Department of Statistics, Sardar Patel University, Vallabh Vidyanagar, Gujarat, India

Transcription

1 Mnu Vaance Unbased Esaon n he Raylegh Dsbuon unde Pogessve Type II Censoed Daa wh Bnoal Reovals. Ashok Shanubhogue and N.R.Jan* Depaen of Sascs, Sada Pael Unvesy, Vallabh Vdyanaga, Gujaa, Inda e-al : a_shanubhogue@yahoo.co, Coespondng auho*: jan_nal@sfy.co. Absac Ths pape concens wh he poble of unfoly nu vaance unbased esaon of he scale paaee of Raylegh dsbuon based on pogessve Type II censoed daa wh bnoal eovals. We oban he unfoly nu vaance unbased esao (UMVUE) fo powes of he scale paaee and s funcons. The UMVUE of he vaance of hese esaos ae also gven. The UMVUE of he () ode () h oen () ean (v) vaance (v) hazad funcon (v) edan (v) p h quanle (v) p.d.f. (x) elably funcon and (x) c.d.f. of he Raylegh dsbuon ae deved. The UMVUE of p.d.f. s ulzed o oban he UMVUE of P( X < Y). An llusave nuecal exaple s pesened. Keywods: pogessve Type II censoed saple, Raylegh dsbuon, bnoal dsbuon, coplee suffcen sasc, UMVUE. Maheacs Subjec Classfcaon: 6N0, 6N0. Inoducon A Type II censoed saple s one fo whch only salles obsevaons n a saple of n es ae obseved. A genealzaon of Type II censong s a pogessve Type II censong. Unde hs schee, n uns of he sae knd ae placed on es a e zeo, and falues ae obseved. When he fs falue s obseved, a nube of suvvng uns ae andoly whdawn fo he es; a he second falue e, suvvng uns ae seleced a ando and aken ou of he expeen, and so on. A he e of h falue, he eanng n uns ae eoved. Balakshnan e.al [5] ndcaed ha such schee can ase n clncal als whee he

2 dop ou of paens ay be caused by gaon o by lack of nees. In such suaons, he pogessve censong schee wh ando eovals s equed. Fo a dealed dscusson of pogessve censong we efe o Balakshnan and Aggawala [4] and Balakshnan []. If 0, hen, hs schee educes o he Type II censong schee. Also noe ha f 0,so ha n,hs schee educes o he case of no censong ha s he case of a coplee saple. In hs pape, we use pogessve Type II censong schee wh bnoal eovals whee he nube of uns eoved a each falue e follows a bnoal dsbuon. The Raylegh dsbuon was fs deved by Lod Raylegh n connecon wh a sudy of acouscal pobles. Snce hen any nvesgaos have used he Raylegh dsbuon o soe elaed fos of n a vaey of engneeng, wave popagaon, adaon and analyss of age daa sudes. The Raylegh dsbuon s also used o odel wave heghs n oceanogaphy, and n councaon heoy o descbe houly edan and nsananeous peak powe of eceved ado sgnals. Seveal such suaons have been dscussed by Polovko [0], Longue-Hggns [9], Zh Ren e.al [], Uvason and Godzenskaya [6], Ynhu Deng e.al [8], Ey Hu [0], Takesh Yaane [7], and any ohes. The Raylegh dsbuon s a specal case of wo paaee Webull dsbuon. The hazad ae of hs dsbuon s lnealy nceasng funcon of e. Fo a evew of leaue on esang paaees of he Raylegh dsbuon one ay efe o Lee e.al [8], Dye [9], Lalha and Msha [5], K and Han [4], Balakshnan N. [3], Solan [4], Raqab and Mad [], and any ohes. Infeence fo Raylegh dsbuon based on pogessve Type II censoed daa wee dscussed by any auhos. Al Mousa e.al [] obaned he axu lkelhood and Bayes esaes fo one and wo paaees and he elably funcon of Raylegh dsbuon unde pogessve Type II censoed saples. K e.al [3] have obaned he axu lkelhood esao, Bayes esao and cedble nevals fo he scale paaee and elably funcon of he Raylegh dsbuon based on geneal pogessve Type II censoed daa. Lee [6] have obaned he UMVUE of lfee pefoance ndex based on he ype II ulply censoed saple and develop he hypohess esng pocedue. Lee e.al [7] have obaned MLE of

3 lfee pefoance ndex and developed Baysan es fo based on pogessve Type II gh censoed saple. In hs pape ou objecve s o sudy he UMVU esaon of he scale paaee and s vaous funcons of Raylegh dsbuon based on pogessve Type II censoed daa wh bnoal eovals. Ths pape s oganzed as follows. In Secon, he lkelhood funcon s gven. In Secon 3, he UMVUE of paaee of θ and s funcons ae deved. Also, he UMVUE of he () ode () h oen () ean and vaance (v) hazad funcon (v) edan (v) p h qunle (v) posve powe of elably funcon (v) p.d.f. (x) c.d.f. ae obaned. In Secon 4 usng he UMVUE of p.d.f. he UMVUE of P( X < Y) s deved. In Secon 5, an llusave nuecal exaple s gven. The odel Le he falue e dsbuon be Raylegh wh pobably densy funcon, x x exp 0 < x <, θ > 0 f( x) θ θ 0 ohewse whee θ s a scale paaee. The densy () s gven n []. The cuulave dsbuon funcon s gven by, x F( x) exp, 0< x< θ The suvval funcon s gven by, () () x S( x) exp, 0< x< θ (3) The densy gven n () can be wen as, [ hθ ] [ g( θ )] d( x) ax ( ) ( ) f( x) (4) whee ax ( ) x, hθ ( ) exp, θ d( x) x and g( θ ) θ (5) such ha ( ) ax>0 and [ ] ( ) g( θ) ax ( ) h( θ) d x dx. 0 3

4 ( X, R ),( X, R ),,( X, R ),denoe a pogessvely Type II censoed saple, Le whee X X: : n, fo,,. and X < X < < X. The condonal lkelhood funcon can be wen as, see Cohen[7], L( θ; x/ R ) c f( x) [ s( x)] (6) whee c n( n )( n ) ( n + ), and n 0 ( ) fo,,. Subsung () and (3) n (6) we ge, c L( θ; x/ R ) x exp ( ) x θ + (7) θ We assue ha X and R ae ndependen fo all.we fuhe suppose ha he nube of uns eoved a each falue e follows a bnoal dsbuon wh pobably p. Fo Tse e al. [8] he jon pobably ass funcon of,, s gven by, ( )( n ) ( ) ( n )! p ( p) PR ( ) n!! Tha s (8) ( n ) ( ) p ( n )! ( p) ( p) PR ( ) The uncondonal lkelhood funcon s, ( θ ) ( θ )!( p) L, p; x, L ; x/ R P( R ) (0) Usng (7) and (9) n (0) we can we he full lkelhood funcon as, (9) 4

5 c L( θ, p; x, ) x exp ( ) x θ + θ p ( n )! ( p) ( p) ( n ) ( )!( ) p 3 Unbased esaon Le Y X,,,, () hen Y have exponenal dsbuon wh ean θ.now consde he followng ansfoaon, Z ny Z ( n + )( Y Y ),,3,,. () In ode o deve he dsbuon of Z,,,, consde he nvese Z Z ansfoaon Y and Y,,3,. The vaables n ( n + ) Z, Z, Z defned n () ae all ndependen and dencally dsbued wh exponenal dsbuon wh ean Z, Z,, Z s, θ, see [5]. The jon densy of I can be seen ha, f ( z, θ / R ) exp z (3) θ θ z ( + ) y (4) Usng () n (4) we have z ( + ) x (5) Le T Z ( ) x (6) + 5

6 Snce (3) s a ebe of exponenal faly of dsbuons, T s a coplee suffcen sasc foθ. The dsbuon of T s gaa wh paaees and, θ whch s agan a ebe of exponenal faly of dsbuons. The p.d.f. of T s gven by, [ hθ ] [ g( θ )] B (, ) ( ) f(, θ ) (7) whee B h g θ (, ), ( θ ) exp, ( θ) θ. Jan and Dave [] have suded he poble of nu vaance unbased esaon n a class of exponenal faly of dsbuons. They have shown ha f X, X, Xn be a ando saple fo densy of he ype gven n ( 4) and he p.d.f. of s coplee suffcen sascs can be wen as he one gven n (7) hen he UMVUE of [ h( θ )] k s gven by, and he UMVUE of [ g( θ )] k s, B ( kn, ) Hkn,, > k (8) Bn (, ) B(, n+ k) Gkn, (9) B(, n) Followng he esuls deved n Jan and Dave [], we ge he UMVUE of soe poan paaec funcons as gven below. () Usng (8) he UMVUE of exp k θ s, k H + x > k K,, ( ) ( + ) x (0) Specal case : Subsung k n (0) we ge he UMVUE of exp θ as, 6

7 H, +, ( ) 0 + x > ( + ) x () Specal case : Subsung k n (0) we ge UMVUE of exp θ as, H,, ( ) + x > ( + ) x () ()Usng (0) he UMVUE of he vaance of H, K s gven by, k k Va [ H K, ] _, (3) ( + ) x ( + ) x ()Usng (9) he UMVUE of ( θ ) k s gven by, ( + ) x > k G K, k ( + ) x k k (4) + Specal case 3: Subsung k n (4) we ge he UMVUE of θ as, G ( ), + x (5) + Snce he ode of Raylegh dsbuon s θ, he equaon gven n (5) s UMVUE of ode. Specal case 4: Subsung k n (4) we ge he UMVUE of θ as, G, ( ) x + (6) 7

8 Specal case 5: The h + oen of he Raylegh dsbuon s. θ Subsung as, k n (4) we ge UMVUE of h oen of he Raylegh dsbuon + EX ( ) ( + x ) (7) + Specal case 6: Subsung n (7) we ge UMVUE of ean as, 3 EX ( ) ( + x ) (8) + Specal case 7: The vaance of Raylegh dsbuon s π θ. Is UMVUE s obaned by subsung k n (4) as follows, π ( ) x + Va [ X ] (9) x Specal case 8: Usng (4) wh k he UMVUE of hazad funcon hx ( ) can θ be obaned as, x( ) hx ( ) (30) ( + ) x Specal case 9: The edan of Raylegh dsbuon s θ ( log 4).Usng (5) he UMVUE of edan s gven by, Medan ( + ) x log 4 ( ) (3) + Specal case 0: The p h quanle of Raylegh dsbuon s ξp θ log [ ] p. Usng (5) he UMVUE of he p h quanle s gven by, 8

9 p ( ) x log [ p ] ξ + (3) + (v) Usng (4) he UMVUE of he vaance of G s gven by, K, k ( ) + x K, k Va [ G ] (33) + k + k (v) The UMVUE of densy f ( x) gven n (), fo fxed x s gven by, φ x, ( + ) x ( + ) x x ( ) x, 0 < x < ( + ) x, > (34) (v) The UMVUE of vaance of φ,, > s gven by, x 4 ( x ) x ( x ) x > 3 x ( x ) x Va [ φ x, ] x x 4 ( x ) x x x < 0 ohewse (35) whee ( + ) x (v) Consdeng x as fxed, he UMVUE of R ( x ) of elably funcon, Rx ( ) PX ( > x), x 0 s obaned as follows. Snce [ ] Rx ( ) hθ ( ) x, whee h( θ ) s gven n (5) and usng (0) wh k x he UMVUE R ( x) of R ( x) s gven by,. 9

10 x R ( x), 0 < x< ( ) x + (36) ( + ) x (v) The UMVUE of he vaance of R ( x) s gven by, x x +,0 x < < ( x ) ( x ) + + ( x ) Va [ R ( x)] ( + x ) (37) x, x ( x ) < < + ( x ) + 0 ohewse. (x) The UMVUE of cuulave dsbuon funcon gven n ( ) s, 0, x < 0 x F ( x), 0 < x< ( + ) x ( ) x +, ohewse. (38) Shanubhouge and Jan [3] have suded he poble of nu vaance unbased esaon n exponenal dsbuon unde pogessve ype II censoed daa wh bnoal eovals. They have gven he UMVUE fo paaee p and vaous funcons of p.snce he jon densy P( R ) gven n (9) s ndependen of θ one ges he sae esaos of p and s vaous funcons as gven n Shanubhouge and Jan [3]. 0

11 4 UMVU esao of P( X < Y) In he followng heoe, we deve he UMVUE of P( X < Y). Le uns (ou of n ) on X and uns (ou of n ) on Y ae ecoded whch follow Raylegh dsbuons, gven n () wh paaees θ and θ especvely. Le,, and s, s, s be coespondng eovals. We denoe, ( ) x (39) + and ( s) y. (40) + Theoe : Unde pogessve Type II censoed daa he UMVU esao of P P( X < Y) fo he densy gven n () s gven by, j (! ) (! ) ( ) ( +! ) (! ) (! ) (! ) ( ) ( ) ( ) j j 0 j j < P (4) > 0! +! whee and ae gven by (39) and (40),especvely. Poof: We have P φ φ dxdy (4) G x, y, whee G { ( x, y):0 < x<, 0 < y<, x< y} Usng (34) n (4) and le <, we have, x ( ) y ( ) x y 0 x P dydx (43) Now y ( ) y x dy (44) x Afe subsung (44) no (43) we ge,

12 x ( ) x x dx (45) 0 n n j j Fuhe splfcaon of (45) and applyng he esul ( w) ( ) we ge, Fuhe splfcaon of (46) gves, j j 0 n j ( ) w ( w) dw j 0 j 0 j j ( ) (46) (! ) (! ) ( ) ( ) j P ( ), < (47) j 0 + j! j! Slaly we can show ha fo he case > he UMVUE of P( X < Y) s, (! ) (! ) ( ) ( ) ( ) P > (48) 0! +! j w 5 Illusave exaple In hs secon we llusae he use of he esaon ehods gven n hs acle. We consde he daa on falue es of 5 ball beangs n enduance es gven n Caon [6]. The daa ae 7.88, 8.9, 33.00, 4.5, 4., 45.60, 48.48, 5.84, 5.96, 54., 55.56, 67.80, 67.80, 67.80, 68.64, 68.64, 68.88, 84., 93., 98.64, 05., 05.84, 7.9, 8.04, These obsevaons ae he nube of llon evoluons befoe falue of 5 ball beangs. Raqab and Mad [], Lee [6], and Lee a el [7], have ndcaed ha a one paaee Raylegh dsbuon fs well fo hese daa. We geneae a pogessve Type II censoed daa wh bnoal eovals fo hese daa. The pogessve censoed saple sze s 3.The dopou nubes have been geneaed usng MYSTAT sofwae as follows: fo B(, 0.05) and,,, have B(,0.05) dsbuon fo, 3, j j and se,

13 f j > 0 j 3 j j 0 o.w. Table I. Obseved es of falue and he dopous easued n he nfoave expeen x Usng he esuls gven n Secon 3, he UMVU esaes of dffeen paaec funcons of θ based on daa gven n Table I ae gven below. 3

14 Table II. The UMVU esaes of dffeen paaec funcons of θ based on daa gven n Table I S. No. Paaec funcon UMVU esae exp θ Vaance of,3 exp θ 3 4 Vaance of,3 H Va H,3 H Va H,3 5 θ G,3 6 7 Vaance of,3 θ 8 Vaance of, G,3 H, [ ] E 09 H, [ ] E Va [ G ] G,3 G, Va [ G ] E 06 9 Medan Thd quale Mean EX ( ) Vaance Va [ X ] Hazad funcon a x hx ( ) Densy a x 80.5 φ 80.5, Vaance of 80.5,3 φ Va 80.5,3 [ φ ].3043E 07 6 Relably a x 80.5 R ( x ) Vaance of R ( x) a x 80.5 Va [ R ( x )] c.d.f. a x 80.5 F ( x )

15 6 Conclusons: In hs pape we have consdeed he esaon poble of paaee and s vaous funcons of Raylegh dsbuon unde he pogessve Type II censoed daa wh bnoal eovals. We have deved an elegan expesson fo he UMVUE esaos fo ode, h oen, ean, vaance, hazad funcon, edan, p h quanle, p.d.f., elably funcon and c.d.f. of he Raylegh dsbuon. Ou ehod of obanng hese esaos s que sple han he adonal appoach. We have also obaned UMVUE of P( X < Y) by usng he UMVUE of p.d.f. These esuls educe o Type II censoed daa (whee eovals ae no allowed) and coplee saple case by subsung 0 fo,, and,,., especvely. Acknowledgen: The auhos would lke o expess deep sense of gaude o he efeee fo nsghful coens and helpful suggesons whch led o a consdeable poveen of he pape. Refeences []. M.A.M. Alousa and S.A.AL.Saghee, Sascal nfeence fo he Raylegh odel based on pogessvely ype II censoed daa, Sascs, 40,(006), pp []. N.Balakshnan, Pogessve censong ehodology: an appasal, Tes,6, (007), pp.-59. [3]. N. Balakshnan, Appoxae MLE of he scale paaee of he Raylegh dsbuon wh censong, IEEE Tans.Relb.,38, (989), pp [4]. N.Balakshnan and R.Aggawala, Pogessve Censong: Theoy, Mehods and Applcaons. Bkhause,Boson.(000). [5]. N.Balakshnan, E.Cae, U.Kaps and N.Schenk, Pogessve ype II censoed ode sascs fo exponenal dsbuons, Sascs, 35, (00) pp [6]. C.Caon, The coec ball beangs daa, Lfee Daa Analyss,8,(00), pp [7]. A.C.Cohen, Pogessvely censoed saples n lfe esng, Technoecs, 5, (963), pp

16 [8]. Ynhu Deng, Yuanyua Wang, Yuzhong Shen, Speckle educon of ulasound ages based on Raylegh ed ansoopc dffuson fle, Paen Recognon Lees, 3,(0) pp [9]. D.D. Dye,, Esaon of he scale paaee of he ch dsbuon based on saple quanles, Technoecs, 5,(973), pp [0]. Ey Hu, Yung He, Yanng Chen, Expeenal sudy on he suface sess easueen wh Raylegh wave deecon echnque, Appled Acouscs, 70,(009), pp []. P.N.Jan,and H.P.Dave, Mnu vaance unbased esaon n a class of exponenal faly of dsbuons and soe of s applcaons, Meon,48, (990), pp []. N.L.Johnson, S.Koz and N.Balakshnan, Connuous Unvaae Dsbuons, Volue I, nd edon, John Wley & Sons, New Yok (994). [3]. Chansoo K and Keunhee Han Esaon of he scale paaee of he Raylegh dsbuon unde geneal pogessve censong, Jounal of he Koean Sascal Socey, 38, (009), pp [4]. Chansoo K, Keunhee Han, Esaon of he scale paaee of he Raylegh dsbuon wh ulply ype-ii censoed saple, Jounal of Sascal Copuaon and Sulaon, 79, (009), pp [5]. S. Lalha, Anand Msha, Modfed axu lkelhood esaon fo Raylegh dsbuon, Councaons n Sascs - Theoy and Mehods, 5, (996), pp [6]. W.-C.Lee, Sascal esng fo assessng lfee pefoance ndex of he Raylegh lfee poducs, Jounal of he Chnese Insue of Indusal Engnees, 5, (008), pp [7]. W.-C.Lee, J.-W.Wu, M.-L.Hong, L.-S.Ln, R.-L.Chan, Assessng he lfee pefoance ndex of Raylegh poducs based on he Bayesan esaon unde pogessve ype II gh censoed saples, Jounal of Copuaonal and Appled Maheacs, 35, (0), pp [8]. Lee,K.R., Kapada,C.H., and Bock,D.B. On esang he scale paaee of he Ralegh dsbuon fo doubly censoed saples, Sascal Papes,, (980), pp 4-9. [9]. Lonhue-Hggns,M.S., On he Sascal dsbuon of he Heghs of sea Waves, Jounal of Mane Reseach,, (95), pp

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