Estimation and Prediction from Inverse Rayleigh. Distribution Based on Lower Record Values

Transcription

1 Applied Matheatical Science, Vol. 4,, no. 6, Etiation and Pediction fo Invee Rayleigh Ditibution Baed on Lowe Recod Value A. Solian, Ea A. Ain,a and Alaa A. Abd-El Aziz a e-ail: e_ain@yahoo.co Abtact Thi aticle dicue Bayeian and non-bayeian etiation poble of the unknown paaete fo the invee Rayleigh ditibution baed on lowe ecod value. Maxiu likelihood etiato of the unknown paaete wee obtained. Alo, Baye etiato have been developed unde quaed eo and zeo one lo function. Thee etiato ae deived uing the infoative pio ditibution. Bayeian and non Bayeian inteval etiation fo the invee Rayleigh paaete ae obtained. Futheoe, Bayeian pediction inteval of the futue ecod value ae dicued and obtained. Finally pactical exaple uing iulated ecod value ae given to illutate the theoetical eult of pediction inteval. Keywod: Bayeian etiation, Lowe ecod value, Maxiu likelihood pocedue, Invee Rayleigh ditibution, Bayeian Pediction. Intoduction The invee Rayleigh ditibution ha any application in the aea of eliability tudie. Voda (97) entioned that the ditibution of lifetie of eveal type of expeiental unit can be appoxiated by the invee Rayleigh ditibution. The pobability denity function of the invee Rayleigh ditibution with cale paaete θ i θ θ f ( x) = ( )exp( ) x, θ > (.) 3 x x The coeponding cuulative ditibution function i, I.S.S.R. Caio univity

2 358 A. Solian, E. A. Ain and A. A. Abd-El Aziz θ F ( x) = exp( ) x, θ > (.) x Voda (97) peented oe popetie of the axiu likelihood etiato, fo invee Rayleigh ditibution, futheoe confidence inteval and tet of hypothee ae developed. Ghaaph (993) deived five eaue of location fo the invee Rayleigh ditibution. Thee eaue ae the ean, haonic ean, geoetic ean, ode, and the edian. He alo, etiated the unknown paaete uing diffeent ethod of etiation. A copaion of thee etiato wa dicued nueically in te of thei bia and oot ean quae eo. Abdel-Mone (3) developed oe etiation and pediction eult fo the invee Rayleigh ditibution. El-Helbawy and Abdel-Mone (5) obtained Bayeian etiato of the paaete of the invee Rayleigh ditibution unde fou lo function. Bayeian of one and two aple pediction ae alo developed including point pediction and pediction inteval. Recod value and aociated tatitic ae of geat ipotance in eveal eal live poble involving weathe, econoic, and uppot data. The tatitical tudy of ecod value tated with Chandle (95) and now pead in diffeent diection. Inteeted eade ay efe to Fote and Stuat (954), Galabo (978), Dunoe(983), Nagaaja (988), Ahanullah (994) and Aonld et al (99) Ahed and Fahad (7) fo a eview of developent in thi aea of eeach. While a lot of wok ha been done on chaacteization, ayptotic theoy and genealization, not uch ha been done on tatitical infeence baed on ecod value. The objective of thi pape i to how how ecod value can be ued to develop a ethodology to contuct and copute Bayeian and non Bayeian etiation and pediction. The lowe ecod value fo invee Rayleigh population baed on a et of lowe ecod value will be conideed. The Baye and axiu likelihood etiato fo the cale paaete wee deived. Baye point and inteval etiato ae deived auing infoative pio on the paaete. Thi can achieved with epect to quaed eo and zeo one eo lo function. Bayeian pediction bound fo the futue ecod value on the bai of fit obeved ecod ae obtained. Siulated ecod value ae ued to illutate the application of the eult though Mathcad ().. Recod Value and Maxiu Likelihood Etiation Let X, X,... be an infinite equence of independent and identically ditibuted ando vaiable having pobability denity function (.). Conide R, R,..., R epeent the fit ( +) lowe ecod value fo the ae denity function the joint pobability denity function of R, R,..., R i;

3 Etiation and pediction fo invee Rayleigh ditibution 359 (θ ) + θ f ( R, R,... R ) = exp( ), < < < L < <, (.) 3 i i = which i the likelihood function baed on the fit ( +) lowe ecod value. Theefoe the pobability denity function of the fit ( +) lowe ecod value fo invee Rayleigh ditibution will be θ ( ) ( ) + θ f = exp( ) >, θ > (.) R! In addition, the joint pobability denity R and R j baed on the invee Rayleigh ditibution i i θ j + f 4θ j i j (, ) =. [ ] e < < < R R i j i j i i + (.3), j i i j!( )! 3 3 i j j i Alo the -th oent about oigin fo ( +) Rayleigh ditibution i given a follow lowe ecod value fo invee k ( k θ k + E ) = Γ, k < + (.4)! The ean and vaiance of the ( +) lowe ecod value can eaily obtained uing equation (.4). Taking the logaith of the likelihood function (.), θ ln( L( θ ; )) = ( + ) ln(θ ) ln( 3 ). (.5) i i = Diffeentiate both ide of equation (.8) with epect to the paaeteθ and equating with zeo, then the axiu likelihood etiate of θ unde lowe ecod value, ayθˆ, i given by ˆ θ = ( + ). (.6) In addition, the expected value fo the etiated paaeteθˆ and it vaiance ae given a follow, + E( ˆ) θ = ( ) θ, (.7) and, + va( ˆ) θ = ( )( θ ). (.8)

4 36 A. Solian, E. A. Ain and A. A. Abd-El Aziz It i clea fo equation (.7) that the axiu likelihood etiate θˆ i biaed etiate fo the paaete θ. Now conide the following pivotal quantity y = ( + θ ). ˆ θ It i eay to pove that y ha a gaa ditibution with paaete ( +,). Uing the fact that y ha a chi-quae ditibution with ( + ) degee of feedo χ ( +), then ( α )% confidence inteval fo the paaeteθ baed on the fit + lowe ecod value i ( L, U ) whee ˆ θ ˆ θ L = χ and U = χ (.9) ( + ) α ( + ), ( + ) α ( + ), whee, θˆ i the axiu likelihood given by equation (.6) 3. Bayeian Etiation Thi ection i concened with the poble of obtaining Bayeian etiato fo the cale paaete fo the invee Rayleigh ditibution. The pio knowledge which i adequately epeented by the natual conjugate pio ditibution unde two lo function will be developed. Let R, R,..., R be the available ( +) lowe ecod value fo the invee Rayleigh ditibution. Conide the following infoative pio ditibution fo the cale paaete θ π ( θ ) = ae aθ, a >, θ > (3.) Since, the poteio pobability denity function, π ( θ ) i obtained by cobining the likelihood given in equation (.4) and the pio pobability denity π ( ) θ, then a(θ ) + π ( θ ) = k exp[ ( + a) θ ], θ > (3.) 3 i i = whee, the noalizing contant k i given by k = / π ( θ ) dθ. Put, A = ( + a). Theefoe,

5 Etiation and pediction fo invee Rayleigh ditibution 36 A( Aθ ) + π ( θ ) = exp( Aθ ), θ >. (3.3) Γ( + ) whee, Γ(.) i the gaa function. It follow that the cale paaete θ ha gaa ditibution with paaete, ( +, A). The quaed eo lo i appopiate when deciion becoe gadually oe daaging fo lage eo. The Bayeian etiato of θ unde quaed eo lo function i the poteio ean and i given by ~ ( + ) θ = E( θ ) =. (3.4) A The Baye etiato of θ with epect to zeo one lo function i the poteio ode which i given by ~ d (ln Π ( θ )) ( ) θ = = +. (3.5) dθ A The highet poteio denity inteval i uch that the poteio denity fo evey point inide the inteval i geate than that fo evey point outide of it o that the inteval include the oe pobable value of the paaete and exclude the le pobable one. Fo the poteio denity fo the paaete θ obtained in equation (3.3) and uing the fact that Aθ ha χ ( + ), then a ( α )% Bayeian inteval etiation fo the paaeteθ baed on the fit + lowe ecod value i L, U ), whee χ ( + 4, α ) L = and θ A ( θ θ χ ( + 4, α ) U = (3.6) θ A 4-Bayeian Pediction In the context of pediction of the futue ecod, the pediction inteval povide bound to contain the eult of a futue ecod, baed upon the eult of the peviou ecod obeved fo the ae ditibution. Pediction poble co up natually in eveal eal life ituation, fo exaple, Ahanullah (98), Nagaja (984) and Doganakooy and Balakihnan (997). Uing genealized odel, Bayian pediction inteval fo inteval fo futue genealized ode tatitic (including ecod value a apecial cae) wa tudied by Al-Huaini and Ahed (3). Madi and Raqab (4) conideed the poble of Bayeian pediction of tepeatue ecod uing Paeto odel. Auing that R, R,..., R ae the lowe ecod value fo the invee Rayleigh ditibution, Balakihnan and Chan (994) obtained the bet linea unbiaed pe- + futue ecod baed on the fit ecod. dicto (BLUP) of the ( )

6 36 A. Solian, E. A. Ain and A. A. Abd-El Aziz Thi ection i devoted to deiving Baye pedictive denity function, which i neceay to obtain bound fo the pedictive inteval of futue ecod. Let R, R,..., R be the fit ( +) obeved lowe ecod value fo invee Rayleigh ditibution. Baed on uch a ecod aple, Bayeian pediction i nedded fo the th futue ecod R, < <. The conditional pdf of R given R i given by Ahanulla (995) in the fo f R R [ln F( ) ln F( )] f ( ) ( ) = < < < (4.)! F( ) Fo the invee Rayleigh ditibution with pobability denity function (.) and cuulative denity function (.), the function f ( ) becoe R R θ f = θ ξ ( ) ( ) [ ξ ( )] e, < < < (4.) R R! 3 whee ξ ( ) =. Uing the fact that the ecod value fo the Makove popety, the conditional denity function of R given R = ( R, R, K, R ) i jut the pobability denity function of R given R. The pedective denity function of R given R i f ( ) = f π θ dθ ( ). ( ). (4.3) R R Fo (3.), (4.) and (4.3), the pedictive denity function can iplified a A + ( ) f ( ) [ ( )] + = ξ [ A + ξ ( )] < (, ) 3 β +, (4.4) whee β (.,.) i the beta function. Pediction bound on equation R i given fo the following

7 Etiation and pediction fo invee Rayleigh ditibution 363 P( (, ; ) ) InBet + η > λ = f ( ) d = (4.5) λ β ( n, ) λ Whee η =, and InBet( z, z; η) i the incoplete beta function defined by λ A η z ( z + z ) InBet ( z, z ; η) = t ( t) dt. The pedictive bound of a two ided inteval with cove ( α), fo the futue lowe ecod value R ay be obtained by nueical olving fo the following two equation, fo the lowe bound L and the uppe bound U : P ( > L ) = α / ; P ( > U ) = α /. (4.6 ) Special cae, when = + then the lowe and uppe bound fo the futue lowe ecod value R with cove ( α) will be + /( + ) / L = ( α / ) A a and U ( α / ) /( ) / = + A a 5 Nueical Illutation In thi ection nueical eult of the Bayeian pedictive inteval fo eveal value of diffeent pio paaete will be obtained. In thi cae, the paaete θ ha the gaa pio given by (3.) with known paaete a. The calculation ae caied out accoding to the following tep: () Fo given value of the invee Rayleigh paaeteθ geneate a ando vaiable X fo the invee Rayleigh ditibution (.) and elected the fit ecod. () Conide the fit 6 ecod a the obeved uppe ecod (=5), while the lat ix ecod a the unobeved ecod, which ae to be pedicted.

8 364 A. Solian, E. A. Ain and A. A. Abd-El Aziz (3) Uing Mathcad () poga applying equation (4.6) to obtain the 95% equal tail Bayeian pediction inteval fo the th uppe ecod value fo =5 and =6 and fo eveal diffeent value of the pio paaete ( a=.5,,,3,5,7) and Table and contained the eult which how (i) the value of the pio paaete a (ii) the 95% Bayeian pediction inteval fo the th ecod, (iii) the length of the pediction inteval. Step and 3 ae epeated fo the value =4, 3, and the eult ae peented in table 3 to 8. Table () =5 Exact pio L U length X6= Bayian pediction inteval fo the =6 futue ecod Table () =4 Exact pio L U length X5= Bayian pediction inteval fo the =5 futue ecod Table (3) =3 Exact pio L U length X4= Bayian pediction inteval fo the =4 futue ecod

9 Etiation and pediction fo invee Rayleigh ditibution 365 Table (4) = Exact pio L U length X3= Bayian pediction inteval fo the =3 futue ecod Refeence Abdel-Mone, A. A. (3). Etiation and Pediction fo the Invee Rayliegh life ditibution. M.Sc. Thei. Faculty of Education, Ain Shae Univeity. Ahed, A.S and Fahad, M. A (7). Bayeian infeence uing ecod value fo Rayling odel with application. Euopean Jounal of Opeation Reeach, doi:.6 / j.ejo. Ahanullah, M. (98). Linea Pediction of Recod Value fo the two Paaete Exponential ditibution. Annal of the Intitute of Statitical Matheatic 3, Ahanullah, M. (994). Recod Statitic. Nova Science Publihe, Inc. Coack. New Yok. Al-Huaini, E. K., Ahed, A. A. (3). On Bayeian Inteval Pediction of Futue Recod. Tet, Anold, C. B., Balakihnan, N. and Nagaaja, H. N. (99). Recod. John Wiley & Son, Inc. Balakihnan, N., Chan, P. S. (994). Recod Value fo Rayleigh and Weibull Ditibution and Aociated Infeence. In: Poceeding of the Confeence on Extee Value Theoy and Application, vol. 3, NIST Special Publication 866, pp Chandle, K. N. (95). The ditibution and fequency of ecod value, Jounal of Royal tattic ociety, eie B, 4, -8.

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