Expectation Identities of Upper Record Values from Generalized Pareto Distribution and a Characterization

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J. Stat. Appl. Pro. 2, No. 2, 5-2 23) 5 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://d.doi.org/.

1 J. Stat. Appl. Pro. 2, No. 2, ) 5 Joural of Statistics Applicatios & Probability A Iteratioal Joural Epectatio Idetities of Upper Record Values from Geeralized Pareto Distributio ad a Characterizatio Devedra Kumar ad Aamika Kulshrestha 2 Departmet of Statistics, Amity Istitute of Applied Scieces,Amity Uiversity, Noida-23, Idia 2 Departmet of Statistics ad Operatios Research, Aligarh Muslim Uiversity, Aligarh-22 2, Idia Received: 8 Mar. 23, Revised: 2 Mar. 23, Accepted: 26 Mar. 23 Published olie: Jul. 23 Abstract: I this paper we cosider geeralized Pareto distributio. Eact epressios ad some recurrece relatios for sigle ad product momets of upper record values are derived. Further a characterizatio of this distributio based o coditioal ad recurrece relatio of sigle momets of record values is preseted. Keywords: Record values; sigle momet; product momet; recurrece relatios; geeralized Pareto distributio; coditioal epectatio; Characterizatio. Itroductio A radom variable X is said to have geeralized Pareto distributio Hall ad Weller [9]) if its probability desity fuctio pd f) is of the form ad the correspodig survival fuctio is f)= +) ) /, >,, >.) ) 2 ) +/), F) = >,, >.2) Where F)= F), >, >, the F is said to be member of geeralized Pareto distributio. It should be oted that for > ad < < this model is, respectively, a Pareto distributio ad a Power distributio. Moreover the survival fuctio.2) teds to the epoetial survival fuctio as teds to zero. This model is a fleible oe due to its properties, i.e. it has a liear mea residual life fuctio its coefficiet of variatio of the residual life is costat ad its hazard rate is the reciprocal of liear fuctio. For more details ad some applicatios of this distributio oe may refer to Hall ad Weller [9] ad Johso et al. []. Record values are foud i may situatios of daily life as well as i may statistical applicatios. Ofte we are iterested i observig ew records ad i recordig them: for eample, Olympic records or world records i sport. Record values are used i reliability theory. Moreover, these statistics are closely coected with the occurreces times of some correspodig o homogeeous Poisso process used i shock models. The statistical study of record values started with Chadler [6], he formulated the theory of record values as a model for successive etremes i a sequece of idepedetly ad idetically radom variables. Feller [8] gave some eamples of record values with respect to gamblig problems. Resick [22] discussed the asymptotic theory of records. Theory of record values ad its distributioal properties have bee etesively studied i the literature, for eample, see, Ahsaullah [], Arold et al. [2, 3], Nevzorov [8] ad Kamps [2] for reviews o various developmets i the area of records. We shall ow cosider the situatios i which the record values e.g. successive largest isurace claims i o-life isurace, highest water-levels or highest temperatures) themselves are viewed as outliers ad hece the secod or Correspodig author devedrastats@gmail.com c 23 NSP

2 6 D. Kumar, A. Kulshrestha: Epectatio Idetities of Upper Record... third largest values are of special iterest. Isurace claims i some o life isurace ca be used as oe of the eamples. Observig successive k largest values i a sequece, Dziubdziela ad Kopociski [7] proposed the followig model of k record values, where k is some positive iteger. Let {X, } be a sequece of idetically idepedetly distributed i.i.d) radom variables with pd f f) ad distributio fuctio d f) F). The j th order statistics of a sample X,X 2,...,X ) is deoted by X j:. For a fi k we defie the sequece{u k), } of k upper record times of{x, } as follows U k) =, The sequece {Y k), } with Y k) {X, }. U k) + = mi{ j> Uk) = X U k) : X j : j+ k+> X k) U :U k) }. +k :U k), =,2,... are called the sequeces of k upper record values of +k For k = ad =,2,... we write U ) = U. The {U, } is the sequece of record times of {X, }. The sequece {Y k), }, where Y k) = X k) is called the sequece of k upper record values of {X U, }. For coveiece, we shall also take Y k) =. Note that k = we have Y ) {X, }. Moreover Y k) = mi{x,x 2,...,X k = X :k }. Let {X k) = X U,, which are record value of, } be the sequece of k upper record values the from.3). The the pd f of X k), is give by ad the joit pd f of X k) m We shall deote f k))= k X )! [ l F))] [ F)] k f).3) ad X k), m<, >2 is give by f k) X m,x k),y)= k m )! m )! [ l F))] m [ l Fy)+l F)] m k f) [ Fy)] fy), <y..4) F) :k = EXk) U) )r ),,2,..., µ r,s) m,:k = EXk) Um) )r,x k) U) )s ), m ad r,s=,2,..., µ r,) m,:k = EXk) µ,s) m,:k = EXk) Um) )r )= m:k U) )s )=µ s) :k, m ad r=,2,...,, m ad s=,2,..., Recurrece relatios are iterestig i their ow right. They are useful i reducig the umber of operatios ecessary to obtai a geeral form for the fuctio uder cosideratio. Furthermore, they are used i characterizig the distributios, which i importat area, permittig the idetificatio of populatio distributio from the properties of the sample. Recurrece relatios ad idetities have attaied importace reduces the amout of direct computatio ad hece reduces the time ad labour. They epress the higher order momets i terms of order momets ad hece make the evaluatio of higher order momets easy ad provide some simple checks to test the accuracy of computatio of momets of order statistics. Recurrece relatios for sigle ad product momets of k record values from Weibull, Pareto, geeralized Pareto, Burr, epoetial ad Gumble distributio are derived by Pawalas ad Szyal [9, 2, 2]. Sulta [24], Sara ad Sigh [23], Kumar [6], Kumar ad Kha [7] are established recurrece relatios for momets of k record values from modified Weibull, liear epoetial, epoetiated log-logistic ad geeralized beta II distributios respectively. Balakrisha ad Ahsaullah [4, 5] have proved recurrece relatios for sigle ad product momets of record values from geeralized Pareto, Loma ad epoetial distributios respectively. Recurrece relatios for sigle ad product momet of geeralized epoetial distributio are derived by Kha et al. [4] ad Kha et al. [5] are characterized the distributios based o geeralized order statistics. Kamps [3] ivestigated the importace of recurrece relatios of c 23 NSP

3 J. Stat. Appl. Pro. 2, No. 2, ) / 7 order statistics i characterizatio. I this paper, we established some eplicit epressios ad recurrece relatios satisfied by the sigle ad product momets of k upper record values from A characterizatio of this distributio based o coditioal epectatio ad recurrece relatios of sigle momets of record values. 2 Relatios for Sigle momet First of all, we may ote that for the geeralized Pareto distributio i.) F)= f). 2.) +) The relatio i 2.) will be eploited i this paper to derive recurrece relatios for the momets of record values from We shall first establish the eplicit epressio for sigle momet k record values EX k) ) r ). Usig.3), we have :k = k r [ l F))] [ F)] k f). 2.2) )! By settig t =[ F)] /+) i 2.2), we get :k = +) r k r+ )! r p= Agai by puttig, w= lt +)/, we obtai :k =[+)k] ) p r p ) r r p= ) ) p r p t k+)+)+ +p r [ lt +)/ ] dt. ) [+)k+p r)]. 2.3) Remark 2.: For k = i 2.3) we deduce the eplicit epressio for sigle momets of upper record values from Recurrece relatios for sigle momets of k upper record values from d f.2) ca be derived i the followig theorem. Theorem 2.: For a positive iteger k ad for ad r=,,2,..., Proof We have from equatios.3) r ) +)k :k = µr) :k + r +)k µr ) :k. 2.4) :k = k r [ l F))] [ F)] k f)d. 2.5) )! Itegratig by parts treatig[ F)] k f) for itegratio ad the rest of the itegrad for differetiatio, we get :k = µr) :k + rk r [ l F))] [ F)] k d k )! the costat of itegratio vaishes sice the itegral cosidered i 2.5) is a defiite itegral. O usig 2.), we obtai :k = µr) :k + rk { r [ l F))] [ F)] k f)d +)k )! } + r [ l F))] [ F)] k f)d ad hece the result give i 2.4). Remark 2.2 Settig k = i 2.4) we deduce the recurrece relatio for sigle momets of upper record values from c 23 NSP

4 8 D. Kumar, A. Kulshrestha: Epectatio Idetities of Upper Record... 3 Relatios for Product momet O usig.4), the eplicit epressio for the product momets of k record values µ r,s) m,:k where µ r,s) m,:k = k m )! m )! ca be obtaied r [ l F))] m f) G)d, 3.) [ F)] G)= y s [ l Fy))+l F))] m [ F)] k fy)dy. 3.2) By settig w = l F)) l Fy)) i 3.2), we obtai G)=+) m ) s s p= ) p s p ) [ F)] k+p s)/+) Γ m) [+)k+p s)] m. O substitutig the above epressio of G) i 3.) ad simplifyig the resultig equatio, we obtai µ r,s) m,:k =[+)k] m ) r+s s r ) ) p+q s r p q) p= q= [+)k+p s)] m [+)k+p+q r s)] m. 3.3) Remark 3. Settig k= i 3.3) we deduce the eplicit epressio for product momets of record values from the geeralized Pareto distributio. Makig use of 2.), we ca drive recurrece relatios for product momets of k upper record values Theorem 3.: For m 2 ad r,s=,2,..., s ) µ r,s) +)k m,:k = µr,s) m, :k + s +)k µr,s ) m,:k. 3.4) Proof: From equatio.4) for m 2 ad r,s=,,2,..., µ r,s) m,:k = k r [ l F))] m f) I)d, 3.5) m )! m )! [ F)] where I)= y s [ l Fy))+l F))] m [ F)] k fy)dy. Itegratig I) by parts treatig [ F)] k fy) for itegratio ad the rest of the itegrad for differetiatio, ad substitutig the resultig epressio i 3.5), we get µ r,s) m,:k = µr,s) m, :k + sk km )! m )! r y s [ l F))] m [ l Fy))+l F))] m [ F)] k f) [ F)] fy)dyd, the costat of itegratio vaishes sice the itegral i I) is a defiite itegral. O usig the relatio 2.), we obtai µ r,s) m,:k = µr,s) m, :k + sk { r y s [ l F))] m k+)m )! m )! [ l Fy))+l F))] m [ F)] k f) [ F)] fy)dyd+ r y s [ l F))] m [ l Fy))+l F))] m [ F)] k f) } [ F)] fy)dyd ad hece the result give i 3.4). Remark 3.2 Settig k = i 3.4) we deduce the recurrece relatio for product momets of upper record values from Oe ca also ote that Theorem 2. ca be deduced from Theorem 3. by puttig s =. c 23 NSP

5 J. Stat. Appl. Pro. 2, No. 2, ) / Characterizatio Let {X, } be a sequece of i.i.d cotiuous radom variables with d f F) ad pd f f). Let X U) be the th upper record values, the the coditioal pd f of X U) give X Um) =, m< i view of.3) ad.4), is fx U) X Um) = )= m )! [ l Fy)+l F)] m fy), >y. 4.) F) Theorem 4.: Let X be a absolutely cotiuous radom variable with d f F) ad pd f f) o the support, ), the for m<, E[X U) X Um) = ]= [)+) m ] 4.2) if ad oly if Proof: From 4.), we have By settig t = l F) Fy) ) +/), F) = >,, >. [ F) E[X U) X Um) = ]= y l m )! Fy) ) +)/ = l y+ ) from.2) i 4.3), we obtai E[X U) X Um) = ]= )] m fy) dy 4.3) F) [)e t/+) ]t m e t dt m )! = ) e [ /+)]t t m dt e t t m dt m )! m )! Simplifyig the above epressio, we derive the relatio give i 4.2). To prove sufficiet part, we have from 4.) ad 4.2) where y[ l Fy)+l F)] m fy)dy= F)H r ), 4.4) m )! H r )= [)+) m ]. Differetiatig 4.4) both sides with respect to, we get which proves that m 2)! m 2 f) y[ l Fy)+l F)] F) fy)dy= f)h r)+ F)H r) f) F) = H r) [H r+ ) H r )] = +) ) ) +/), F) = >,, >. Theorem 4.2: Let k is a fi positive iteger, r be a o- egative iteger ad X be a absolutely cotiuous radom variable with pd f f) ad cd f F) o the support, ), the if ad oly if r ) +)k :k = µr) :k + r +)k µr ) :k. 4.5) ) +/), F) = >,, >. c 23 NSP

6 2 D. Kumar, A. Kulshrestha: Epectatio Idetities of Upper Record... Proof: The ecessary part follows immediately from equatio 2.4). O the other had if the recurrece relatio i equatio 4.5) is satisfied, the o usig equatio.3), we have k r [ l F))] [ F)] k f)d )! = k r [ l F))] 2 [ F)] k f)d 2)! rk + +) )! rk + +) )! r [ l F))] [ F)] k f)d r [ l F))] [ F)] k f)d. 4.6) Itegratig the first itegral o the right had side of equatio 4.6), by parts ad simplifyig the resultig epressio, we fid that rk r [ l F))] [ F)] k k )! { F) + + ) } f) d= 4.7) + Now applyig a geeralizatio of the Mütz-Szász Theorem Hwag ad Li, [] to equatio 4.7), we get f) F) = + which proves that ) +/), F) = >,, >. 5 Applicatios The results established i this paper ca be used to determie the mea, variace ad coefficiets of skewess ad kurtosis. The momets ca also be used for fidig best liear ubiased estimator BLUE) for parameter ad coditioal momets. Some of the results are the used to characterize the distributio. 6 Coclusio I this study some eact epressios ad recurrece relatios for sigle ad product momets of record values from the geeralized Pareto distributio have bee established. Further, coditioal epectatio ad recurrece relatio of sigle momets of record values has bee utilized to obtai a characterizatio of the geeralized Pareto. Ackowledgemet The authors are grateful to the referee for givig valuable commets that led to a improvemet i the presetatio of the study. c 23 NSP

7 J. Stat. Appl. Pro. 2, No. 2, ) / 2 Refereces [] Ahsaullah, M., 995) Record Statistics. Nova Sciece Publishers, New York. [2] Arold, B.C., Balakrisha, N. ad Nagaraja, H.N.,992) A First course i Order.Joh Wiley ad Sos, New York. [3] Arold, B.C., Balakrisha, N. ad Nagaraja, H.N., 998) Record,Joh Wiley ad Sos, New York. [4] Balakrisha, N. ad Ahsaullah, M., Relatios for sigle ad product momets of record values from epoetial distributio, Ope joural of Statistics. J. Appl. Statist. Sci., 2 993), [5] Balakrisha, N. ad Ahsaullah, M., Recurrece relatios for sigle ad product momets of record values from geeralized Pareto distributio. Comm. Statist. Theory Methods, 23994), [6] Chadler, K.N., The distributio ad frequecy of record values. J. Roy. Statist. Soc., Ser B, 4952), [7] Dziubdziela, W. ad Kopociski, Limitig properties of the k th record value. Appl. Math., 5976), [8] Feller, W., A itroductio to probability theory ad its applicatios. 2966), Joh Wiley ad Sos, New York [9] Hall, W. ad Weller, J.A., Mea residual life. I: Csorgo, M., Dawso, D. A., Rao, J.N.K., Saleh, A.K. Md. E., eds Statistics ad Related Topics, 98) [] Hwag, J.S. ad Li, G.D. O a geeralized momets problem II Proc. Amer. Math. Soc., 9984), [] Johso, N.L., Kotz, S. ad Balakrisha, N.M., Cotiuous Uivariate Distributios. 994) Joh Wiley New York. [2] Kamps, U., A cocept of geeralized Order Statistics, J. Stat. Pla. Iferece, 48995), -23. [3] U. Kamps, Characterizatios of distributios by recurrece relatios ad idetities for momets of order statistics. I: Balakrisha, N. Ad C.Rao, C.R. eds., Had book of statistics, 6, Order Statistics, Theory ad Methods, Amsterdam: North Holad, 6 998), [4] Kha, R.U. Kulshrestha ad Kumar, D., Lower geeralized order statistics from geeralized order statistics distributio, J. Stat. Appl. Pro., 222), -3. [5] Kha, A.H., Haque, Z. ad Fazia, M., A Characterizatio of Distributios based o Coditioal Epectatios of Geeralized Order Statistics, J. Stat. Appl. Pro., 22) to appear). [6] Kumar, D., Relatios for momets of th lower record values from epoetiated log-logistic distributio ad a characterizatio, Iteratioal Joural of Mathematical Archive, 62), -7. [7] Kumar, D. ad Kha, M.I., Recurrece relatios for momets of th record values from geeralized beta distributio ad a characterizatio, Seluk Joural of Applied Mathematics. 322), [8] Nevzorov, V.B. Records., Theory probab. Appl., 32987), Eglish traslatio). [9] Pawlas, P. Ad Szyal, D., Recurrece relatios for sigle ad product momets of k th record values from Weibull distributio ad a characterizatio., J. Appl. Stats. Sci., 2), [2] Pawlas, P. ad Szyal, D., Relatios for sigle ad product momets of k th record values from epoetial ad Gumbel distributios, J. Appl. Statist. Sci., 7998), [2] Pawlas, P. ad Szyal, D., Recurrece relatios for sigle ad product momets of k th record values from Pareto, geeralized Pareto ad Burr distributios, Comm. Statist. Theory Methods, 28999), [22] Resick, S.I., Etreme values, regular variatio ad poit processes, 973) Spriger-Verlag, New York. [23] Sara, J. ad Sigh, S.K., Recurrece relatios for sigle ad product momets of k th record values from liear epoetial distributio ad a characterizatio, Asia J. Math. Stat., 28), [24] Sulta, K. S., Record values from the modified Weibull distributio ad applicatios, Iteratioal Mathematical Forum, 427), c 23 NSP

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION

STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy