Characterization and Estimation of Weibull-Rayleigh Distribution with Applications to Life Time Data

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1 App. Math. f. Sc.Lett. 5, No., 7-79 (7) 7 Apped Mathematcs & fomato Sceces Lettes A teatoa Joua Chaactezato ad Estmato of Webu-Rayegh Dstbuto wth Appcatos to Lfe Tme Data Afaq Ahmad,*, S. P Ahmad ad A. Ahmed Depatmet of Statstcs, Uvesty of Kashm, Saga, da. Depatmet of Statstcs ad Opeato Reseach, Agah Musm Uvesty, da. Receved: 5 Sep. 6, Revsed: Nov. 6, Accepted: 5 Nov. 6. Pubshed oe: May 7. Abstact: ths pape, a ew famy of dstbutos caed T- dstbuto s defed. Some of ts popetes ad speca cases ae dscussed. A membe of the famy, amey, the thee-paamete Webu-Rayegh dstbuto s defed ad studed. Some of ts popetes cudg dstbuto shapes, mt behavo, hazad fucto, momets, ad chaactestc fucto ae dscussed. The method of mamum kehood estmato, method of momets ad L-momet estmato s used fo estmatg the mode paametes ad the obseved Fshe s fomato mat s deved. The febty of the Webu-Rayegh dstbuto s assessed by appyg t to the ea data set ad compag t wth othe dstbutos. Keywods: Webu-Rayegh dstbuto, Hazad fucto, Momets, Ode statstcs, Mamum kehood estmato, R Softwae. toducto Statstca dstbutos ae mpotat fo paametc feeces ad appcatos to ft ea wod pheomea. Athough may dstbutos have bee deveoped, thee ae may methods fo geeatg statstca dstbutos the teatue. Some we-kow methods the eay days fo geeatg uvaate cotuous dstbutos cude methods based o dffeeta equatos deveoped by Peaso [], methods of tasato deveoped by Johso [], ad the methods based o quate fuctos deveoped by Tukey [3]. The teest deveopg ew methods fo geeatg ew o moe febe dstbutos cotues to be actve the mode decades. Lee et a. [4] dcated that the maoty of methods deveoped afte 98s ae the methods of combato fo the easo that these ew methods ae based o the dea of combg two estg dstbutos o by addg addtoa paametes to a estg dstbuto to geeate a ew famy of dstbutos. As a esut, may ew fames of dstbutos have bee deveoped ad studed by eseaches. Mudhoka ad Svastava [5] poposed the epoetated Webu dstbuto to aayze bathtub faue data. Gupta et a. [6] defed the epoetated epoeta dstbuto by takg F() to be the cumuatve dstbuto fucto (CDF) of a epoeta dstbuto. The epoetated Webu dstbuto Mudhoka ad Svastava [5] s a membe of the cass of epoetated dstbutos by * Coespodg autho e-ma:badeaafaq@gma.com takg F() to be the CDF of a Webu dstbuto. Eugee et a. [7] toduced a ew cass of dstbutos geeated fom the beta dstbuto. The cumuatve dstbuto fucto F(), the cass of beta-geeated dstbutos s defed as G B ( ) B(, ) F ( ) t ( t) whee s ay cotuous adom vaabe wth CDF F(). Eugee et a. [] deveoped ad studed the beta-oma dstbuto by takg F() to be the CDF of a oma dstbuto. Azaateh, Lee ad Famoye [8] poposed a method fo geeatg ew dstbutos, amey, the T-famy. Let (t) be the PDF of a o-egatve cotuous adom vaabe T defed o [, ), ad et F() deote the CDF of a adom vaabe. The the CDF fo the T- cass of dstbutos fo a adom vaabe s og( F( )) G( ) ( t) dt R og( F( )) () whee R(t) s the CDF of the adom vaabe T. The coespodg PDF of the epoetated T- dstbuto s gve by dt f( ) g( ) og( F( )) F( ) () 7 NSP Natua Sceces Pubshg Co.

2 7 A. Ahmad et a.: Chaactezato ad Estmato of Webu-Rayegh ths ew cass, the dstbuto of the adom vaabe T s the geeato. The ew famy of dstbutos geeated fom () s caed T- dstbuto. Azaateh, Famoye ad Lee [9] defed the Webu-Paeto dstbuto fom () by takg (t) to be the Webu dstbuto ad F() to be the Paeto dstbuto. Note that the uppe mt fo geeatg the T- dstbuto s og( F()). t s cea that oe ca defe a dffeet uppe mt fo geeatg dffeet types of T- dstbutos. Some cotuous dstbutos of the T- fames that have bee studed ae Webu-epoeta dstbuto (Azagha et a. []) ad Kaeema ad Bosh [], deveoped the Epoeta Paeto dstbuto. ths atce we peset a ew geeazato of the Rayegh dstbuto caed the Webu-Rayegh dstbuto. Ths Rayegh mode was fst toduced by Rayegh []. The Rayegh dstbuto has a wde age of appcatos cudg fe testg epemets, opeatos eseach eabty aayss, apped statstcs, agcutue ad cca studes. Ths dstbuto s a speca case of the two paamete Webu dstbuto wth the shape paamete equa to. Sddqu [3] dscussed the og ad popetes of the Rayegh dstbuto. Meovc et a [4] deveoped the tasmuted Rayegh dstbuto whe Ahmad et a [5] studed the tasmuted vese Rayegh dstbuto usg Quadatc tasmutato map ad dscussed some popetes of ths famy. Sevea authos have cotbuted to ths mode, amey, Howade ad Hossa [6] ad Abd Efattah et a. [7]. The pape s outed as foows. Secto, we defe the cumuatve, desty ad hazad fuctos of the Webu-Rayegh (WR) dstbuto. Secto 3, we toduced the statstca popetes cude, skewess ad kutoss, th momet ad momet geeatg fucto. The dstbuto of ode statstcs s epessed Secto 4. Fay, the Secto 5 gves the estmato of the mode paametes usg Least squaes ad weghted east squaes estmatos, method of momets ad the Mamum kehood estmato. T- Webu-Rayegh Dstbuto f the adom vaabe T foows the Webu dstbuto wth paamete ad the ts pobabty desty fucto (PDF) s gve as ad the coespodg pobabty desty fucto s gve by f( ) ( ) og( F( )) ep og( F( )) g F ( ) Thus the CDF of the Webu-Rayegh dstbuto (WRD) whe foows the Rayegh dstbuto equato (5) s gve by G ( ) ep ad the coespodg PDF of the Webu-Rayegh dstbuto s gve by g ( ) ep whe ad dstbuto wth paamete. (5) (6) (7), the WRD educes to the Rayegh Fgue (.) ad Fgue (.) epesets pdf s ad cdf s of T-Webu-Rayegh dstbuto whee t t ( t) e ; t,, ad ae shape ad scae paamete espectvey. Thus by usg the cumuatve dstbuto fucto (CDF) of Webu dstbuto the cdf of Webu- famy usg () s defed as G ( ) ep og( F ( )) (3) (4) Fgue. ad. ustates some of the possbe shapes of Webu-Rayegh dstbuto fo dffeet vaues of the 7 NSP Natua Sceces Pubshg Co.

3 App. Math. f. Sc. Lett. 5, No., 7-79 (7) / 73 paametes, ad. Fgue. shows that the desty fucto of Webu-Rayegh s umoda ad, fo fed, t becomes moe ad moe peaked as the vaue of s deceased. ad. Suvva ad hazad fuctos The WRD ca be a usefu chaactezato of the suvva tme of a gve system because of t aaytca stuctue. The suvva fucto s gve by S() = G(). Thus usg (6), S( ) ep Aothe chaactestcs of teest of a adom vaabe s the hazad fucto defed by g( ) h( ) S( ) Thus usg (6) ad (8), the hazad fucto s gve by h ( ) Fom the hazad fucto the foowg ca be obseved: By settg, the hazad fucto (9) educes to the hazad fucto of the Rayegh dstbuto. 3 Statstca Popetes ths secto, we peset the statstca popetes of WRD especay mea, vaace, coeffcet of vaato, momet, Skewess, Kutoss, Momet geeatg fucto ad Chaactestc fucto. 3. Momets Theoem 3.: f s a adom vaabe dstbuted as a WRD( ;,, ) th, the the o-ceta momet s gve by Poof: ( ) g( ;,, ) ep (8) (9) ep Substtutg t, we have ( ) t ep( t ) whch competes the poof. ( ) () Substtute =, equato () we get mea ad vaace fo Webu Rayegh dstbuto. Vaace = Mea = () By puttg, equato () we get mea of the Rayegh dstbuto. 3. Momet geeatg fucto ths sub secto we deved the momet geeatg fucto of Webu-Rayegh dstbuto. Theoem 3.: f has the WRD( ;,, ) the the M momet geeatg fucto () t has the foowg fom M t ( t) ( )! Poof: We beg wth the we kow defto of the momet geeatg fucto gve by t t M ( t) E e e g( ;,, ) ( t) t... g( ;,, )! 7 NSP Natua Sceces Pubshg Co.

4 74 A. Ahmad et a.: Chaactezato ad Estmato of Webu-Rayegh t! t g( ;,, )! t M ( t) ( )! 3.3 Chaactestc fucto ths sub secto we deved the chaactestc fucto of Webu-Rayegh dstbuto. Theoem 3.3: f has the WRD( ;,, ) the the Chaactestc fucto ( t) (t) ( t) ( )! has the foowg fom Poof: We beg wth the we kow defto of the chaactestc fucto gve by t ( t ) E e e t g ( ;,, ) ( t ) t... g( ;,, )! ( t) ( t)!! g( ;,, ) ( t) ( t) ()! 4 Ode Statstcs ths secto, we deve cosed fom epessos fo the pdfs of the k th ode statstc of the Webu-Rayegh dstbuto. statstcs, the k th ode statstc of a statstca sampe s equa to ts k th smaest vaue. Togethe wth ak statstcs, ode statstcs ae amog the most fudameta toos o-paametc statstcs ad feece. We kow that f (), (),..., ( ) deotes the ode statstcs of a adom sampe,,..., fom a G () ad pdf () cotuous popuato wth cdf the the pdf of th ode statstcs () s gve by g,! G ( k )( ) g ( ) G ( ) G ( ) ( k )!( k)! k k () Substtutg equato (6) ad equato (7) equato (), we get pdf of k th ode statstc gve as! g ( k) ( ) (,, ) (,, ) (,, ) ( k )!( k)! whee (,, ) ep k k (3) Note that at, (3) yeds the pdf of the k th ode statstc of Rayegh dstbuto. Theefoe, the pdf of the fst (smaest) ode statstc () s gve by g () ( ) (,, ) (,, ) ad the pdf of the agest ode statstc () s gve by g ( ) ( ) (,, ) (,, ) 5 Estmato of Paametes (4) (5) ths secto, we dscuss the vaous methods of estmato ke momet method, L momet estmato, ad Mamum kehood estmato fo Webu-Rayegh dstbuto ad vefyg the effceces. 5. Method of Momet Estmatos ths secto, we study the method of momet estmatos (MMEs) of the paametes of Webu-Rayegh dstbuto. f foows WRD (,, ), the Mea = Vaace = (6) (7) t s we kow that the pcpe of the momet s method s to equate the sampe momets wth the subsequet 7 NSP Natua Sceces Pubshg Co.

5 App. Math. f. Sc. Lett. 5, No., 7-79 (7) / 75 popuato momets. Fom (6) ad (7), we obta the coeffcet of vaato (C.V) as CV. (8) The C.V. s depedet of the paametes λ ad. Theefoe, equatg the sampe C.V. wth the popuato C.V., we obta Whee S S ( ) ad (9). We eed to sove (9) to obta the estmate of, say MME. Oce we estmate, we ca use (6) ad (8) to estmate ad say MME ad MME 5. L-Momet Estmato ths secto, we popose a method of estmatg the ukow paamete of Webu- Rayegh dstbuto by L- momets estmatos, whch ca be obtaed as the ea combato of ode statstcs. The L-momets estmatos wee ogay poposed by Hoskg [8], ad t s obseved that the L-momets estmatos ae moe obust tha the usua momet estmatos. t s obseved (see, Gupta ad Kudu [9] that the LMEs have ceta advatages ove the covetoa momet estmatos. The stadad method to compute the L-momet estmatos s same as the commo momet estmatos,.e. to equate the sampe L-momets wth the popuato L- momets. The fst two sampe L-momets ae ( ) ad ( ) ( ) ( ) ad the fst two popuato L-momets ae E( ) E( ) : ( : ) E( : ) E [G( ) ] g( ) Thus fo Webu-Rayegh dstbuto, we obta Aaogousy to the usua method of momets, the L- momet method aso cossts of equatg the fst few popuato L-momets (λ ) to the coespodg sampe L- momets ( ), thus obtag as may equatos as ae eeded to sove fo the ukow popuato paametes,.e. fo the p paametes. =,,..,p Theefoe, LMEs ca be obtaed by sovg the foowg thee equatos. Substtutg the vaues of the popuato L- momets λ, λ, ad λ 3 fo the sampe L-momets, we get () () 3 () 3 3 Sovg these equatos do ot yed epct souto fo the estmates of paametes, we used the L-skewess measue to estmate Fst, we obta the LME of, say LME, as the souto of the foowg oea equato Oce (3) LME s obtaed, the LME of λ ad say LME 7 NSP Natua Sceces Pubshg Co.

6 76 A. Ahmad et a.: Chaactezato ad Estmato of Webu-Rayegh ad LME, ca be obtaed fom () ad () 5.3 Mamum Lkehood Estmato,,..., be a adom sampe of sze fom Let Webu-Rayegh dstbuto. The the kehood fucto s gve by L (4) ep By takg ogathm of (4), we fd the og kehood fucto og L og og og og ( ) og Theefoe, the MLE of must satsfy the foowg oma equatos og L og L (5), ad whch mamzes (5) og ( ) og L ( ) 3 og The souto of the o-ea system of equatos obtaed by dffeetatg equato (5) wth espect to, ad gves the mamum kehood estmates of the mode paametes. The souto ca aso be obtaed decty by usg R softwae whe data sets ae avaabe. ode to compute the stadad eo ad asymptotc cofdece teva we use the usua age sampe appomato whch the mamum kehood estmato of a paameteca be teated as beg appomatey mutvaate oma. Hece as the asymptotc dstbuto of the mamum kehood estmatos (,, ) s gve by v v v ~ N, v v v v v v v v v whee v v v s the v v v appomate vaace covaace mat wth ts eemets obtaed fom E og L E og L E og L 6 Appcatos E og L E og L E og L To compae the febty of the Webu-Rayegh dstbuto ove the we-kow Rayegh ad sub modes, two ea data sets ae used ad aayss pefomed wth the hep of R softwae. Data set : The fst data set (=63) s o the stegths of.5 cm gass fbes. The data was ogay obtaed by wokes at the UK Natoa Physca Laboatoy ad t has bee used by Smth ad Nayo [] ad Bougugo et a. [] apped the Webu G famy to ft the data. The data s as foows:.55,.74,.77,.8,.84,.93,.4,.,.3,.4,.5,.7,.8,.9,.3,.36,.39,.4,.48,.48,.49,.49,.5,.5,.5,.5,.53,.54,.55,.55,.58,.59,.6,.6,.6,.6,.6,.6,.6,.63,.64,.66,.66,.66,.67,.68,.68,.69,.7,.7,.73,.76,.76,.77,.78,.8,.8,.84,.84,.89,.,.,.4. The summay of the data s gve Tabe. The MLEs of WRD paametes ad the goodess of ft statstcs ae epoted Tabe. Tabe, dspays the Mamum Lkehood estmates of the mode paametes. t was obvous that T- Webu- Rayegh povdes a bette ft as compaed to othe Rayegh modes sce t has owest vaue of -ogl, Akake fomato Cteo (AC), Bayesa fomato Cteo (BC). Hece, the Webu Rayegh dstbuto pefomed bette tha othe geeazatos of Rayegh dstbuto. The dstbuto of the data s skewed to the eft wth skewess Ths shows that T- Webu- Rayegh has the abty to ft the eft skewed data. 7 NSP Natua Sceces Pubshg Co.

7 App. Math. f. Sc. Lett. 5, No., 7-79 (7) / 77 Tabe : Data summay fo stegth of.5 cm gass fbes Statstcs Vaues Statstcs Vaues 63 Meda.59 Mmum.55 Mamum.4 Fst Quate.375 Thd Quate.685 Mea.57 Vaace.5 Skewess Kutoss 3.93 Tabe :Estmates ad Pefomace of the dstbutos Dstbuto Paametes (S.E) -ogl AC BC =.893 (.88) Webu Rayegh dstbuto Rayegh dstbuto Epoetated Rayegh dstbuto Tasmuted Rayegh dstbuto =.396 (.7) =.837 (6.3) =.894 (.686) =.4564 (55.47) =.395 (5.8) = (.848) =.673 (76.7) =.334 ( ) =-. (.383) Data set : The data set s o the beakg stess of cabo fbes of 5 mm egth (GPa). The data has bee pevousy used by Codeo ad Lemote [] ad A-Aqtash et a. [3]. The data s as foows:.39,.85,.8,.5,.47,.57,.6,.6,.69,.8,.84,.87,.89,.3,.3,.5,.,.35,.4,.43,.48,.5,.53,.55,.55,.56,.59,.67,.73,.74,.79,.8,.8,.85,.87,.88,.93,.95,.96,.97, 3.9, 3., 3., 3.5, 3.5, 3.9, 3., 3., 3.7, 3.8, 3.3, 3.3, 3.33, 3.39, 3.39, 3.56, 3.6, 3.65, 3.68, 3.7, 3.75, 4., 4.38, 4.4, 4.7, 4.9. The summay of the data s gve Tabe 3. The MLEs of WRD paametes ad the goodess of ft statstcs ae epoted Tabe 4. Tabe 4, dspays the Mamum Lkehood estmates of the mode paametes. t was obvous that T- Webu- Rayegh povdes a bette ft as compaed to othe Rayegh modes sce t has owest vaue of -ogl, Akake fomato Cteo (AC), Bayesa fomato Cteo (BC). Hece, the Webu Rayegh dstbuto pefomed bette tha othe geeazatos of Rayegh dstbuto. The dstbuto of the data s skewed to the eft wth skewess -.8. Ths shows that T- Webu- Rayegh has the abty to ft the eft skewed data. Tabe 3:Data summay Statstcs Vaues Statstcs Vaues N 66 Meda.835 Mmum.39 Mamum 4.9 Fst.78 Thd 3.78 Quate Quate Mea.76 Vaace.794 Skewess -.8 Kutoss 3. Tabe 4:Estmates ad Pefomace of the dstbutos Paametes Dstbuto (S.E) -ogl AC BC Webu Rayegh dstbuto Rayegh dstbuto Epoetated Rayegh dstbuto Tasmuted Rayegh dstbuto 7 Cocuso =.75 (.654) =.534 (3.597) =.934 (8.54) =.49 (.6) =.98 (5.5533) =.45 (9.8959) =.3483 (.43) =.6653 (7.3984) =.844 (5.687) = (.99) ths atce we popose a ew mode of T- famy, caed the Webu-Rayegh dstbuto whch eteds the Rayegh dstbuto the aayss of data wth ea suppot. A obvous easo fo geeazg a stadad dstbuto s because the geeazed fom povdes age febty modeg ea data. We deve epasos fo the momets, momet geeatg fucto, chaactestc fucto ad Ode statstcs. The estmato of paametes s appoached by the method of L-momet estmato, method of momets ad mamum kehood estmato. A appcato of the Webu-Rayegh dstbuto to ea data s povded whch show that the ew dstbuto ca be used qute effectvey to povde bette fts tha the Rayegh dstbuto, Epoetated Rayegh dstbuto ad Tasmuted Rayegh dstbuto. 7 NSP Natua Sceces Pubshg Co.

8 78 A. Ahmad et a.: Chaactezato ad Estmato of Webu-Rayegh Ackowedgemet The Authos woud ke to offe scee thaks to the efeees ad the edto fo the hepfu commets ad suggestos, whch mpoved the eae daft of the mauscpt. Refeeces [] K Peaso, Cotbutos to the mathematca theoy of evouto.. Skew Vaato Homogeeous matea Phos Tas R Soc Lod A vo. 86, pp , (895). [] N.L Johso, Systems of fequecy cuves geeated by methods of tasato, Bometka, vo. 36, pp , (949). [3] J.W Tukey, The Pactca Reatoshp Betwee the Commo Tasfomatos of Pecetages of Couts ad Amouts, Techca Repot 36, Pceto Uvesty, Pceto, NJ, Statstca Techques Reseach Goup, (96). [4] C Lee, F Famoye, ad A Azaateh, Methods fo geeatg fames of uvaate cotuous dstbutos the ecet decades, WREs Computatoa Statstcs, vo. 5(3), pp.9-38, (3). [5] G Mudhoka ad D Svastava, Epoetated Webu famy fo aayzg bathtub faue data, EEE Tasactos o Reabty, vo. 4, pp. 99-3, (993). [6] R. C Gupta, P Gupta ad R.D Gupta, Modeg faue tme data by Lehma ateatves, Commucatos Statstcs-Theoy ad Methods, vo.7, pp , (998). [7] N Eugee, C Lee ad F Famoye, Beta-oma dstbuto ad ts appcatos, Commucatos Statstcs-Theoy ad Methods, vo.3 (6), pp.497-5, (). [8] A Azaateh, C Lee ad F Famoye, A ew method fo geeatg fames of cotuous dstbutos, Meto,vo. 7(), pp.63-79, (3a) [9] A Azaateh, F Famoye ad C Lee, Webu-Paeto dstbuto ad ts appcatos, Commucatos Statstcs-Theoy ad Methods, vo. 4(9), pp , (3b) [] A Azagha, A Famoye ad C Lee, Epoetated T- Famy of Dstbutos wth Some Appcatos, teatoa Joua of Statstcs ad Pobabty, vo. (3), pp. 3-49, (3). [] A. K Kaeema ad M.A Bosh, Epoeta Paeto Dstbuto, Mathematca Theoy ad Modeg, vo. 3(5), pp.35-46, (3). [] J Rayegh, O the esutat of a age umbe of vbatos of the same ptch ad of abtay phase, Phos. Mag., vo., pp.73 78, (98). [3] M. M Sddqu, Some pobems coected wth Rayegh dstbutos, J. Res. Nat. Bu. Stad, 6D, pp.67 74, (96). [4] F Meovc, Tasmuted Rayegh dstbuto, Austa Joua of Statstcs, vo. 4(), pp.-3, (3). [5] A Ahmad, S.P Ahmad ad A Ahmed, Tasmuted vese Rayegh dstbuto, Mathematca Theoy ad Modeg, vo.4(7), pp. 9-98, (4). [6] H. A Howade ad A Hossa, O Bayesa estmato ad pedcto fom Rayegh dstbuto based o type- cesoed data, Commucatos Statstcs-Theoy ad Methods, vo. 4(9), pp.49 59, (995). [7] A. M Abd Efattah, A.S Hassa ad D.M Zeda, Effcecy of Mamum Lkehood Estmatos ude Dffeet Cesoed Sampg Schemes fo Rayegh Dstbuto, testat, (6). [8] J. R. M Hoskg, L-Momemts: Aayss ad estmato of dstbutos usg ea combato of dstbutos usg ea combatos of ode statstcs, Joua of the Roya Statstca Socety B, vo.5 (), pp.5-4, (99). [9] R.D Gupta ad D. Kudu, Geeazed epoeta dstbuto: Dffeet methods of estmato Joua of Statstca Computato ad Smuato, vo. 69(4), pp , (). [] R. L Smth, ad J.C Nayo, A compaso of mamum kehood ad bayesa estmatos fo the thee-paamete Webu dstbuto, Apped Statstcs, vo.36, pp , (987). [] M Bougugo, R.B Sva ad G.M Codeo, The Webu-G famy of pobabty dstbutos J. Data Sc., vo., pp , (4). [] G.M Codeo ad A.J. Lemote, The β-bbaum- Saudes dstbuto: A mpoved dstbuto fo fatgue fe modeg, Computatoa Statstcs ad Data Aayss, vo.55, pp , (). [3] R A-Aqtash, C. Lee ad F Famoye, Gumbe-webu dstbuto: Popetes ad appcatos, Joua of Mode Apped Statstca Methods, vo.3, pp.-5, (4). A. Ahmad s a Ph.D. eseach schoa of Depatmet of Statstcs, at the Uvesty of Kashm at Saga Kashm, da. He eceved hs M.SC degee Statstcs fom Uvesty of Kashm, da. Hs ma eseach teests ae the aeas of Bayesa Statstcs, Optmzato, ad Pobabty dstbuto theoy. He has pubshed moe tha 5 pubcatos. 7 NSP Natua Sceces Pubshg Co.

9 App. Math. f. Sc. Lett. 5, No., 7-79 (7) / 79 A. Ahmed s a Pofesso of Depatmet of Statstcs ad Opeatos Reseach at Agah Musm Uvesty, Agah, da. He eceved hs Ph.D degee Statstcs fom the..t Rookee. Pof. Ahmad has ogazed two teatoa cofeeces coucto wth da Socety of Pobabty ad Statstcs (8) ad Opeatoa Reseach Socety of da (3). Hs ma eseach teests ae the aea of Optmzato ad Bayesa Statstcs. He has pubshed ad coauthoed moe tha 8 pubcatos. S.P Ahmad s a S. Assstat Pofesso of Depatmet of Statstcs, at the Uvesty of Kashm at Saga Kashm, da. He eceved hs Ph.D degee Statstcs fom Uvesty of Kashm, da. Hs ma eseach teests ae the aeas of Bayesa Statstcs ad Pobabty dstbuto theoy. He has pubshed moe tha 3 pubcatos. 7 NSP Natua Sceces Pubshg Co.