THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Sees A, OF THE ROMANIAN ACADEMY Volume 8, Numbe 3/27,. - L-MOMENTS EVALUATION FOR IDENTICALLY AND NONIDENTICALLY WEIBULL DISTRIBUTED RANDOM VARIABLES Roxaa CIUMARA Academy of Ecoomc Studes, Deatmet of Mathematcs I ths ae we comute the L-momets fo detcally ad odetcally Webull dstbuted adom vaables. The esults ae based o the evaluato of momets of ode statstcs. Fst, we deve the ode statstcs ad L-momets fo geealzed Webull dstbuted adom vaables. As a secal case, we obta the exesso fo the ode statstcs ad L-momets fo the Webull dstbuto. Moeove the L-momets fo detcally dstbuted Webull adom vaabes could be obtaed fom the L-momets fo odetcally Webull dstbuted adom vaables. Key wods: L-momets; ode statstcs; Webull dstbuted adom vaables.. INTRODUCTION It s a stadad actce statstcs to use desctve measues (quattes ode to descbe the shae of the dstbuto of a oulato. Such measues has bee defed by meas of classcal momets of dffeet odes: the mea to estmate locato, the vaace to measue the sead, the stadadzed measues fo sewess ad utoss. Though t s well-ow that deste the oulaty both data descto ad moe fomal statstcal ocedues, samle momets suffe fom seveal dawbacs. Fo examle, they ae sestve to exteme obsevatos. Moeove, the asymtotc vaaces of momet-based estmatos ae maly detemed by hghe ode momets, whch ae athe lage o eve ubouded fo heavy tal dstbutos. As a cosequece, asymtotc effcecy of samle momets s athe oo esecally fo dstbutos wth fat tals. Momet-based measues ae just atcula, but ot exhaustve meas of summazg qualtatve featues of the shae of a dstbuto. The otos of dseso, sewess ad utoss ae athe abstact ad theefoe ca be descbed may ways. A alteatve set of vey effectve desctve measues called L-momets ae based o atal odegs ad seem to lagely ovecome the samlg dawbacs of classcal measues. The L-momets aeaed fo the fst tme wthout ame quatle exaso of Slltto [] ad the Hosg s [5] eseach eot. Fomally, the L-momets wee toduced by Hosg [4] as a lea combato of the ode statstcs of a oulato. Le classcal momets, L-momets ovde tutve fomato about the shae of a dstbuto, whch ca be cosstetly estmated fom the samle values. Hosg, Walls ad Wood [7] ad Hosg ad Walls [6] use L-momets estmato method to exteme value dstbuto. They foud that t efoms bette tha method of momets ad, moeove, that both momet- based methods do well small samles comaed to maxmum lelhood estmato. Tag to accout the moe satsfactoy samlg behavo of L-momets estmatos, the sueo emcal efomace ove some data sets ad the esults of fomal smulato exemets ove some selected dstbutos, some authos (fo examle, Hosg [3] ecommed the use of L-momets stead of the classcal momets. I what follows we wll emd the fomal defto of L-momets ad some motat esults coceg them. I Sectos 3 we deve the L-momets fo geealzed Webull dstbuted adom vaables. As a atcula case, Secto 4 we fd the L-momets exesso fo detcally Webull
Roxaa Cumaa 2 dstbuted adom vaables. I Secto 5 we evaluate the L-momets fo odetcally Webull dstbuted adom vaables. Also we ote that the esults fom Secto 4 could be obtaed as a secal case of those fom Secto 5. 2. SOME PRELIMINARY RESULTS Let X be a eal-valued adom vaable wth cumulatve dstbuto fucto ad let : X 2:... X be the ode statstcs of a adom samle of sze fom the dstbuto of X. X : fo Defto 2. ([3]. The L-momets ae defed to be the quattes,2,...,, whee C j, N!( *.! j j ( C E( X j: F( x, (2. Rema 2.. The L-momets as well as oday momets ae secal cases of obablty weghted momets toduced by Geewood et al. [2] as M, s s ( X ( F( X ( F( X whee,, s. Obvously ae the oday momets,,,, E, (2.2 M,, X + : + ad + M E( ( j!( j! E( X M, j, j j:.! The use of L-momets to descbe obablty dstbutos s justfed by the followg esult. Theoem 2. (Hosg [3]. ( The L-momets,,2,..,, of a eal-valued adom vaable X exst f ad oly f X has fte mea. ( A dstbuto whose mea exsts s chaactezed by ts L-momets. (,2,... Thus a dstbuto ca be secfed by ts L-momets eve f some of ts covetoal momets do ot exst. Futhemoe, such a secfcato s always uque, whch s of couse ot tue fo covetoal momets. The followg esult wll be used Secto 4. Lemma 2. (Kha et al. [8]. If X s a ostve adom vaable wth cumulatve dstbuto fucto F ( x ad ( b a J a, b x F( x ( F( x dx, whee a, b ae eal umbes chose such that J exsts ( a, b ad s fte, the J ( +, ( C J ( I Secto 5 we wll use the followg esult.. +,
3 L-momets evaluato fo Webull dstbuto Theoem 2.2 (Baaat ad Abdelade []. Let X X,..., be odetcally dstbuted adom vaables ad X : X 2:... X : the coesodg ode statstcs. The fo ad,2,... the th ode momet of ca be evaluated ecusvely as whee ( X :, 2 ( ( j ( + ( E X : ( C j I j, j + (... α ( j, ( ( α ( : m x F x dx s the th momet of the mmum of I j : j < j2 <... < j X 2 m F ( x ( X, X,..., m ( t t ad s the cumulatve dstbuto fucto of X. X 3. GENERALIZED WEIBULL DISTRIBUTION CASE Let cosde a adom samle X, X 2,..., X of sze fom a geealzed Webull dstbuto wth cumulatve dstbuto fucto γ x F ( x ex, (3. θ * whee x >, θ, γ R. The obablty desty fucto ths case s, + ( γ F( f ( x x ( F( x ( x γ. (3.2 θγ The obablty desty fucto of the th ode statstcs Deote ( (! X : s γ [ F( x F( x ]( F( x f ( x! : x! θγ. (3.3 ( ( X Tag to accout elato (3.3, the exesso of ( a b s s α : E :. (3.4 ( E( X x f ( x α : : : dx J, fom Lemma 2., the fact that ad Defto 2., we get the followg esult fo the th ode momet of the th ode statstc ad the L- momets fo the geealzed Webull dstbuto. fo Theoem 3.. Ude the evously metoed codtos, ( α : [ J (, γ J (, ] (3.5!! θγ ( ( j j 2 ( ( C [ J ( j, j γ J ( j j ] θγ j + +,,2,...,, f the quattes volved the above exesso exst ad ae fte., (3.6
Roxaa Cumaa 4 4. THE WEIBULL DISTRIBUTION CASE I ths secto we cosde a adom samle X, X 2,..., X of sze fom a two-aamete Webull dstbuto wth cumulatve dstbuto fucto F x ( x ex, (4. θ * whee x > ad, θ R +. Obvously, ths coesods to the case γ fom Secto 3. Theefoe, the obablty desty fucto ths case s f ( x x ( F( x. (4.2 θ Alyg Theoem 3.. fo γ we get ( α : [ J (, J (, ].!! θ ( ( It s easy to see fom elato the exesso of ( a b ad, theefoe, ( a, b J ( a, b J ( a +, b J, gve Lemma 2. that J (4.3 ( : J ( +,, (4.4!! θ α ( ( a esult also obtaed dectly by Kha et al. [8]. Moeove, ths atcula case we have the followg esults egadg L-momets. ad Theoem 4.. The L-momets fo the Webull dstbuto satsfy j ( ( j 2 θ j j j 2 ( ( C J ( j +, j θ + Γ ( a(, whee a, C C, fo,2,...,. j + +, (4.5, (4.6 Poof. We use Defto 2. ad elato (4.4 o Theoem 3.. ad elato (4.3. By Lemma fom Kha et al. [8], we get Futhemoe, we ote that J ( j j + j +, j ( C j J + ( +,. (4.7 J θ,. (4.8 ( a Γ a
5 L-momets evaluato fo Webull dstbuto Now, usg Theoem 4.., ad elatos (4.7 ad (4.8, we get elato (4.6. The ext esult gves a smle exesso fo the L-momets a secal case. Poosto 4.. Fo θ ad, the L-momets fo the Webull dstbuto ae gve by whee 2,...,. Poof. Saha [9] showed that fo, (4.9 ( θ ad we have α : + By the defto of L-momets, we the have. j j j j j ( C ( C ( ( j + + j 2 C 2 fom whch we get elato (4.9 fo 2,...,. Rema 4.. Obvously, ( α θ X : E fo. 5. L-MOMENTS FOR NONIDENTICALLY WEIBULL DISTRIBUTED RANDOM VARIABLES Let F F,...,, 2 X X,..., F, 2, whee X be deedet ostve adom vaables wth cumulatve dstbuto fuctos * wth x >, θ, θ,..., θ., 2 R + As the evous sectos, let ( F x, (5. ( x ex θ X : X 2:... X : deote the coesodg ode statstcs, α : E X :, ad the L-momets defed by (2.. The ext esult gves a chaactezato of L-momets fo odetcally Webull dstbuted adom vaables. Theoem 5.. The L-momets fo odetcally dstbuted Webull adom vaables ae gve by whee ( j Γ ( b(, ~ I, (5.2 j j ~ b, C C ad I.... j j... j j j... < 2 < < θ + θ + + θ 2 j Poof. We aly Theoem 2.2 of Baaat ad Abdelade [] fo. Thus, we have to evaluate ot ( I j I j. Tag to accout the exesso fo ( ~ I j ad elato (5., we obta I j Γ I j, whee I ~ j has the fom metoed above. Now, we get
Roxaa Cumaa 6 α j: Γ j+ ( j C ~ j I, ad the, fom Defto 2., afte some calculato, Γ ( j j ~ C C I j, hece, fally, elato (5.2. Rema 5.. Fo detcally Webull dstbuted adom vaables, that s, Theoem 5., we get elato (4.6 fom Theoem 4.. θ θ fo,, fom REFERENCES. BARAKAT, H.M., ABDELKADER, Y.H., Comutg the momets of ode statstcs fom odetcal adom vaables, Statstcal Methods ad Alcatos, 3,. 3-24, 24. 2. GREENWOOD, J.A., LANDWEHR, J.M., MATALA, N.C., WALLIS, J.R., Pobablty weghted momets: Defto ad elato to aametes of seveal dstbuto exessable the vese fom, Wate Resouces Reseach, 5, 5,. 49-54, 979. 3. HOSKING, J.R.M., Some theoetcal esults coceg L-momets, Reseach Reot RC 4492 (6497 3/22/89, evsed 7/5/96, IBM Reseach Dvso, Yotow Heghts, 996. 4. HOSKING, J.R.M., L-momets: aalyss ad estmato of dstbutos usg lea combatos of ode statstcs, Joual of the Royal Statstcal Socety, Sees B, Methodologcal 52,. 5-24, 99. 5. HOSKING, J.R.M, The theoy of obablty weghted momets, Reseach Reot RC22, IBM Reseach, Yotow Heghts, 986. 6. HOSKING, J.R.M., WALLIS, J.R., Paamete ad quatle estmato fo the geealzed Paeto dstbuto, Techometcs, 29, 3,. 339-349, 987. 7. HOSKING, J.R.M., WALLIS, J.R., WOOD, E.F., Estmato of the geealzed exteme-value dstbuto by the method of obablty weghted momets, Techometcs, 27, 3,. 25-26, 985. 8. KHAN, A.H., KHAN, R.U., PARVEZ, S., Ivese momets of ode statstcs fom Webull dstbuto, Scadava Actuaal Joual,. 9-94, 984. 9. SARHAN, A.E., Estmato of the mea ad stadad devato by ode statstcs, A. Math. Statst., 25,. 37-328, 954.. SILLITTO, G.P., Devato of aoxmats to the vese dstbuto fucto of a cotuous uvaate oulato fom the ode statstcs of a samle, Bometa, 56,. 64-65, 969. Receved May 2, 27