γ). Here I γ) is the observed iformatio matrix evaluated at γ. The observed iformatio matrix is give by El ) El ) El ) El µ ) I γ) = El ) El ) El ) El µ ) El ) El ) El ) El µ ). El µ ) El µ ) El µ ) El µµ ) The expected values of the secod derivatives ca be foud by usig some methods of itegratio. Now, without loss of geerality, we assume that =. The MLE of, whe the other 5

6 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 parameters are kow is obtaied by solvig the o-liear equatio = 2 exp[ x i µ ) ] exp[ x i µ ) ]. 4.2) The MLE s of ad ca be obtaied directly by maximizig 4.) with respect to,, respectively, give that other parameters are kow Method of Momet Estimatio I order to use method of momet estimators MME), it is essetial to compute the momets of MOEW distributio. We use the idea of biomial expasio i Eqs. 2.4) ad 2.5) i order to obtai the followig: EX ) = { w j EY j η j < < ), w j = j= ν j >, 4.3) where Y j f WE x;,, µ). So we divide our work ito two cases: Case a): < < : The expected value i Eq. 4.3) is writte as EX ) = η j EY j ). 4.4) j= The th momet for a three parameter Weibull distributio was give by [23]. Usig their formula we ca rewrite Eq. 4.4) as EX ) = j= i=η j i )µ i j + ) / ) i) Γ + i ). Usig the above formula we ca fid the first four momets. We equate sample momets with the populatio momets of MOEW to obtai the followig four equatios: m = X = EX) = µ + Γ + )ploylog,), 4.5) m 2 = EX 2 ) = µ µγ + )ploylog,) + 2 Γ + 2 )ploylog 2,), 4.6) m 3 = EX 3 ) = µ µ2 Γ + )ploylog,) µγ + 2 )ploylog 2,) + 3 Γ + 3 )ploylog 3,), 4.7) m 4 = EX 4 ) = µ µ3 Γ + )ploylog,) µ 2 Γ + 2 )ploylog 2,) +2 3 µγ + 3 )ploylog 3,) Γ + 4 )ploylog 4,), 4.8) 6

7 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 where polylogs,z) is a special fuctio Li s z) which is the polylogarithm fuctio defied by the power series Li s z)= z k k. k= s Case b): > : I this case the expected value i Eq. 4.3) ca be writte as ) i EX ) = ν j EY j ) = j )µ j= j= i=ν i i j + ) / Γ + i ) ) = j ) k k + ) )µ j + ) j )k i i j + ) / j= i= k= j ) i Γ + i ). Usig this formula ad by applyig the same argumet used for case < <, we ca fid the first four momets for case >. I fact, after some mathematical maipulatio, we fid that the first four momet for this case are the same as the first four momets for case < < give i Eq. s 4.5)-4.8), respectively. We use umerical methods to solve the above four equatios i order to estimate the eeded parameters. Sometimes populatio ad sample variace ca be used to obtai the secod momet istead of Eq. 4.6), hece S 2 = σ 2 = 2 Γ + 2 )ploylog 2,) ) 2 2 Γ 2 + )ploylog,))2. 4.9) Now we discuss the asymptotic distributio of the MME s of,, ad µ, let us deote γ =,,,µ) ad let f γ) = X µ Γ + )ploylog,), f 2 γ) = S 2 2 Γ + 2 )ploylog 2,) + ) 2 2 Γ 2 + )ploylog,))2, f 3 γ) = EX 3 ) µ 3 3 µ2 Γ + )ploylog,) 3 2 µγ + 2 )ploylog 2,) 3 Γ + 3 )ploylog 3,), f 4 γ) = EX 4 ) µ 4 4 µ3 Γ + )ploylog,) 2 2 µ 2 Γ + 2 )ploylog 2,) 2 3 µγ + 3 )ploylog 3,) 4 4 Γ + 4 )ploylog 4,). Usig Taylor expasio of f γ MM ) about the true value of γ =,,,µ), where f γ) = f γ), f 2 γ), f 3 γ), f 4 γ)), we obtai f ) f 2 ) f 3 ) f 4 ) f γ MM ) f γ) = [ MM, MM, f MM, µ MM µ] ) f 2 ) f 3 ) f 4 ) f ) f 2 ) f 3 ) f 4 ), f µ ) f 2 µ ) f 3 µ ) f 4 µ ) γ=γ 7

8 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 where γ is a poit betwee γ MM ad γ. Its clear that as, γ MM γ ad γ γ. The derivatives metioed i the above matrix ca be obtaied simply, let us cosider the first oe as a example: f Γ + /) = 2 polylog + /,) polylog/,) + polylog + /,)), where polyγz) = ψz) = d dz lγz)) = Γ z) Γz) ad polylog,) a/,) = 2 polyloga/,) a. Usig cetral limit theorem we obtai X EX)) N,σ 2 ) ad Therefore S 2 ES 2 )) N, )2 m 4 ) 3)m ). [ X EX)), S 2 ES 2 ))] N, [ σ 2 CovX,S 2 ) CovX,S 2 ) )2 m 4 ) 3)m where CovX,S 2 ) = m3 2m m 2 + m σ 2 + m 3 ) m m 2 m 2 ). As a special case, we assume that µ = ad σ =, the EX k ) = = k=o x k x e x e x ) 2 dx x m x k + ) ) k e k+)x dx Usig the substitutio k + )x = t, we coclude EX k ) = k=o k+ ) m ) k Γ + m ) Estimators Based o Percetiles This method was itroduced by [] ad [2], which ca be used whe the data has a distributio fuctio with closed form. The idea depeds o estimatig the ukow parameters by fittig straight lie to the theoretical percetile poits obtaied from the distributio fuctio ad the sample percetile poits. [] ad [2] foud that this method ca be useful i Weibull ad expoetial distributios. I this sectio we use the same techique for the MOEW distributio. Cosider the cdf) of MOEW distributio Gx;γ) = exp[ x µ ) ], where γ =,,,µ). Therefore [ ] / x = l Gx;γ) + ) + µ. Let X i) deotes the i th order statistic from a sample of size. If p i deotes some estimate of Gx i) ;γ), the the estimate of γ =,,,µ) ca be obtaied by miimizig λ = [ ] ) / 2 x i) µ l + ). 4.) p i ], 8

9 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 This method is used by several authors, see for example [2], [9] ad [4]. Sice Eq. 4.) is a oliear equatio it is appropriate to use o-liear optimizatio techique to fid the miimum values of the eeded estimators. These estimators are called percetile estimators PCE s). It is possible to use several p i s as estimators of Gx i) ). For example p i = i + is the most used estimator of Gx i)), i as. Some of the other choices + is the expected value of Gx i)). I this paper we also use p i = i + of p i s are p i = i 3/8))/ + /4)) or p i = i /2))/, see [5]. We assume that is kow, the the PCE s of,,µ) ca be obtaied by takig partial derivatives with respect to the other three parameters respectively. Let π i = l p i + ), the the three ormal equatios that should be solved are: λ µ = 2 λ = 2 λ = 2 2 x i) µ π i ) / ) 4.) x i) µ π i ) / ) π i ) / 4.2) x i) µ π i ) / ) π i ) / lπ i ). 4.3) By solvig these ormal equatios after equatig them to zero, we have π i ) /, hece µ PCE = x π i ) /. x i) = µ + It is easy to fid µ PCE whe ad are kow. Now, whe µ ad are kow ad if we deote Gx) = Gx;,,, µ), the which is equivalet to = x µ [ ]) /. Therefore, the PCE of ca be calculated by mii- l Gx) + mizig exp x µ ) = Gx), x i) µ l[ p i + with respect to. Hece, the PCE of is give by PCE = ]) / x i) µ ]) / l[ p + i Similarly, whe µ ad are kow, the PCE of is foud to be PCE = ) l l[ p + ]x i i) [ xi) µ ] l ). 4.5) 9

10 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 Without loss of geerality, we ca assume = ad µ ad are kow. If we deote Gx) = Gx;,,, µ) the Gx) = ex µ) or equivaletly = ex µ) ) Gx)). Gx) Therefore, the PCE of ca be obtaied by miimizig ex i) µ) ) p i ) p i with respect to. I this case the PCE of is foud to be PCE = ) 2 e x i) µ) ) p i ) p i. 4.6) Iterestigly, all the PCE s estimators have a closed forms whe assumig that the other parameters are kow Least Squares Estimators ad Weighted Least Squares Estimators The method of least squares estimate or regressio estimate was first suggested by [22] to estimate the parameters of beta distributio. The method ca be described as follows: Suppose Y,Y 2,...,Y is a radom sample of size from a distributio fuctio G.) ad Y ),Y 2),...,Y ) deote the order statistics of the observed sample. Cosider GY i) ) to be the distributio fuctio of the i th order statistics from the observed sample, the GY i) ) has U,) distributio. Therefore we have EGY i) )) = i +, VarGY i i + ) i))) = + ) 2 + 2), 4.7) ad CovGY i) ),GY j) )) = i j + ) + ) 2, f or i < j. 4.8) + 2) See, for example, []. The least squares estimators of the ukow parameters ca be obtaied by miimizig GY i) ) i ) 2 + with respect to the ukow parameters,,, µ ad hece miimizig e Y i) µ ) e Y i) µ ) ) 2 i ) Numerically, we ca obtai the least squares estimate of the four parameters which are deoted by LSE, LSE, LSE, µ LSE.

11 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 For the weighted least square estimators of the ukow parameters we eed to miimize VarGY i) )) GY i) ) i ) 2. + Equivaletly, the weighted least square estimators ca be evaluated by miimizig + ) 2 + 2) i i + ) e Y i) µ ) e Y i) µ ) ) 2 i +, 4.2) to obtai what are deoted by WLSE, WLSE, WLSE, µ WLSE. Numerical argumet are eeded to evaluate the so obtaied estimators. 5. Some Characterizatio Results I this sectio, we preset four characterizatio results based o MOEW distributio. To prove the mai results, we eed to cosider some Lemmas. Lemma 5.. Suppose that X is a absolutely cotiuous radom variable ad its cdf Fx) with F) = ad Fx) > for all x >. Assume that f x) is the pdf of X ad f x) exists for all x >. For a cotiuous fuctio gx) defied o < x < with fiite EgX)). If EgX) X x) = hx)rx), where hx) is differetiable i x > ad rx) = f x) is the hazard rate, the f x) = cexp x gu)+h u) hu) Fx) du), where c is determied by the coditio Proof. Let hx) = obtai gx) f x) = f x)h x) + f x)hx). After some simplificatios, we have x gu) f u)du f x) f x)dx =., the x gu) f u)du = f x)hx). Differetiatig both sides, we f x) f x) = gx) + h x). hx) Itegratig both sides of the above equatio, we obtai f x) = cexp x determied by the coditio f x)dx =. gu)+h u) hu) du) ad c is Lemma 5.2. Suppose that X is a absolutely cotiuous radom variable ad its cdf Fx) with F) = ad Fx) > for all x >. Assume that the pdf of X is f x) ad f x) exists for all x >. For a a cotiuous fuctio gx) defied o < x < with fiite EgX)). If EgX) X x) = hx)τx), where hx) is a differetiable fuctio i x > ad τx) = cexp x gu)+h u) hu) du) ad c is determied by the coditio f x)dx =. x gu) f u)du f x) Proof. We have hx) = obtai gx) f x) = f x)h x) + f x)hx). After some simplificatios, we have f x) Fx), the f x) =, the x gu) f u)du = f x)hx). Differetiatig both sides, we f x) f x) = gx) h x). hx) Itegratig both sides of the above equatio, we obtai f x) = cexp x determied by the coditio f x)dx =. gu) h u) hu) du), where c is

12 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 Theorem 5.. Suppose that X is a absolutely cotiuous radom variable ad its cdf Fx) with F) = ad Fx) > for all x >. Assume that the pdf of X is f x) ad f x) exists for all x > ad EX m ) exists for m. The EX m X x) = hx)rx), where rx) = f x) Fx) ad hx) = e x ) 2 cx e x k= k+ ) m Γ c m+ k+)x ) where Γ c ) = x e x dx if ad oly if f x) = cx e x for <. e x ) 2 Proof. We have f x)hx) = = = x x k= where Γ c ) = x e x dx. Thus u m u e u e u ) 2 du u m u e u k=k + ) ) k e ku du k + ) m Γ c k+)x m + ), Suppose the hx) = e x ) 2 cx e x hx) = e x ) 2 cx e x k= k= k + ) m Γ c k+)x m + ). k + ) m Γ c k+)x m + ), x m + h x) hx) Usig Lemma 5., we coclude = u x 2 x e u e u. f x) f x) = x 2 x e u u e. u By itegratig the above equatio, we obtai f x) = c cx e x coditio f x)dx =, we obtai f x) = cx e x e x ) 2, <., where c is a costat. Usig the e x ) 2 Theorem 5.2. Suppose that X is a absolutely cotiuous radom variable ad its cdf Fx) with F) = ad Fx) > for all x >. Assume that the pdf of X is f x) ad f x) exists for all x > ad assume EX m ) exists for m. The EX m X x) = hx)τx), where τx) = f x) Fx) ad 2

13 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 hx) = e x ) 2 cx e x k= k+ ) m Γ k+)x m+ cx e x e x ) 2, <. Proof. We have f x)hx) = = = x o x o k= where Γ ) = x e xk+)x dx. Thus Suppose ow that the hx) = e x ) 2 cx e x hx) = e x ) 2 cx e x x m h x) hx) Usig Lemma 5.2, we coclude that ), where Γ ) = x e x dx if ad oly if f x) = u m u e u e u ) 2 du u m u e u k=k + ) ) k e ku du k + ) m Γk+)x m + ), = u k= k= k + ) m Γk+)x m + ) k + ) m Γ c k+)x m + ), x 2 x e u e u. f x) f x) = x 2 x e u u e. u By itegratig the above equatio, we obtai f x) = c cx e x the coditio f x)dx =, we obtai f x) = cx e x e x ) 2, <., where c is a costat. Usig e x ) 2 Based o Eq. s 2.) ad 2.2) ad characterizatio purpose of this distributio, we ca assume w.l.o.g µ = ad =. 3

14 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 6. Numerical Computatios ad Discussios I this sectio, we perform some umerical computatios to compare the performaces of the differet estimators proposed i the previous sectios. We perform a extesive simulatio study to compare the performaces of the differet methods i the sese of bias ad mea square error MSE) for differet sample sizes ad for differet parameter values. The geeratio of the MOEW ca be easily obtaied through the trasformatio X = µ + log U) + ))/, where U is a uiform distributio deviate o, ). Mathematica 7 codes are used for geeratig the MOEW radom variables ad for solvig the o-liear equatios as well as for computig the miimizatio or maximizatio of the related fuctios. Sice is the scale parameter ad all the estimators are scale ivariat, we take = i all cases cosidered. We cosider various choices of the parameters,, µ ad sample sizes =,5 ad. We compute the average relative biases ad average relative MSE s over, rus. This umber of rus will give the accuracy i the order ±,) 5 = ±. see [3]). Therefore, we report all the results up to three decimal places. First we cosider the estimatio of whe other parameters are kow. If, ad µ are kow, the MLE s ad PCE s of ca be obtaied directly from 4.2) ad 4.6) respectively. The MLE s of ad ca be obtaied directly from Eq. 4.). The PCE s of, ad µ ca be computed directly from Eq. 4.). The MME s of all parameters ca be obtaied by solvig the o-liear equatio 4.5) whe other parameters are kow. The LSE s ad WLSE s ca be obtaied by miimizig 4.9) ad 4.2), respectively, with respect to the eeded parameter oly. If is a estimate, the we preset the average value of /) ad the average MSE of /). The relative average bias ad relative average MSE of γ/γ), where γ is a estimate of γ, are defied, respectively, as follows: RelativeBias γ/γ) = /k) k γ i/γ) ad RelativeMSE γ/γ) = /kγ 2 ) k γ i ρ) 2, where ρ = γrelativebias γ/γ) ad k is the umber of iteratios. We are calculated the results for =.2,.4,.8, for, µ =.5,.,2. ad for =,5 ad. The results are preseted i 2 differet tables. But sice the umber of tables are very large, we preset oly 4 tables each of them whe =.8. The other tables are to be give upo request from the correspodig author of this paper. For each method, the average value of /) is give i each box ad the correspodig MSE is reported withi parethesis. It is observed from Table that all of the estimators usually overestimate for small values of,.4). For >.4, most of the estimators teds to be uderestimates for µ >. It is also observed that all estimates decrease as the value of µ icrease. Oe ca also observe that for each estimatio method, the average relative MSE s decreases as the sample size icreases ad also as the value of icreases. It is observed from Table 2 that all of the estimators usually overestimate for all values of except PCE s that appear to be always uderestimate. All estimates ted to be uderestimate for large values of µ ad large sample sizes. It is also observed that all estimates decrease as the value of µ icrease. Oe ca also observe that for each estimatio method, the average relative MSE s decreases as the sample size icreases ad also as the values of ad µ icrease. It is observed from Table 3 that all of the estimators usually uderestimate for all values of except MLE s that ted to be uderestimate for >.4. All estimates ted to be uderestimate for large values of µ ad large sample sizes. It is also observed that all estimates decrease as the value 4

15 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 Table. Average relative estimators ad average relative MSE s of.,, µ) method.8,.5,.5).8,.5,.).8,.5, 2.).8,.,.5).8,.,.).8,., 2.) MLE.9.78).3.57).79.4).39.7).9.98) ) MME ).83.33) ).3.38) ) ) PCE ) ).96.27) ) ) ) LSE ).8.22).87.87) ) ) ) WLSE ).3.92).783.5) ) ).86.53) 5 MLE.3.48).7.38) ).2.76).2.49).82.32) MME ).95.28).887.) ) ).82.53) PCE ) ).28.84) ).4.29) ) LSE )..94).84.52) ).75.28).83.36) WLSE ).3.59).73.62) ).37.23).822.4) MLE..63) ).637.7)..64) ).72.29) MME.272.4).6.2).842.9) ).423.9).76.27) PCE.93.29).37.23).84.5).7.232).3.26).745.) LSE ).99.28) ) ).78.27).76.93) WLSE.5.22).98.4).6.7).3.92).92.3).625.6) Table 2. Average relative estimators ad average relative MSE s of.,, µ) method.8,.5,.5).8,.5,.).8,.5, 2.).8,.,.5).8,.,.).8,., 2.) MLE.3.2).58.92).3.87).27.63).25.6)..52) MME.36.6).28.).3.73).3.7).26.62).5.58) PCE ).92.2) ) ) ).9.5) LSE.75.9).28.25).8.).9.92).4.89)..82) WLSE.582.).3.23).82.2).44.87).3.8).3.77) 5 MLE.83.3).8.92).72.82).62.68).59.65).4.56) MME.7.5).4.)..76).4.62)..57) ) PCE.94.).92.97).97.54) ).936.6).98.45) LSE.62.65).3.8) ).52.88).7.8) ) WLSE.4.97)..95).592.8) ).87.62).9.57) MLE.55.87).2.85).7.8).46.55).3.43) ) MME.2.89).99.79) )..64).992.6).92.43) PCE.98.94).92.88) ).923.5).97.42).9.5) LSE.59.97).2.87).88.82).5.7).9.67).97.5) WLSE.2.8)..73).56.63).56.6)..54).524.4) of µ icreases. Oe ca also observe that for each estimatio method, the average relative MSE s decreases as the sample size icreases ad also as the values of ad µ icrease. It is observed from Table 4 that most of the estimators usually uderestimate µ for all values of especially for large values of µ. All estimates ted to be uderestimate for large values of µ ad large sample sizes except MLE s that appear to be always overestimate. It is also observed that all estimates decrease as the values of µ ad icrease but they icrease as the sample sizes icrease. Oe ca also observe that for each estimatio method, the average relative MSE s decreases as the sample size icreases ad also as the values of ad µ icrease. The MLE s provide the best results for all sample sizes. The WLSE s work better tha the LSE s for all sample sizes ad all values of. I the cotext of computatioal issues, the MLE s, MME s 5

16 Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 Table 3. Average relative estimators ad average relative MSE s of.,, µ) method.8,.5,.5).8,.5,.).8,.5, 2.).8,.,.5).8,.,.).8,., 2.) MLE ).98.59) ).992.6).98.58).97.53) MME.9.39) ).82.28).6.2).595.9).589.) PCE.2.72).92.7) ) ).56.63).489.7) LSE.69.6).987.2).7.5).57.8).349.).3.5) WLSE..39)..9).699.4).55.8).322.9).295.5) 5 MLE.979.6) ).966.5).98.5) ) ) MME.65.65).982.6).85.58).585.4).58.9).543.6) PCE ) ).85.55).73.66).5.59) ) LSE.89.54) ).664.2).482.5).34.).293.4) WLSE.3.27) ).639.).486.2).37.5).28.4) MLE ) ) ) ).96.4).94.32) MME.5.56).97.57).843.5).6.4).572.9).523.5) PCE ) ) ).75.5).49.46).43.23) LSE ).95.42).6.7).478.5).332.9).284.6) WLSE ).98.9).6.).473.5).34.4).275.4) Table 4. Average relative estimators ad average relative MSE s of µ.,, µ) method.8,.5,.5).8,.5,.).8,.5, 2.).8,.,.5).8,.,.).8,., 2.) MLE.32.6).6.48).9.34).35.4).8.25).7.9) MME.94.25).37.27).746.9) ) ) ) PCE.93.7) ) ).95.48).94.4) ) LSE.93.99).92.46) ).73.3).628.8).64.2) WLSE.87.77).93.52).86.3).77.3).663.8).628.6) 5 MLE.34.59).24.39).9.32).45.39).22.3).7.7) MME.7.9).45.2).85.9) ) ).547.8) PCE ).97.45).892.4) ).96.37).9.29) LSE.42.23).964.2).79.9).734.6).68.4).636.3) WLSE.92.2).72.8).868.9).789.5).733.3).7.2) MLE.82.7).76.5).3.9).52.2).4.7).32.2) MME.27.7).9.93).87.82).62.) ) ) PCE ).958.4).99.33).942.3) ).94.9) LSE.92.2).87.9).9.7).782.6).79.4).683.2) WLSE.99.5).85.7).97.4).87.4).749.2).72.) ad PCE s could ot be easily implemeted sice they ivolve o-liear equatios. The LSE s ad WLSE s ivolve o-liear fuctios that should be miimized. Ackowledgmets The authors would like to thak the aoymous referees for their very costructive suggestios that cotributed to the improvemet of this versio of the paper. 6

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