Minimax Estimation of the Parameter of the Burr Type Xii Distribution

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1 Australa Joural of Basc ad Appled Sceces, 4(1): , 1 ISSN Mmax Estmato of the Parameter of the Burr Type X Dstrbuto Masoud Yarmohammad ad Hassa Pazra Departmet of Statstcs, Payame Noor Uversty, Tehra, Ira Abstract: I ths paper the classcal estmators of the shape parameter for the Burr Type XII dstrbuto, such as, the Maxmum Lkelhood Estmator (MLE), the Uformly Mmum Varace Ubased Estmator (UMVUE), ad the Mmum Mea Squared Error (MMSE) estmator are obtaed. The the problem of fdg the mmax estmators of ths parameter uder the squared log error, precautoary, ad weghted balaced squared error loss fuctos by applyg the theorem of Lehma [195] s cocered. The obtaed results have bee terpreted the lght of two-perso zero sum game. All these estmators are compared emprcally usg Mote Carlo smulato. Key words: Mmax estmator, Burr type XII dstrbuto, Precautoary loss fucto, Weghted Balaced type loss fucto, Game theory, Mote-Carlo smulato. INTRODUCTION The mmax estmato s a upgraded o-classcal approach the estmato area of statstcal ferece, whch was troduced by Wald (195) from the cocept of game theory. It opes a ew dmeso statstcal estmato ad erches the method of pot estmatos. Vo Neuma (1944) troduced the word mmax game theory whch s the optmum strategy of the secod player the two perso zero game. Accordg to Wald (1945), mmmax approach tres to guard agast the worst by requrg that the chose decso rule should provde maxmum protecto agast the hghest possble rsk. A estmator havg ths property s called a mmax estmator. The Burr system of dstrbutos was costructed by Burr (194). The Burr-XII dstrbuto s qute flexble as a lfetme model. The flexblty of the dstrbuto arses from the fact that t has a o-mootoe hazard fucto whch makes t approprate for represetg the lfetme for may products, Solma (). The probablty desty fucto of a Burr-XII dstrbuted radom varable s gve by 1 x f ( x;, ), x,,, (1.1) 1 (1 x ) where ad are shape ad scale parameters, respectvely. If 1, f( x;, ) s strctly decreasg. O the other had, f 1, f( x;, ) creases o (, x ] ad the decreases o [ x, ) where 1 x ( 1) ( 1). The dstrbuto fucto (.. cd f) s; F ( x;, ) 1 (1 x ), x,,. (1.) The desty fuctos of the Burr-XII dstrbuto ca take dfferet shapes. For =1 ad dfferet values of t s a decreasg fucto ad for = ad dfferet values of t s a umodal, skewed, rght. Podder et al. (4) studed the mmax estmator of the parameter of the Pareto dstrbuto uder Quadratc ad MLINEX loss fuctos. Also, Dey (8) studed the mmax estmator of the parameter for the Raylegh dstrbuto uder quadratc loss fucto. Shadrokh ad Pazra (1) studed the mmax estmator of the parameter for the mmax dstrbuto uder several loss fuctos. I ths paper, we shall estmate the parameter (whe s kow) by usg the techque of mmax approach whch s essetally a Bayesa approach. The most mportat elemet the mmax approach s the specfcato of a dstrbuto fucto o the parameter space, whch s called pror dstrbuto. I addto to the pror dstrbuto, the mmax estmator for a partcular model depeds strogly o the loss fucto assumed. The basc dfferece betwee the phlosophy of the mmax ad classcal estmato s that the parameter of the Correspodg Author: Hassa Pazra, Departmet of Statstcs, Payame Noor Uversty, Tehra, Ira 6611

2 Aust. J. Basc & Appl. Sc., 4(1): , 1 dstrbuto s assumed to be a radom varable, the frst approach, whereas a fxed pot the secod oe for small sample sze. It has bee observed that the o-classcal mmax approach s better tha the classcal approach. I may of the o-classcal estmatos the symmetrcal loss fuctos are cosdered. But there are some real lfe stuatos where the use of the symmetrcal loss fuctos may be approprate. I some cases a gve postve error may be more serous tha a gve egatve error ad vce-versa. Ths paper s orgazed as follows: I Secto, we fd the classcal estmators of the shape parameter for the Burr-XII dstrbuto. The mmax estmators of uder three type of loss fuctos,.e., Squared Log Error (whch s symmetrc), Precautoary (whch s asymmetrc), ad Weghted Balaced Squared Error are obtaed secto 3. I Secto 4, we terpret the mmax estmators wth two-perso zero-sum game. I Secto 5, a smulato study s carred out to compare these estmators. The smulato results ad dscussos are provded Secto 6.. Classcal Estmatos: I ths secto, we obta the classcal estmators of the parameter for the Burr-XII dstrbuto ad compare these estmators based o ther mea squared errors (MSE's). Let X1, X,..., X be a radom sample from desty (1.) (whe s kow). The lkelhood fucto s gve by L( ) (1 x ) 1 the the log-lkelhood fucto s ( 1) ( ) l ( 1) l(1 x ) 1, (.1), (.) hece, ( ) l (1 x ) 1 thus the MLE of s, (.) ˆ MLE l (1 X ) 1 T l (1 X ) T where. 1, (.3) Here, we obta the Uformly Mmum Varace Ubased Estmator (UMVUE) of. Sce the famly of desty (1.) belogs to the expoetal famly, therefore, statstc T s a complete suffcet statstc for. It s easy to show that statstc T s dstrbuted as gamma dstrbuto wth parameters ad, wth the desty 1 1 t gt () ( ) t e ; t, 1 E ( ) T 1,. Thus hece, the UMVUE of s 1 ˆUMVUE. (.4) T 661

3 u T Aust. J. Basc & Appl. Sc., 4(1): , 1 We ca fd the Mmum Mea Squared Error (MMSE) estmator the class of estmators of the form. Therefore u u u u MSE ( ) E[( ) ] Var ( ) [ E ( ) ] T T T T r ( r) Whereas E ( T ), r, thus r ( ) u 1 ( 1) u E( ) ue( T ) u T ( ) 1 ad u 1 u Var ( ) u Var( T ) T ( 1) ( ) the u u u MSE ( ) 1 T ( 1) ( ) ( 1) ru ( ). The dervatve of r(u) s u u 1 r( u) 1 ( 1) ( ) ( 1) ( 1) that thereby u=-1. Thus, the M MSE estmator of s ˆMMSE T From (.5), the MSE of the classcal estmators of are calculated as follow: ˆ ( ) MSE ( MLE ), ( 1)( ) MSE ( ˆ UMVUE ), ad MSE ( ˆ MMSE ). 1 It s easy to show that MSE ( ˆ ) MSE ( ˆ ) MSE ( ˆ ). MMSE UMVUE MLE. Mmax Estmatos: I ths secto, we obta the mmax estmators of the parameter for the Burr-XII dstrbuto. The dervato depeds prmarly o a theorem, whch s due to Hodge ad Lehma (195) ad ca be stated as follows: 3. Lehma's Theorem: Let { ; } be a famly of dstrbuto fuctos ad D a class of estmators of. Suppose that F (.5) (.6) 6613

4 Aust. J. Basc & Appl. Sc., 4(1): , 1 d D s a Bayes estmator agast a pror dstrbuto ( ) o the parameter space, ad the rsk fucto R d (, ) costat o ; the d s a mmax estmator of. The ma results are cotaed the followg Theorems. THEOREM 1: Let X X1 X X (,,..., ) be depedetly ad detcally dstrbuted radom varables draw from the desty (1.). The ˆ ( ) w l(1 X ) s the Mmax estmator of the parameter MWBSE for the Weghted Balaced Squared Error type loss fucto gve by: 1 L (, d ) wq( )( d d ) (1 w) q( )( d ) wd, w[,1) d where, q( ) 1, ad s a "target" estmator, obtaed from the crtero of maxmum lkelhood. Proof: To prove the theorem we have to use Lehma's theorem ad Theorem 1 of Jafar Joza et al. (6a). I order to prove the theorem t wll be suffcet to show that 1 1 (3.1) d ( w) l(1 x ) s a mmax estmator of for the weghted balaced squared error loss fucto (3.1). Ths loss, whch depeds o the observed value of d, reflects a desre of closeess of d 1 both to: () terms of weghted squared error loss, ad () the target estmator d terms of weghted squared dstace. Jafar Joza et al. (6b) troduced a exteded class of balaced type loss fuctos of the form gve (3.1), wth q() beg a sutable postve weght fucto. For the case of q()=1 ad a least squares d (3.1) s equvalet to Zeller's (1994) balaced loss fucto. For the more detals about ths loss fucto see Jafar Joza et al. (6a). If we ca show that the rsk of d 1 s costat, the the Theorem 1 wll be proved. Sce, d l (1 x ) has costat rsk uder, Theorem 1 of 1 L, d (, d 1) ( d1 1) ( 1)( ) Jafar Joza et al. (6a) by usg the equato (.6) tell us that estmator wth mmax rsk (1 ) w w 1 1 1w So accordg to the Lehma's theorem t follows that (1 w) w, whch s a costat w.r.t.. d ˆ 1 MWBSE ( w) l (1 X ) 1 s a mmax s the mmax estmator of the parameter for the Burr-XII dstrbuto uder the Weghted Balaced Squared Error loss fucto of the form (3.1). Note that: for w= the mmax estmator estmator,.e., ˆ ˆ # ˆMMSE MWBSE MMSE s detcal to the classcal THEOREM : Let (1.). X X1 X X (,,..., ) be depedet ad detcally dstrbuted radom varables from the desty 6614

5 The Aust. J. Basc & Appl. Sc., 4(1): , 1 ˆ exp{ ( ) ( )} l(1 X ) MSLE 1 the Squared Log Error loss fucto of the type s the mmax estmator of the parameter for L(, d ) ld l, (3.) where d s the estmate of. Proof: Usg Lehma's theorem, t wll be suffcet to show that d exp{ ( ) ( )} l (1 x ) s a 1 mmax estmator of for the symmetrc loss fucto (3.). Therefore, we have to fd the Bayes estmator d of. The f we ca show that the rsk of d s costat, the the Theorem wll be proved. Let us assume that has Jeffrey's o-formatve pror desty defed as 1 g( ) ;. (3.3) The the posteror dstrbuto of for the gve radom sample X ( X1, X,..., X ) s 1 T g( x) T e ;, x, (3.4) ( ) where l (1 x ). Whch mples that x s dstrbuted as Gamma dstrbuto wth parameters 1 ad T ad mea of the dstrbuto s /T. Now the Bayes estmator of uder the squared log error loss fucto (3.) s ˆ E [l ], BSLE e where E [l ] l g( x) d t 1 t (l ) e d ( ) u 1 Usg the relato t u d du t t 1 u 1 u E l l ( ) u e du ( ) t 1 1 u l ul tu e du ( ), we have 1 u t 1 u uu e du u e du 1 l l ( ) ( ) ( ) ( ) l t l t ( ) u 1 where s the dgamma fucto,.e., ( ) ( ), ad ( ) l( u) e u du s the frst dervatve of () wth respect to. Usg ths result we get ˆ ( ) e BSLE exp{ l t} ( ), 6615

6 Aust. J. Basc & Appl. Sc., 4(1): , 1 where l (1 x ). R SLE 1 Now the rsk fucto uder the squared log error loss fucto (3.) s gve by ( ) EL( ˆ BSLE, ) E l ˆ BSLE l l ˆ ˆ BSLE (l )( l BSLE ) l E l (l )( l ) l E l (l ) E l l E E E E l l l (l ) l l. Hece we get, t 1 El (l t) e t dt ( ), 1 1 usg the relato t y t y dt dy we have ( ) y 1 E (l ) l( y) e ( y) dy l 1 ( ) ( ) y 1 y 1 e y dy l y e y dy ( ) l l. ( ) (3.5) Also we have, t 1 E (l ) (l t) e t dt, ( ) v 1 usg the relato t v t dt dv we obta v v 1 (l ) l( ) ( ) ( ) 1 1 E e v dv 1 ( ) v 1 l l v e v dv 1 v 1 l v 1 (l ) v 1 l v e v dv l ve v dv e v dv ( ) ( ) ( ) l (l ) v 1 v e v dv ( ) (l ) where ( ) ( ) ad s the secod dervatve of () wth respect 6616

7 to. Usg these results the rsk fucto becomes Aust. J. Basc & Appl. Sc., 4(1): , 1 RSLE ( ) ( l ) l (l ) l (l ) ( l ) l where s the trgamma fucto, amely the frst dervatve of at,whch s a costat w.r.t.. So from Lehma's theorem t follows that d ˆ exp{ ( ) ( )} l(1 X ) MSLE 1 s the mmax estmator of the parameter of the Burr-XII dstrbuto uder the squared log error loss fucto (3.).# THEOREM 3: Let (,,..., ) X X1 X X the desty (1.). The ˆ ( 1) l(1 X ) MP be depedetly ad detcally dstrbuted radom varables draw from the Precautoary loss fucto (see Norstrom (1996)) of the type ( d3 ) L(, d3) d3 where d 3 s the estmate of. 1 s the Mmax estmator of the parameter for Proof: Usg Lehma's theorem, t wll be suffcet to show that d ( 1) l(1 x ) 3 1 a mmax estmator of for the asymmetrc loss fucto (3.6). For ths, frst we have to fd the Bayes estmator d 3 of. The f we ca show that the rsk of d 3 s costat, the the proof s complete. Regardg to the Theorem ad usg the o- formatve pror, we get the posteror dstrbuto of as equato (3.4). Now the Bayes estmator of uder the precautoary loss fucto (3.6) s ˆ E [ ] BMP. E [1 ] Whereas, (3.6) s E [ ] ad, E [1 ] ( 1) hece,. ˆ BMP 1 ( 1) ( 1) l (1 ) X The rsk fucto of the estmator R MP ( ) EL( ˆ BMP, ) ˆBMP s 6617

8 Aust. J. Basc & Appl. Sc., 4(1): , 1 ˆ BMP E ˆ BMP 1 ˆ 1 E BMP E ˆ BMP ( 1) 1 E E ( 1) Whereas, 1 E ad E, 1 therefore, RMP ( ) 1, 1. whch s a costat w.r.t.. So from Lehma's theorem t follows that d ˆ ( 1) l(1 X ) 3 MP 1 s the mmax estmator of the parameter for the Burr-XII dstrbuto uder the precautoary loss fucto (3.6).# 4. Iterpretato of Mmax Estmators wth Two-Perso Zero-Sum Game: Accordg to Wald (195) the followg statstcal problem s equvalet to some two-perso zero-sumgame betwee the statstca (Player-II) ad ature (Player-I). Here the pure strateges of ature are the dfferet values of the terval (,) ad the mxed strateges of ature are the pror destes of the terval (,). The pure strateges of statstca are all possble decso fuctos the terval (,). Expectato of the loss fucto L(,d) s the rsk fucto, R(, d) E L(, d) whch s the ga of player-i. R(, d) s the value of R(, d) d( ), where ( ) s the pror desty. If the loss fucto s cotuous both d ad ad covex d for each the there exst measures ad for all ad d so that the followg relato holds. R d R d R d (, ) (, ) (, ) The umber R(, d ) s kow to be the value of the game ad ad are the correspodg optmum strateges of Player-I ad Player-II. I statstcal terms ad d d s the least favourable pror desty of s a mmax estmator of. I fact, the value of the game s the loss of the statstca. It has bee show that, here (I) d ˆ ( ) w l(1 X ) s the optmum strategy of 1 MWBSE 1 player-ii for the weghted balaced squared error loss fucto (3.1) ad the value of the game s d R WBSE (1 w) w (, d1 ) w (II) ˆ exp{ ( ) ( )} d l(1 X ) s the MSLE 1 optmum strategy of player-ii for the squared log error loss fucto (3.) ad the value of the game s R SLE ( ) ( ) (, d ). ( ) ( ) (III) d ˆ ( 1) l(1 X ) 3 MP 1 s the optmum 6618

9 Aust. J. Basc & Appl. Sc., 4(1): , 1 strategy of player-ii for the precautoary loss fucto (3.6) ad the value of the game s R MP (, d3 ) 1. 1 I all the cases g( ) ;, s the optmum strategy for 1 Player-I. 5. Smulato Study: The estmators, ad are classcal estmators of the parameter for the Burr-XII ˆMLE ˆUMVUE ˆMMSE dstrbuto; whereas, ad are mmax estmators uder weghted balaced squared ˆMSLE error, squared log error, ad precautoary loss fuctos, respectvely. ˆMP I secto we showed that, for the classcal estmators, MSE ( ˆ ) MSE ( ˆ ) MSE ( ˆ ), MMSE UMVUE MLE but ths secto our ma am s to compare the (classcal ad mmax) estmators terms of ther Bases ad MSEs. The Bases ad MSEs of the estmators are computed usg the Mote-Carlo smulato study. The smulato s carred out for 1 ad, ad wthout loss of geeralty we take, wth sample sze =,4,5,7,1(1)5. All results are based o 1 replcatos. The obtaed results are demostrated Tables 1 ad ad also preseted them Fgures 1 ad. From Table 1 ad, we ca see that betwee the classcal estmators, estmator s better tha ˆMMSE the rest, ths s true for all values of, see also Fgure 1. From Table 1 ad, we ca see that betwee the mmax estmators, for all values of, the mmax estmator uder weghted balaced squared error loss fucto ( ) s better tha the rest, see Fgure. I geeral, estmator s better tha estmator for small sample sze (.e for 1). But, for large sample sze ( 1 ), the ˆMMSE classcal ad mmax estmators ( ad ), are detcal. Also we ca see that betwee the ˆMMSE mmax estmators, the mmax estmators uder precautoary loss fucto ( ˆMP ) have the smallest estmated Bases as compared wth the rest. O the other had, the classcal estmator estmators, ad, are overestmato, but the rest are uderestm ato. Table 1: Sample Sze ˆMSLE ˆMP, ad the mmax ˆMLE Bases ad MSEs of dfferet estmators for the parameter of the Burr-XII dstrbuto whe =,=1 ad w=.5 (MSE parethess). ˆMLE ˆUMVUE ˆMMSE ˆMSLE (16.883) (3.951) (1.) (.8459) (9.5118) (8.968) (1.18) (.57) (.335) (.375) (.715) (.6974) (.643) (.3544) (.591) (.571) (.473) (.463) (.564) (.1731) (.153) (.15) (.74) (.53) (.1348) (.135) (.984) (.977) (.116) (.1154) ˆMP 6619

10 Aust. J. Basc & Appl. Sc., 4(1): , 1 Table 1: Cotue (.66) (.54) (.51) (.513) (.576) (.575) (.378) (.344) (.334) (.334) (.358) (.358) (.3) (.76) (.66) (.66) (.86) (.86) (.11) (.198) (.193) (.193) (.3) (.3) Fg. 1: MSE's of the classcal estmators, for dfferet value of. Table : Sample Sze Bases ad MSEs of dfferet estmators for the parameter of the Burr-XII dstrbuto whe =,=1 ad w=.5 (MSE parethess). ˆMLE ˆUMVUE ˆMMSE ˆMSLE (67.533) (15.84) (4.) (3.3838) (38.47) (3.387) (4.513) (.17) (1.3) (1.31) (.8861) (.7897) (.497) (1.4178) (1.84) (1.63) (1.8811) (1.8413) (1.55) (.695) (.61) (.689) (.895) (.81) (.5391) (.4139) (.3935) (.398) (.464) (.4618) (.53) (.159) (.5) (.5) (.33) (.3) (.1511) (.1375) (.1338) (.1337) (.1431) (.143) (.1198) (.115) (.166) (.167) (.1145) (.1144) (.845) (.791) (.77) (.77) (.814) (.814) ˆMP 66

11 Aust. J. Basc & Appl. Sc., 4(1): , 1 Fg. : MSE's of the mmax estmators, for small values of. 6. Cocluso: I ths paper we obtaed the Classcal ad Mmax estmators for the shape parameter of the Burr-XII dstrbuto. We derved the mmax estmators uder symmetrc, asymmetrc ad balaced loss fuctos. We have show that betwee the classcal estmators, estmator ˆMMSE s better tha the rest, ad betwee the mmax estmators, the mmax estmator uder weghted balaced squared error loss fucto ( ) s better tha the rest, these are true for all values of. I geeral, betwee ths two estmators ( ad ), estmator s better tha estmator for small sample sze (1). But, ˆMMSE for large sample sze (>1), the classcal ad mmax estmators ( ad ), are detcal. Thus, ˆMMSE ˆMMSE we suggest to use the mmax estmator uder weghted balaced squared error loss fucto for estmatg the shape parameter of the Burr-XII dstrbuto. REFERENCES Burr, H.W., 194. Cumulatve Frequecy Fuctos. A. Math. Stat., 13: Dey, S., 8. Mmax Estmato of the parameter of the Raylegh Dstrbuto uder Quadratc loss fucto, Data Scece Joural, 7: 3-3. Hodge, Z.I., E.L. Lehma, 195. "Some Problem Mmax Estmato," A. Math. Stat., 1: Jafar Joza, M., E. Marchad ad A. Parsa, 6a. O estmato wth weghted balaced-type loss fucto, Statst. Probab. Lett, 76: Jafar Joza, M., E. Marchad ad A. Parsa, 6b. Bayes estmato uder a geeral class of balaced loss fuctos, Rapport de recherché 36, Departemet de mathematques, Uverste de sherbrooke ( Norstrom, J.G., The use of precautoary loss fucto rck aalyss. IEEE Tras. Relab., 45(3): Podder, C.K., M.K. Roy, K.J. Bhuya ad A. Karm, 4. Mmax estmato of the parameter of the Pareto dstrbuto for quadratc ad MLINEX loss fuctos, Pak. J. Statst., (1): Roy, M.K., C.K. Podder ad K.J. Bhuya,. Mmax estmato of the scale parameter of the Webull dstrbuto for quadratc ad MLINEX loss fuctos, Jahagragar Uversty Joural of Scece, 5: Shadrokh, A. ad H. Pazra, 1. Mmax estmato o the mmax dstrbuto, Iteratoal Joural of Statstcs ad Systems, 5():

12 Aust. J. Basc & Appl. Sc., 4(1): , 1 Solma, A.A.,. Relablty Estmato a Geeralzed Lfe Model wth Applcato to the Burr-XII. IEEE Tra. o Relblty, 51: Vo Neuma, J., O. Morgestem, Theory of Games ad Ecoomc Behavour, Prceto Uversty Press. Wald, A., 195. Statstcal Decso Theory, Mc Graw-Hll, New York. 66