Estimation of the Loss and Risk Functions of Parameter of Maxwell Distribution

Transcription

1 Scece Joural of Appled Mathematcs ad Statstcs 06; 4(4): 9- do: 0.648/j.sjams ISSN: (Prt); ISSN: (Ole) Estmato of the Loss ad Rsk Fuctos of Parameter of Maxwell Dstrbuto Guobg Fa Departmet of asc Subjects, Hua Uversty of Face ad Ecoomcs, Chagsha, Cha Emal address: To cte ths artcle: Guobg Fa. Estmato of the Loss ad Rsk Fuctos of Parameter of Maxwell Dstrbuto. Scece Joural of Appled Mathematcs ad Statstcs. Vol. 4, No. 4, 06, pp. 9-. do: 0.648/j.sjams Receved: May, 06; Accepted: Jue 6, 06; Publshed: Jue 9, 06 Abstract: I statstcal decso-makg, whe ayes estmator s used as the ukow parameter s estmato, there ofte exsts certa loss. The the am of ths paper s to study the ayes estmato for the loss ad rsk fuctos of parameter of Maxwell dstrbuto uder Rukh s loss fucto. ayes estmator s derved o the bass of the verse gamma pror dstrbuto uder squared error loss fucto. The ayes estmators of loss ad rsk fucto are obtaed, respectvely. Fally, the codtos of ayes estmators beg coservatve are also derved. Keywords: ayes Estmator, Loss Fucto, Rsk Fucto, Maxwell Dstrbuto. Itroducto I statstcal decso makg process, whe a estmator d δ ( x) s used as the ukow parameters estmato, t ca brg certa loss. Here we ote the loss fucto s w(, δ ), the w(, δ ) s also ot a crsp result due to t s also ukow. So we eed estmate the loss fucto w(, δ ). Classcal decso theory advocates makg some decso d δ ( x), where x s the observato, wth frequetst rsk R(, δ Ths approach has bee ofte crtczed because t mplctly assumes that R(, δ ) s a good measure of accuracy of the procedure used (or of a measure of coclusveess). I 988, Rukh proposed a loss fucto [] L( ; δ, γ ) w(, δ ) γ + γ, () Here γ s a estmator of loss w(, δ The loss fucto L( ; δ, γ ) combes the o-egatve decso loss w(, δ ) wth the accuracy of δ. Ths loss fucto s a very coveet tool the problem of smultaeous decsoprecso reportg. The correspodg rsk fucto has frequetst terpretablty terms of log-ru frequeces. Loss ad rsk fuctos are mportat parts statstcal ferece. May valuable results are obtaed uder dfferet loss fuctos, ad the performaces of estmators are compared uder rsk fucto [-6]. For the estmatos of loss ad rsk fuctos of parameter of bomal dstrbuto, Rukh [] attaed the ayes estmators uder the loss L( ; δ, γ ), ad he also gave ts propertes; Later, may authors studed estmato of loss ad rsk fuctos of paramerers for varous lfetme dstrbutos models, such as, Posso dstrbuto ad expoetal dstrbuto [7], ormal ad logormal dstrbuto [8, 9], Raylegh dstrbuto [0]. The codtos of coservatve propertes are also dscussed these refereces. The Maxwell dstrbuto was frst troduced by Maxwell 860 as a lfetme dstrbuto model, ad sce the, the study ad applcato of Maxwell dstrbuto have bee receved great atteto. Tyag ad hattacharya [] frstly cosdered Maxwell dstrbuto as a lfe tme model, ad they obtaed the mmum varace ubased estmator (UMVUE) ad ayes estmator of the parameter ad relablty of ths dstrbuto. Chaturved ad Ra [] studed ayesa relablty estmato of the geeralzed Maxwell falure dstrbuto. Podder ad Roy [] studed the estmato of the parameter of ths dstrbuto uder modfed lear expoetal loss (MLINEX) fucto. ekker ad Roux [4] dscussed the maxmum lkelhood estmator (MLE), ayes estmators of the trucated frst momet ad hazard fucto of the Maxwell dstrbuto. Dey ad Mat

2 0 Guobg Fa: Estmato of the Loss ad Rsk Fuctos of Parameter of Maxwell Dstrbuto [5] derved ayes estmators of Maxwell dstrbuto by cosderg o-formatve ad cojugate pror dstrbutos uder three loss fuctos, amely, quadratc loss fucto, squared-log error loss fucto ad MLINEX fucto. Let X be a radom varable dstrbuted wth Maxwell dstrbuto wth the followg probablty desty fucto (pdf) (Krsha ad Malk [6]): x 4 f ( x; ) x e, x > 0, > 0 () π Here, s the ukow parameter. The purpose of ths paper s to study ayes estmato of the loss ad rsk fuctos of Maxwell dstrbuto o the bass of the verse gamma pror dstrbuto uder squared error loss fucto. Some coservatve of ayes estmators wll also be obtaed.. Prelmary Kowledge Ths secto wll gve some prelmary kowledge for the further dscusso of the estmato of loss ad rsk fucto of Maxwell dstrbuto (). For a estmator, rsk fucto s ofte used as a method to measure the performace of ths estmator. The rsk fucto s stll a fucto of ukow parameter, thus t s also ukow. Thus t has bee ofte crtczed whe we use the rsk fucto to measure the performace of estmators. Rukh s loss fucto () ca overcome ths shortcomg, ad t has the followg propertes: () For a fxed d δ (x), L ( ; δ, γ ) gets ts (uque) mmum γ at the pot γ m w(, δ A coveet coveto s to put γ m 0 whe w (, δ ) 0. () For ay fxed γ 0, the loss fucto L( ; δ, γ ) s just a lear trasformato of w(, δ ), so that the ayes estmator δ ( X ) of for L( ; δ, γ ) s the ayesa procedure for w(, δ The ayes estmator γ ( X ) of accuracy s γ ( x) E[ w(, δ ) X ] () Defto Supposed γ ( X ) s estmator of w(, δ ) wth ukow parameter, γ ( X ) s called a coservatve estmator, f t satsfes: E ( γ ( X )) R(, δ ) E [ w(, δ )]. Let X ( X, X,, X ) be a sequece of depedet ad detcally dstrbuted radom varables of Maxwell dstrbuto wth pdf (), ad x x, x,, x ) s the ( observato of X. The lkelhood fucto of for the gve sample observato s x x 4 l( ; x) x e π x 4 ( x ) e (4) π K e 4 ( x ) π Here K s a proportoalty costat wth respect to the parameter. Ad ayesa statstcs, we usually use to demostrate the lkelhood kerel. That s l( ; x) ca also be rewrrtte as x t l( ; x) e e (5) Where t x s the observato of T X. y solvg log lkelhood equato, the maxmum lkelhood estmator (MLE) of s easly derved as follows: ˆM (6) T Ad by Eq. (), we ca also show that T s dstrbuted wth Gamma dstrbuto Γ (, ), whch has the followg probablty desty fucto: t ft ( t; ) t e, t > 0, > 0 Γ( ). ayes Estmato of Loss ad Rsk Fucto I ths secto, we shall cocer the estmato of the loss ad rsk fucto of Maxwell dstrbuto (). For later use, we cosder the ayes estmato of uder the followg square error loss fucto w(, δ ) ( δ ) (7) (8) Theorm Let X ( X, X,, X ) be a sequece of depedet ad detcally dstrbuted radom varables of Maxwell dstrbuto wth pdf (), ad x ( x, x,, x) s the observato of X. The cojugate pror dstrbuto for s the verse Gamma dstrbuto, ad γ ( X ) s a estmator of w(, δ ), the () Uder squared error loss fucto (8), the ayes

3 Scece Joural of Appled Mathematcs ad Statstcs 06; 4(4): 9- ad ertmator of s: + T δ + () The ayes estmator of w (, δ ) based o Rukh s loss fucto () s: γ ( X ) ( + T) ( + ) ( + ) ( ) E ( ) γ X () The rsk fucto of δ s: ( + ) ( + ) (9) (0) () [ + ( ) ] + ( ) + R(, δ ) () ( + ) Proof () Suppose that the cojugate famly of pror dstrbutos for s the verse Gamma dstrbutos, IΓ (, ), wth the probablty desty fucto: π Γ( ) ( + ) (, ) e, > 0,, > 0 () The the posteror dstrbuto of s also verse Gamma dstrbuto IΓ ( +, + T), Thus ˆ + T δ E( X ) + () The ayes estmator of w (, δ ) uder the Rukh s loss fucto () s: γ ( X ) E[ w(, δ ) X ] E X X [( δ( )) ] Var( X ) ( + T) ( + ) ( + ) ; Thus Eγ X E T X ( + ) ( + ) ( + ) + + ( + ) ( + ) ( ) [( + ) ) () ) The Rsk fucto of δ : R(, δ ) E [ W (, δ )] + T E [( ) ] + [ + ( ) ] + ( ) + ( + ) ecause R(, δ ) E [ W (, δ )] s the mea value of W (, δ ), the we ca regard R(, δ ) as a estmator of W (, δ ut the rsk fucto R(, δ ) E [ W (, δ )] as the average loss of decso d δ (X ) s the fucto of the ukow parameter, thus t also ukow. The we eed gve a estmator of R(, δ Uder the squared error loss, ayes estmator Φ ( δ ) of R(, δ ) s just the posteror mea of R, δ ), the ( Φ ( δ ) E[ R(, δ ) X ] T [ ( )( + T) + + ] + ( + ) ( + ( ) ) ( + ) ( + )( + ) 4. Coservatve Propertes of Estmato for Loss ad Rsk Fucto (4) For coveece, the followg dscusso we always suppose 0, 0 ad >. I ths secto, we obta some codtos coservatve propertes of estmators uder the loss fucto (8). Theorem. The estmator γ ( X ) s coservatve uder the loss fucto (8) wheever oe of the followg codtos holds: ) 0,,

4 Guobg Fa: Estmato of the Loss ad Rsk Fuctos of Parameter of Maxwell Dstrbuto ) 0, 0 C, where C ) 0, 0. Proof. To see () ote that from Theorem,, ( + ) + + E ( ) γ X, (5) ( ) ( ) ( + ) + + R(, δ ) ( ) To show that γ (x ) s a coservatve estmator, we eed prove Eγ ( X ) R(, δ Substtute (5) ad (6) the equalty: ( ) 4 ( ) 0 (6) + (7) Obvously, f, the equalty (7) holds. So the case () s proved. Whe,{ 0 C} s the soluto of (6), so the case () also be proved. For the case (), whe 0, from Theorem, we have E γ ( X ) ( ) + ( + ) ( + ) [ + ( ) ] R(, δ ) ( + ) γ (x prove Eγ ( X ) R(, δ, (8) (9) To show that ) s a coservatve estmator, we eed Substtute (8) ad (9) the equalty, the we have: ( )[ + ( ) + ( + )] 0 We ote that the former equalty always holds where 0, so the case () s obtaed. Theorem The estmator Φ ( δ ) s coservatve uder the loss fucto (8) wheever oe of the followg codtos holds: () 0, 0 () 0, 0 Proof. To see (), whe 0, from () we have 9 ( + ) + ( ) + ( ) E ( ) 4 Φ δ ( ) ( ) (0) To show that γ ) s a coservatve estmator, we eed (x prove E Φ( δ ) R(, δ Substtute () ad (0) the equalty ad smplfy t, we have: ( ) 0 ( + )( ) + ( + ) We ote that the former qualty always holds where 0, so the case () s obtaed. For the case () whe 0, from (4) we have: Φ ( δ ) T [ + ( ) ] ( + ) ( + ) ( + )[ + ( ) ] E ( ) Φ δ ( + ) ( + ) () () To show that γ (x ) s a coservatve estmator, we eed prove E Φ( δ ) R(, δ Substtute () ad () the former equalty ad smplfcato t, we have: ( )( + ) 0 We ote that t s always holds where 0, so () s obtaed. 5. Cocluso Ths paper studes the ayesa estmato of loss ad rsk fuctos Maxwell dstrbuto uder Rukh s loss fucto. The codtos of coservatve estmators are also dscussed. Accordg to the former dscusso ad cosderg the coveece of estmator γ ( X ), for the estmato of loss fucto w(, δ ), we ca get coclusos as followg: () If the pror parameter 0, the γ ( X ) s more reasoable;

5 Scece Joural of Appled Mathematcs ad Statstcs 06; 4(4): 9- () If the pror parameter 0, the whe s a smaller umber, we use γ ( X ), otherwse we propose Φ ( δ ) ; () Especally, f the pror dstrbuto of s oformatve pror dstrbuto (.e. 0, 0 ), or the matchg pror dstrbuto (.e., 0 ), the the estmators γ ( X ) ad Φ ( ) are both coservatve estmators, but γ ( X ) s more reasoable because of ts smplcty. Ackowledgmet δ Ths study s partally supported by Socal Scece Foudato of Hua Provce (No. 05YA065). The author also gratefully ackowledges the helpful commets ad suggestos of the revewers, whch have mproved the presetato. Coflct of Iterest The author has declared that o coflct of terests exsts. Refereces [] Rukh A. L., 988. Estmatg the loss of estmators of bomal parameter. ometrka, 75 (): [] Hu G., L Q., Yu S., 04. Optmal ad mmax predcto multvarate ormal populatos uder a balaced loss fucto. Joural of Multvarate Aalyss, 8 (4): [] Cao L., Tao J., Sh N. Z. ad Lu, W., 05. A stepwse cofdece terval procedure uder ukow varaces based o a asymmetrc loss fucto for toxcologcal evaluato. Australa & New Zealad Joural of Statstcs, 57 (), [4] Zakerzadeh H. ad Zahrae S. H. M., 04. Admssblty o-regular famly uder squared-log error loss. Metrka, 78 (): 7-6. [5] Ahmed E. A., 04. ayesa estmato based o progressve Type-II cesorg from two-parameter bathtub-shaped lfetme model: a Markov cha Mote Carlo approach. Joural of Appled Statstcs, 4 (4): [6] Xu M. P. ad Xog L. C., 009. ayes ferece for the loss ad rsk fucto Levy dstrbuto parameter estmato. Mathematcs Practce & Theory, 9 (0): -6. [7] Xa Z. H., 99. ayes ferece for loss fucto. Mathematcal Statstcs ad Appled Probablty, 8 (): 5-0. [8] Xa Y. F. ad Ma S. L., 008. ayes ferece of loss ad rsk fucto logarthmc ormal dstrbuto parameter estmato. Joural of Lazhou Uversty of Techology, 4 (): -. [9] Dg X. Y. ad Xu M. P., 04. The ayes ferece for the loss ad rsk fuctos of parameters ormal ad logormal dstrbuto. Joural of Jagx Normal Uversty (Natural Sceces Edto), 8 (): [0] Xu M. P., Dg X. Y. ad Yu J., 0. ayes ferece for the loss ad rsk fuctos of Raylegh dstrbuto parameter estmator. Mathematcs Practce & Theory, 4 (): [] Tyag, R. K. ad hattacharya S. K., 989. ayes estmato of the Maxwell s velocty dstrbuto fucto, Statstca, 9 (4): [] Chaturved, A. ad Ra U., 998. Classcal ad ayesa relablty estmato of the geeralzed Maxwell falure dstrbuto, Joural of Statstcal Research, : -0. [] Podder C. K. ad Roy M. K., 00. ayesa estmato of the parameter of Maxwell dstrbuto uder MLINEX loss fucto. Joural of Statstcal Studes, : -6. [4] ekker, A. ad Roux J. J., 005. Relablty characterstcs of the Maxwell dstrbuto: a ayes estmato study, Comm. Stat. Theory & Meth., 4 (): [5] Dey S. ad Sudhasu S. M., 00. ayesa estmato of the parameter of Maxwell dstrbuto uder dfferet loss fuctos. Joural of Statstcal Theory & Practce, 4 (): [6] Krsha H. ad Malk M., 0. Relablty estmato Maxwell dstrbuto wth progressvely Type-II cesored data. Joural of Statstcal Computato & Smulato, 8 (4): -9.