Estimating a Bounded Normal Mean Under the LINEX Loss Function

Transcription

1 Journl of Sciences, Islic Republic of Irn 4(): (03) University of Tehrn, ISSN Estiting Bounded Norl Men Under the LINEX Loss Function A. Krinezhd * Deprtent of Sttistics, Fculty of Mthetics, Sttistics nd Coputer Science, University of Tehrn, Tehrn, Islic Republic of Irn Received: 8 Jnury 03 / Revised: 3 My 03 / Accepted: July 03 Abstrct Let X be rndo vrible fro norl distribution with unknown en θ nd known vrince σ. In ny prcticl situtions, θ is known in dvnce to lie in n intervl, sy [,], for soe > 0. As the usul estitor of θ, i.e., X under the LINEX loss function is indissible, finding soe copetitors for X becoes worthwhile. The only study in the literture considered the proble of inix estition of θ In this pper, by constructing dointing clss of estitors, we show tht the xiu likelihood estitor is indissible. Then, s copetitor, the Byes estitor ssocited with unifor prior on the intervl [,] is proposed. Finlly, considering risk perfornce s coprison criterion, the estitors re copred nd depending on the vlues tken by θ in the intervl [,], the pproprite estitor is suggested. Keywords: Adissibility; Byes estitor; LINEX loss function; Mxiu likelihood estitor; Norl distribution Introduction In the sttisticl literture it is often ssued tht the preter spce is unbounded which sees to be never fulfilled in prctice. In vrious physicl, industril nd biologicl experients, the experienter hs often soe prior knowledge bout the preter(s) of the underlying popultion. The verge per cpit incoe of developing country is likely to lie between those of n underdeveloped nd developed country. The verge fuel efficiency of new odel of pssenger cr will lie between those of n old odel nd forul one rcing cr. Exples of siilr nture where en of rel phenoen lies in bounded intervl bound in prctice (e.g., physicl ttributes such s height or weight of people, verge life of nils), [4]. Therefore there is prcticl interest to include such dditionl infortion into sttisticl procedures. Surprisingly, while the ssuption of boundedness cn be useful in prctice, it introduces soe chllenging probles in theory. Such probles first rose with the prcticl proble in 950 in which two probbilities nd known to stisfy the restriction, needed to be estited. Mxiu likelihood estition ws used for this purpose. Lter Mxiu Likelihood Estitors (s) were shown to be indissible under Squred Error Loss (SEL) function L ) (, ) (, * Corresponding uthor, Tel.: +98()664, Fx: +98()66478, E-il:.krinezhd@khy.ut.c.ir 57

2 Vol. 4 No. Spring 03 Krinezhd J. Sci. I. R. Irn tht is, it ws shown tht there exist estitors which re better thn the in the sense tht their expected loss, i.e., R(, ) E[ L(, )], s function of the preter to be estited, is nowhere lrger nd soewhere sller thn tht of the. This then led to the serch for dointors for these indissible estitors s well s for dissible estitors with good properties. One such property is tht of inixity where n estitor is inix when there does not exist n estitor with sller xiu expected loss. Exples of probles ddressed in the restricted preter spces, cn be found in [], [7], [6] nd the recent tretise by ven Eeden [7] for detiled discussion. Let X ~ N (, ) denote rndo vrible hving norl distribution with the probbility density function ( ) x f ( x, ) e, x, where, it is supposed tht the vrince is known nd the unknown en, lies in n intervl of the for [, ], for soe known > 0. The first study in estiting the bounded norl en under the SEL function dtes bck to 98. Csell nd Strwdern [4] showed tht, when , there exists unique dissible nd inix estitor of, ssocited with syetric two-point prior on, nd proved tht it dointes the of, when 0. They lso gve clss of dissible nd inix estitors for the cse when.4.6. These estitors re inix w.r.t. syetric three-point prior on,0,. Bickel [] presented n estitor which is syptoticlly inix s, nd showed tht the wek liit of the lest fvourble prior (rescled to [,] ) hs the density cos, nd the inix risk is o( ). After these initil works, severl uthors considered the estition proble in restricted preter spces under the SEL function. Moors [8, 9] ssued bounded estition proble is invrint w.r.t. finite group of trnsfortions nd constructed dointors of boundry estitor. He then pplied his results into the estition proble of bounded norl en. Gtsonis et l. [0] considered the Byes estitor w.r.t. the unifor prior on the intervl [, ], s copetitor for the sple en X, nd showed tht it dointes X. They further nuericlly copred risk perfornce of their Byes estitor, the, the inix estitor, nd the Byes estitor w.r.t. the Bickel s prior, nd finlly recoended the use of their proposed estitor. In ddition to [8, 9], the estition in restricted preter spces under the SEL function, in very generl setting, ws considered in [5, 6, 7]. DsGupt [7] in estiting vector h ( ) when is restricted to sll bounded convex subset k of derived sufficient conditions under which the Byes estitor w.r.t. lest fvourble prior on the boundry of is inix. He then pplied his results in soe distributions including the norl distribution nd showed tht the Byes estitor w.r.t. the two-point prior considered in [4] is inix when Conditions for either indissibility or ethods of constructing dointors within given clss of estitors were presented in [5, 6]. Kur nd Tripthi [4], on the bsis of, proposed nother estitor nd copred the risk perfornce of it with the boveentioned estitors. Dou nd vn Eeden [8], showed the indissibility conditions in [5, 6] re stisfied for the bounded norl en proble nd hence, by giving n explicit for of dointing estitor, derived indissibility of of the en. Ltely, the generl theory of estiting preters of syetric distribution which is subject to n intervl constrint, under the SEL function developed in [6]. See lso [5] for siilr developent. It is worth entioning tht the proble of estiting norl en in the cse where is bounded below, i.e.,, for soe constnt, lso received considerble ttention in the literture. Estition of positive norl en ws first considered in Ktz [3]. Ktz proposed the generlized Byes estitor of w.r.t. the unifor prior on [0, ) nd proved its dissibility nd inixity under the SEL function. He lso proved tht the restricted, is inix. The results of Ktz were independently proved in [] nd generlized in [9] to generl loction preter fily under certin conditions. Therefter, the proble of estiting positive norl en hs developed in the literture, see for exples [5,6] nd references there in. All the bove-entioned studies re bsed on the SEL function which penlizes eqully overestition nd underestition of desired preter, while it does not occur in prctice. For exple, in food processing industries, if the continers re underfilled, it is possible to incur uch ore severe penlty rising fro isrepresenttion of the products ctul weight or volue, see []. As nother exple, in d construction, n underestition of the pek wter level is usully uch ore serious thn n overestition, see [9]. 58

3 Estiting Bounded Norl Men Under the LINEX Loss Function Soe uthors hve considered n estition proble under n syetric LINEX (LINer EXponentil) loss function L s e () ( ) s, (, ) { ( ) }, where 0 nd s 0 re fixed rel nubers. This loss function which ws first introduced by Vrin [8], rises exponentilly on one side nd pproxitely linerly on the other side, nd is useful when overestition is perceived s ore serious thn underestition or vice-vers. For ore infortion on point estition under LINEX loss function, see [0, 9]. In the norl en estition proble with no restriction on, Zellner [9] showed tht the usul estitor X under the LINEX loss function is indissible nd dointed by * ( X ) X. () Hence, X is neither dissible nor inix. Also, Rojo [] nd Sdooghi-Alvndi nd Netollhi [3] proved tht in the clss of estitors of the for * cx d, is the only inix nd dissible estitor of. In the bounded norl cse under the LINEX loss function (), the only study dtes bck to 995. Bischoff et l. [3] considered inix estition of nd showed tht the Byes estitor w.r.t. the two-point prior i.e., ( ) ( ), g( X ) ( X ) ln g ( X ) x x where g ( x ) e ( ) e, is inix when 0, nd (0, 0], 0 in 3, ln 3 In this pper, we consider the LINEX loss function () nd with no loss of generlity, we tke s, i.e., L (, ) e, 0. (3) We then propose soe estitors of the bounded norl en nd copre risk perfornce of the proposed ones with the inix estitor derived in [3]. To this end, first, we obtin conditions under which the of, i.e., X ( X ) X X X. is indissible nd hence, we derive clss of dointing estitors. We then propose the Byes estitor of the en w.r.t. unifor prior s copetitor for. Finlly, we wish to copre risk perfornce of the derived estitors. Notice tht the concept of dissibility, Byesinity, invrince nd inixity highly depends on the choice of loss function. It is worth noting tht there exist soe other works relted to restricted preter estition proble, considering other losses. For brief history nd soe relted references, see the recent work done by Krinezhd [] which considered the bounded norl en estition proble reltive to the SEL function. Indissibility of In this section, we construct dointing clss of estitors of bsed on the work done by Chrrs [5] nd Chrrs nd vn Eeden [6] (reders y refer to [8] s convenience version of [5, 6]). Let Ch be shrinkge estitor of the following for X Ch ( X ) X X X, (4) where (0, ]. Due to the entioned i, we first provide soe required terils below. Le. Let x ux ( ) e x ( x) e ( x ) ( x), 0, x v( x) e x ( x) e ( x ) ( x), 0, (5) (6) where (.) nd (.) denote the distribution nd density function of stndrd norl rndo vrible. Then ux ( ), v( x ) nd their first nd second derivtives hve 59

4 Vol. 4 No. Spring 03 Krinezhd J. Sci. I. R. Irn the following properties: () ux ( ) nd v( x ) re decresing for x 0 nd incresing for x 0. (b) ( ) x u x e ( x ) nd hs the se sign s x. x (c) u ( x) e ( x) for x 0. (d) ( ) x v x e ( x ) nd hs the se sign s x. x (e) v ( x ) e ( x) for x 0. Proof. The proof is strightforwrd nd oitted. Le. Let hx (, ) u( x) v( x) where ux ( ) nd v( x ) re given by (5) nd (6). Further suppose is positive, then () For fixed (0, ) nd for (, ], h (, ) is decresing function of. x [0, ], (b) For let ( x ) h ( x, ) u( x) v( x). (7) Then (0) 0 nd ( ) 0. Furtherore, if 0 stisfy one of the following conditions: 0 (8) or where 0 x ( ), ( ) 0 nd w w (9) w e (0) ( ) nd w ( ) e ( ) () ( e ) e ( ), then ( x ) 0, for x (0, ). Hence, there exists unique root (0, ) of the eqution ( x ) 0 such tht under conditions (8) or (9), for x [0, ), ( x ) 0, nd for x (, ], ( x ) 0. Proof. () Differentiting h (, ) w.r.t., we hve h (, ) u( ) () v ( ). Now using Le (c) nd (e), the reinder is esily obtined. (b) Differentiting ( x ) w.r.t. x, we hve ( x ) u ( x ) v ( x). On the other side, for x [0, ] u ( x) v ( x) ( x) Now let w x e e e ( x ) x x ( ). (3) We discuss the sign of w( x ). Due to condition (8), nd hence w( x ) is positive. Moreover under condition (9), it cn be esily seen tht w( x ) is n incresing function. So using eqution (0), for x [0, ], w ( x ) w (0) e w ( ). Further, using the following inequlities ( x ) e x e ( x ) x e x e ( ) ( ), ( x) e ( ) e, x ( x) e 0, ( ) (0), And equtions () nd (3), we conclude tht u ( x ) v ( x) w ( ). Hence, if 0 nd e ( x ) x e ( x) ( ) x x e () x () x e x e ( x) ( ) x x e () x () x e x e ( x) ( ) x ( x) e e x w( ), w ( ) 0, u ( x) v ( x) will be positive nd the proof is copleted. The in result of this section is s follows. Theore. Let X ~ N(,) when, 0. Then under conditions (8) nd (9), :0 Ch is clss of dointing estitors of w.r.t. the LINEX loss function (3), where is the unique root of eqution (0). x ( x) e e e ( x) x x 60

5 Estiting Bounded Norl Men Under the LINEX Loss Function Proof. Due to equtions (5) nd (6), it cn be esily verify tht the risk function of nd Ch hve the following for nd R(, ) u( ) v( ) Let (, ;, ) R(, ) R(, ). Ch Ch (4) (5) Now, we discuss the sign of (, Ch ;, ) in two cses. (i) when [, ), using Le () the desired result follows. (ii) when (, ], differentiting (, Ch ;, ) w.r.t., ccording to Le (), we hve () 0, 0 (, Ch ;, ) h (, ) ( ) 0,. Consequently, for [0, ] nd (, ], (, ;, ) (, ;,0) 0 Ch Ch And this copletes the proof. The next le which is siilr to the work done by Moors [8, 9] nd Bischoff et l. [3], provides useful extension of the in result. Le 3. Suppose n estition proble is invrint w.r.t. the finite group of trnsfortions G e, g, where for x, ex ( ) x is n identity trnsfortion nd g ( x) x. Further suppose for x, i( x ) i( x ), i,. Then, in the fily of norl distributions N (,), nd under the LINEX function (3), for given 0 nd [, ], dointes if nd only if for 0 nd [, ],.. dointe. R(, ) u( ) v( ). Ch Proof. The following reltion is held ssuing the entioned ssuptions E L, i ( X ) E L, i( X ), i, nd becuse of tht,, ( ), ( ) E L X E L X E L, ( X ) E L, ( X ). This gives the desired result. Now, using Theore nd Le 3, the next theore is ieditely derived. Theore. Let X ~ N(,) when, 0. Then under conditions or 0 nd x w ( ), w ( ) 0 where w () nd w () re given in equtions (0) nd () respectively, for 0 nd [, ], is indissible nd :0 Ch is clss of dointing estitors of, where Ch is given in (4) nd is the unique root stisfying Theore. In Tble, vlues of w(), w () (given by equtions (0) nd ()) for different vlues of nd hve been tbulted. As it cn be seen, when is positive, t lest on of w () nd w () is positive nd the condition x w( ), w ( ) 0 is not liiting one. In the se wy, it cn be inferred tht when is negtive, the condition x w( ), w ( ) 0 is not n encubrnce condition. In Tble, vlues of the unique root of eqution (7) for vrious vlues of nd hve been clculted. Notice tht under conditions entioned in Theore, for (0, ] nd [, ], Ch dointes but becuse of the theticl difficult coputtions, it is not esy to prove this property explicitly. This subject in Theore for (0, ] nd [, ] holds but becuse of the se reson, we cnnot prove tht. This desired property is obviously seen fro Fig. Tble. Vlues of (w(),w()) for different vlues of nd (0.0896,0.0997) (0.408,0.79) (0.488,0.0305) 0.50 (0.087,0.056) (0.065,0.043) (0.493,0.33) 0.80 ( ,0.038) ( ,0.004) (0.0780,0.004).00 (-0.97,0.005) (-0.3,0.009) (0.009, ).00 ( ,0.0373) ( ,0.0453) (-0.59,0.090) 5.00 ( ,0.0368) ( ,0.005) (-0.997,0.0073) 0.00 (-0.787,0.035) ( ,0.007) (-0.000,0.007) 6

6 Vol. 4 No. Spring 03 Krinezhd J. Sci. I. R. Irn to Fig. 6. Moreover when tends to, the difference between risk functions of Ch nd becoes significnt. This desired property is obvious fro Fig.. Besides in Fig., risk functions of Ch nd cut ech other in [, ]. This is due to the vlue of. In fct the vlues 0.4 nd 0.7 is out of the intervl (0, ) (see Tble ) nd hence ccording to Theore there is no reson for the doinnce of Ch on. The se thing hppens in Fig. 3. Additionlly, copring Fig. nd Fig. 3 revels the iportnt of selection for vlues of (difference between overestition nd underestition). Underlying the iportnt of overestition nd underestition, it cn be seen tht in Fig. ll the risk functions, between nd, tke their iniu t the point nd in Fig. 3 this issue hppens conversely. Byes Estitor Associted with the Unifor Prior In this section we propose nother copetitor for. This copetitor is the Byes estitor ssocited with the unifor prior u ( ), [, ]. The Byes estitor w.r.t. the LINEX loss function (3) is given by, ( X ) lne e X h,0( X ) X ln, h ( X ), where h, ( X ) ( X ) ( X ). Copring, with * (given by ()), we re interested in the behviour of risk function of,. In the next section, we crry out siultion study to see the perfornce of,. Coprisons of Risk Perfornce In this section we coproise the risk function of ll the estitors, nely *,,,, Ch nd,. * The risk function of is constnt ( /) nd the risk of nd Ch hve been given in (4) nd (5),, respectively. The risk function of nd, hs not n nlyticl for nd hence, it is ipossible to copre their risks explicitly. So, we hve coputed their estited risks nd figured the in Fig. 4 to Fig. 6. Since Bischoff et l. [3] hve proved inixity of, for 0, we hve chosen soe positive vlues for. Further, we hve considered two-point prior used in this pper, is syetric one, i.e., /. Notice tht our coprisons re on the bsis of crefully nlysis nd here, we hve just figured the risk functions for soe selected vlues of nd. Our conclusion is s follows: * () Risk function of is constnt ( /) but s it cn be seen especilly fro Fig. 4, other the estitors hve sller risk vlues. Moreover * is not rngepreserving. So, becuse of these resons, we prefer not to use it ny longer., (b) under the inixity condition, i.e., (0, 0], where 0 in{ ( 3 ), ln 3}, hs quite good risk perfornce when is close to. (c) Ch hs good risk perfornce w.r.t.. The risk difference of these two estitors for sll vlues of nd is stisfctory. In ddition, nd Ch hs sller risk vlues when is close to. (d), for oderte vlues of in [, ], hs rerkble risk perfornce. It lso is less sensitive to, chnges in vlues of nd. Moreover, tkes the xiu vlue its risk function t point which is close to the risk function of the inix estitor,. These desirble behviours of the risk function of cn be observed siplicity fro Fig. 4 to Fig. 6., Results The proble considered in this pper is tht of estiting the en of norl distribution under the dditionl infortion tht the en lies in bounded intervl [, ] under the LINEX loss function (3). Soe estitors for the en were proposed, nely, X, *,,,, Ch nd,. It ws first shown tht, under ild conditions, Ch dointes. Then, considering risk perfornce s coprison criterion, the estitors were copred. We do not recoend the use of the usul estitor X nd *, which re not rnge preserving. We then bsed on our nuericl, results, recoend the use of when is close to, nd or Ch when is close to. The use of, is recoended when hs oderte vlues in [, ]. 6

7 Estiting Bounded Norl Men Under the LINEX Loss Function Figure. Risk functions of δ nd δch for = = 0.5 nd different vlues of. Figure. Risk functions of δ nd δch for = = 0.75 nd different vlues of. Figure 3. Risk functions of δ nd δch for = = 0.75 nd different vlues of. Figure 4. The risk functions for = = Figure 5. The risk functions for = 0.5 nd =. Figure 6. The risk functions for = nd =.5. 63

8 Vol. 4 No. Spring 03 Krinezhd J. Sci. I. R. Irn Tble. Vlues of the unique root of eqution (7) for vrious vlues of tken by estitors when, Tble. Continued Acknowledgents The uthor pprecites Prof. Ahd Prsin nd Dr. Nder Netollhi for their vluble rerks. The uthor lso would like to thnk Dr. J. J. A. Moors for sending copy of his thesis. References. Bder G. nd Bischoff W. Old nd new spects of inix estition of bounded preter. IMS Lecture Notes-Monogrph Series, 4: 5-30, Institute of theticl sttistics, Hywrd, Cliforni, USA (003).. Bickel P.J. Minix estition of the en of norl distribution when the preter spce is restricted. Ann. Sttist., 9: (98). 3. Bischoff W., Fieger W. nd Wulfert S. Minix nd Γ- inix estition of bounded norl en under LINEX loss. Sttist. Decisions, 3: (995). 4. Csell G. nd Strwdern W.E. Estiting bounded norl en. Ann. Sttist., 9: (98). 5. Chrrs A. Propriete Bysienne et dissibilite p destiteurs dns in sousenseble convex de, Ph.D. Thesis, Universite de Montrel, Montrel, Cnd, (979). 6. Chrrs A. nd vn Eeden C. Byes nd dissibility properties of estitors in truncted preter spces. Cnd. J. Sttist., 9: 34 (99). 7. DsGupt A. Byes inix estition in ultipreter filies when the preter spce is restricted to bounded convex set. Snkhy Ser. A, 47: (985). 8. Dou Y. nd vn Eeden, C. Coprisons of the perfornces of estitors of bounded norl en under squred-error loss. REVSTAT, 7: 03-6 (009). 9. Frrell R.H. Estitors of loction preter in the bsolutely continuous cse. Ann. Mth. Sttist., 35: (964). 0. Gtsonis C., McGibbon B. nd Strdern, W.E. On the estition of restricted norl en. Sttist. Prob. Lett., 3: -30 (987).. Hrris T.J. Optil controliers of nonsyetric nd nonqudrtic loss functions. Technoetrics, 34: (99).. Krinezhd A. Estiting bounded norl en reltive to squred error loss function. J. Sci., I.R. Irn, (3): (0). 3. Ktz M.W. Adissible nd inix estites of preters in truncted spces. Ann. Mth. Sttist., 3: 36 4 (96). 4. Kur S. nd Tripthi Y.M. Estiting restricted norl en. Metrik, 68: 7-88 (008). 5. Mrchnd E., Oussou I., Pyndeh A.T. nd Perron F. On the estition of restricted loction preter for syetric distributions. J. Jpn Sttist. Soc., 38: (008). 6. Mrchnd E. nd Perron F. Estiting bounded preter for syetric distributions. Ann. Inst. Sttist. Mth. 6: 5-34 (009). 7. Mrchnd E. nd Strwdern, W. E. Estition on restricted preter spces: A review, IMS Lecture Notes- Monogrph Series, 45: -4, Institute of theticl sttistics, Hywrd, Cliforni, USA (004). 8. Moors J.J.A. Indissibility of linerly invrint estitors in truncted preter spces. J. Aer. Sttist. Assoc., 76: (98). 9. Moors J.J.A. Estition in truncted preter spces, Ph.D. Thesis, Tilburg University, The Netherlnds (985). 0. Prsin A. nd Kirni S.N.U.A. Estition under LINEX function. In: An Ullh, Wn A.T.K. nd Chturvedi A. (Eds.), hndbook of pplied econoics nd sttisticsl inference, pp (00).. Rojo J. On the dissibility of cx d w.r.t. the LINEX loss function, Co. Sttist. - Theory Methods, 6: (987).. Scks J. Generlized Byes solutions in estition probles, The Ann. Mth. Sttist., 34: (963). 3. Sdooghi-Alvndi. S.M. nd Netollhi N. On the dissibility of cx d reltive to LINEX loss function. Co. Sttist. - Theory Methods, : (989). 4. Sho P. Y-S. nd Strwdern W. E. Iproving on the of positive norl en, Sttist. Sinic, 6: (996). 5. Tripthi Y.M. nd Kur S. Estiting positive norl en, Sttist. Ppers, 48: (007). 6. vn Eeden C. Estition in restricted preter spcesoe history nd soe recent developents. CWI Qurterly, 9: (996). 7. vn Eeden C. Restricted preter spce estition probles: dissibility nd inixity properties. Springer, New York (006). 8. Vrin A. Byesin pproch to rel stte ssessent. In studies in Byesin Econoetrics nd sttistics in honour of Leonrd J. Svge, Eds. Stephn E., Fienberg nd Arnold Zellner, Asterd: North-Holnd, (975). 9. Zellner A. Byesin estition nd prediction using ssyetric loss function. J. Aer. Sttist. Assoc., 8: (986). 64