Scece Joural of Appled Mathematcs ad Statstcs 06; 4(5): 9-35 http://www.scecepublshggroup.com/j/sjams do: 0.648/j.sjams.060405.6 ISSN: 376-949 (Prt); ISSN: 376-953 (Ole) Mmax stmato of the Parameter of ЭРланга Dstrbuto Uder Dfferet Loss Fuctos Lapg L Departmet of Basc Subjects, Hua Uversty of Face ad coomcs, Chagsha, Cha mal address: Llapg98@63.com o cte ths artcle: Lapg L. Mmax stmato of the Parameter of ЭРланга Dstrbuto Uder Dfferet Loss Fuctos. Scece Joural of Appled Mathematcs ad Statstcs. Vol. 4, No. 5, 06, pp. 9-35. do: 0.648/j.sjams.060405.6 Receved: August 3, 06; Accepted: September, 06; Publshed: October 8, 06 Abstract: he am of ths artcle s to study the estmato of the parameter of ЭРланга dstrbuto based o complete samples. he Bayes estmators of the parameter of ЭРланга dstrbuto are obtaed uder three dfferet loss fuctos, amely, weghted square error loss, squared log error loss ad etropy loss fuctos by usg cojugate pror verse Gamma dstrbuto. he the mmax estmators of the parameter are derved by usg Lehma s theorem. Fally, performaces of these estmators are compared terms of rsks whch obtaed uder squared error loss fucto. Keywords: Bayes stmator, Mmax stmator, Squared Log rror Loss Fucto, tropy Loss Fucto. Itroducto I relablty ad supportablty data aalyss feld, the most commoly used dstrbuto are the expoetal dstrbuto, ormal dstrbuto ad Webull dstrbuto, etc. But some practcal applcato, such as the repar tme, guaratee the dstrbuto delay tme, the above several dstrbutos does ot just as oe wsh. At ths tme ЭРланга dstrbuto was proposed as a sutable alteratve dstrbuto []. Suppose that the repar tme obeys the ЭРланга dstrbuto wth the followg probablty desty fucto (pdf) ad dstrbuto fucto respectvely: t f(; t ) = 4 t e, t 0, > 0 () t F(; t ) = ( + t) e, t 0, > 0 () Here, s the ukow parameter. It s easly to see that =, ad the the parameter s also ofte referred to as the mea tme to repar equpmet. Lv et al. [] studed the characterstc parameters, such as mea, varace ad meda ad the maxmum lkelhood estmato of ЭРланга dstrbuto was also derved. Pa et al. [] studed the terval estmato ad hypothess test of ЭРланга dstrbuto based o small sample, ad the dfferece of expoetal dstrbuto wth З рланга dstrbuto was also dscussed. Log [3] studed the estmato of the parameter of З рланга dstrbuto based o mssg data. Yu et al. [4] used the Эрланга dstrbuto to ft the battlefeld jury degree, ad establshed the smulatg model, the proposed a ew method to solve the problem the producto ad dstrbuto of battlefeld jury campag macrocosm. Log [5] studed the Bayes estmato of Эрлангa dstrbuto uder type-ii cesored samples o the bass cojugate pror, Jeffreys pror ad o formato pror dstrbutos. he mmax estmato was troduced by Abraham Wald 950, ad the mmax approach has receved great atteto ad applcato may aspects by researchers [6-9]. Mmax estmato s oe of the most aspect statstcal ferece feld. Uder quadratc ad MLINX loss fuctos, he refereces [0-3] studed the mmax estmato of the Webull dstrbuto, Pareto dstrbuto ad Raylegh dstrbutos ad Mmax dstrbuto, respectvely. Rasheed ad Al-Shareef [4] dscussed the mmax estmato of the scale parameter of Laplace dstrbuto uder squared-log error loss fucto. L [5] studed the mmax estmato of the parameter of expoetal dstrbuto based o record values. L [6] obtaed the mmax estmators of the parameter of Maxwell dstrbuto uder dfferet loss fuctos. he purpose of ths paper s to study maxmum lkelhood estmato (ML) ad Bayes estmato of the parameter of ЭРланга dstrbuto. Further, by usg Lehma s theorem we derve mmax estmators uder three loss fuctos,
30 Lapg L: Mmax stmato of the Parameter of ЭРланга Dstrbuto Uder Dfferet Loss Fuctos amely, weghted squared error loss, squared log error loss ad etropy loss fuctos.. Maxmum Lkelhood stmato LetX = ( X, X,, X ) be a sample draw from ЭРланга dstrbuto wth pdf (), ad x = ( x, x,, x ) s the observato of X. For gve sample observato, we ca get the lkelhood fucto of the parameter as follows: x x = l( ; x) = f( x; ) = 4 xe = ( 4 x) e (3) hat s ( x ) t l( ; x) = ( 4 x) e = ( 4 x) e (4) Here t = x s the observato of = X. he the log-lkelhood fucto s L[( l ; x)] = l(4 x) + l t By solvg log-lkelhood equato dll [( ; x)] = 0, d the maxmum lkelhood estmator of ca be easly derved as follows: ˆM = (5) Ad by q. (), we ca easly show that s a radom varable dstrbuted wth the Gamma dstrbuto Γ (, ), whch has the followg probablty desty fucto: t f(; t ) = t e, t > 0, > 0 Γ( ) 3. Bayesa stmato I Bayesa statstcal aalyss, loss fucto plays a mportat role the Bayes estmato ad Bayes test problems. May loss fucto are proposed Bayesa aalyss, ad squared error loss fucto s the most commo loss fucto, whch s a symmetrc loss fucto. I may practcal problems, especally the estmato of relablty ad falure rates, symmetrc loss may be ot sutable, because t s to be thought the overestmato wll brg more loss tha uderestmato [7]. he some asymmetrc loss fuctos are developed. For example, Zeller [8] proposed the LINX loss Bayes estmato, Brow [9] put forward the squared log error loss fucto for estmatg ukow (6) parameter, Dey et al. [0] proposed the etropy loss fucto the Bayesa aalyss. I ths paper, we wll dscuss the Bayes estmato of the ukow parameter of ЭРланга dstrbuto uder the followg loss fuctos: () Weghted squared error loss fucto ( δ ) L(, δ) = (7) Uder the weghted squared error loss fucto (7), the Bayes estmator of s ˆ δ [ X] = (8) [ X] () Squared log error loss fucto Squared log error loss fucto s a asymmetrc loss fucto, whch frst proposed by Brow for estmatg scale parameter. hs loss fucto ca also be foud Kapoura ad Nematollahb [] wth the followg form: L(, δ) (lδ l ) = (9) Obvously, L(, δ) as δ 0or. he loss fucto δ (9) s ot always covex, ad t s covex for e ad cocave otherwse. But the rsk fucto of ths fucto has mmum value, whch we also call t the Bayes estmator ˆ δ uder squared log error loss fucto. hat s ˆ δ = exp[ (l X)] (0) () tropy loss fucto I may practcal stuatos, t appears to be more realstc ˆ to express the loss terms of the rato. I ths case, Dey et al. [0] poted out a useful asymmetrc loss fucto amed etropy loss fucto: L( ˆ δ δ, ) = l () Whose mmum occurs at δ =. Also, ths loss fucto has bee used Sgh et al. [], Nematollah ad Motamed-Sharat [3]. he Bayes estmator uder the etropy loss () s deoted by ˆB, obtaed by ˆ δb [ ( X)] = () I ths secto, we wll estmate the ukow parameter o the bass of the above three metoed loss fuctos. We further assume that some pror kowledge about the parameter s avalable to the vestgato from past experece wth the ЭРланга model. he pror kowledge ca ofte be summarzed terms of the so-called pror destes o parameter space of. I the followg
Scece Joural of Appled Mathematcs ad Statstcs 06; 4(5): 9-35 3 dscusso, we assume the followg Jeffrey s oformatve quas-pror desty defed as, π( ), > 0 (3) d Hece, d = 0 leads to a dffuse pror ad d = to a oformatve pror. LetX = ( X, X,, X ) be a sample draw from ЭРланга dstrbuto wth pdf (), ad x = ( x, x,, x ) s the observato of X. Combg the lkelhood fucto (3) wth the pror desty (3), the posteror probablty desty of ca be derved usg Bayes heorem as follows ( ) ( ; ) π( ) t d d t h x l x e e (4) heorem. Let X = ( X, X,, X ) be a sample of Э Рланга dstrbuto wth probablty desty fucto (), ad = (,,, ) s the observato of X. t = x s the x x x x observato of = X he () Uder the weghted square error loss fucto (7), the Bayes estmator s ˆ δ X d = = (5) [ ] [ X] () he Bayes estmator uder the squared log error loss fucto (9) s Ψ( ) ˆ e δ = exp[ (l X)] = (6) () he Bayes estmator uder the etropy loss fucto () s ˆ d δb = [ ( X)] = (7) Proof. () Form quato (4), t s obvously cocluded that the posteror dstrbuto of the parameter s Gamma dstrbuto Γ( d +, t). hat us he X ~ Γ( d +, ), [ X] =, [ X] d = ( d)( d ) (8) hus, the Bayes estmator uder the weghted square error loss fucto (7) s derved as ˆ δ X d = = For the case (): By usg (4), [ ] [ X] (l X) Γ( + ) e d d = d + ( d + ) l 0 d = l Γ( d + ) l( ) = Ψ( d + ) l( ) d Where l y d + y y e Ψ ( ) = l Γ ( ) = 0 dy d Γ( ) s a Dgamma fucto. he the Bayes estmator uder the squared log error loss fucto (9) s come out to be ˆ e δ = exp[ (l X)] = Ψ( d + ) () By qs. () ad (7), the Bayes estmator uder the etropy loss fucto () s gve by ˆ d δb = [ ( X)] = 4. Mmax stmato of ЭРланга Dstrbuto hs secto wll derve the mmax extmators of Э Рланга Dstrbuto by usg Lehma s heorem, whch depeds o specfc pror dstrbuto ad loss fuctos of a Bayesa method. he Lehma s heorem s stated as follows: τ = ; Θ be a class dstrbuto F Lemma Let { } fuctos ad Dbe the estmators of. Suppose thatδ D s a Bayes estmator, whch derved o the bass of a pror π o Θ. he f the Bayes rsk fucto dstrbuto ( ) (, ) R δ equals costat o Θ, the δ s a mmax estmator of. heorem Let X = ( X, X,, X ) be a sample draw from ЭРланга dstrbuto wth pdf (), ad x = ( x, x,, x ) s the observato of = (,,, ). Suppose that X X X X observato of the statstcs = X he t = x s the () Uder the weghted square error loss fucto (7), d ˆ δ = s the mmax estmator of parameter
3 Lapg L: Mmax stmato of the Parameter of ЭРланга Dstrbuto Uder Dfferet Loss Fuctos () Uder the squared log error loss fucto, Ψ( d + ) ˆ e δ = s the mmax estmator of parameter 3 d () Uder the etropy loss fucto, ˆB δ = s the mmax estmator of parameter Proof. o use Lehma s heorem for the proof of the results. We eed calculate the rsk fucto of Bayes estmators ad prove these rsk fuctos are costats. For the case (), we ca derve the rsk fucto of the Bayes estmator ˆ δ uder the weghted square error loss fucto (7) as follows: ( ) (, ˆ δ ) d R = =, = ( ) ( ) ( ) ( ) + L L d d From equato (6), we have ~ Γ (, ), the we have Cosequetly, = = ( )( ) ( ), ( ) d ( d ) R( ) = ( ) ( ) d d + = + ( )( ) ( )( ) he, for Bayes estmator ˆ δ, the rsk fucto R( ) s a costat o the parameter So, Accordg to Lemma, ˆ δ s the mmax estmator of parameter uder the weghted square error loss fucto (7). For the case (). he rsk fucto of the Bayes estmator ˆ δ s By ~ Γ (, ), we ca easly get the result he R L ( ) ( ˆ ) ( ˆ ) ˆ ˆ =, δ = l δ l = [l δ ] l l[ δ ] + (l ) (l ) = Ψ( ) l [l ˆ δ ] = ( Ψ( d + ) l ) = Ψ( d + ) [ Ψ( ) l ] = Ψ( d + ) Ψ ( ) + l ˆ δ = Ψ d + = Ψ d + Ψ d + + [l ] [ ( ) l( )] ( ) ( ) [l( )] [(l ) ] Let Y~ Γ (,), the we ca prove that Y = ~ Γ (,). he dervatve of Ψ ( ) s he From above results, we ca the fact Further, we have (l y) y e (l y) y e Ψ = Ψ = Ψ Γ( ) Γ( ) ( ) y ( ) [(l ) ] ( ) 0 y dy dy Y 0 Ψ ( ) + Ψ ( ) = [(l Y) ] = [(l( ) + l ) ] = [(l( )) ] + l (l( )) + (l ) = [(l( )) ] + l ( Ψ( ) l ) + (l ) [(l ) ] = Ψ ( ) + Ψ ( ) l Ψ ( ) + (l ) ˆ δ = Ψ d + Ψ d + + [l ] ( ) ( ) (l( )) [(l ) ]
Scece Joural of Appled Mathematcs ad Statstcs 06; 4(5): 9-35 33 = Ψ ( d + ) Ψ( d + )[ Ψ( ) l ] + Ψ ( ) + Ψ ( ) l Ψ ( ) + (l ) ( ) ˆ ˆ R = [l δ ] l l[ δ ] + (l ) = Ψ ( d + ) Ψ( d + )[ Ψ( ) l ] + Ψ ( ) + Ψ ( ) l Ψ( ) + Ψ + Ψ + + = Ψ + Ψ Ψ + + Ψ + Ψ (l ) l [ ( d ) ( ) l ] (l ) ( d ) ( ) ( d ) ( ) ( ) he R ( ) s also a costat about the parameter. So, accordg to Lemma, we kow that, ˆ δ s a mmax estmator for parameter uder the squared log error loss fucto. For the case (). he rsk fucto of the Bayes estmator ˆB δ ca be obtaed as follows: ( ) ˆ (, δ ) d d d R = L B = l + l( ) l (l ) = d + + + d d = l( d) + l + Ψ( ) l + = l( d) + Ψ ( ) + he R( ) s also a costat about the parameter. So, accordg to lemma, we kow that, ˆB δ s a mmax estmator for the parameter uder the etropy loss fucto. 5. Performaces of Bayes stmators o llustrate the performace of these Bayes estmators, squared error loss fucto L( ˆ, ) = ( δ ) s used as a loss fucto to compare them. We oter( ˆ δ ), ( ˆ R δbl) adr( ˆ δb) are the rsk fuctos of estmators ˆ δ, ˆ δ ad BL ˆB δ relatve to the squared error loss, respectvely. hey ca be easly derved as follows: ˆ ˆ ˆ δ δ δ d R L ( ) = [ (, )] = [( ) ] = [( ) ] ( d ) ( d ) ( ) ( ) () d d = + = + 0 f t dt, t t ( )( ) Ψ( d + ) Ψ( d + ) Ψ( d + ) ˆ ˆ e e e R( δ) = [( δ ) ] = [( ) ] = + ( )( ) ˆ ˆ ˆ ( ) ( ) δ δ δ d = = = = d d + B B B R( ) [ L(, )] [( ) ] [( ) ] ( )( ) Let L, L ad L 3 are the rato of the rsk fuctos to, whch are plotted Fgs. -4 wth dfferet sample szes, (=0, 0, 30, 50) Fgure. Performace of estmators wth =0. Fgure. Performace of estmators wth =0.
34 Lapg L: Mmax stmato of the Parameter of ЭРланга Dstrbuto Uder Dfferet Loss Fuctos helpful commets ad suggestos of the revewers, whch have mproved the presetato. Refereces [] Lv H. Q., Gao L. H. ad Che C. L., 00. Э рланга dstrbuto ad ts applcato supportablty data aalyss. Joural of Academy of Armored Force geerg, 6 (3): 48-5. [] Pa G.., Wag B. H., Che C. L., Huag Y. B. ad Dag M.., 009. he research of terval estmato ad hypothetcal test of small sample of З рланга dstrbuto. Applcato of Statstcs ad Maagemet, 8 (3): 468-47. Fgure 3. Performace of estmators wth =30. [3] Log B., 03. he estmatos of parameter from З рланга dstrbuto uder mssg data samples. Joural of Jagx Normal Uversty (Natural Scece), 37 (): 6-9. [4] Yu C. M., Ch Y. H., Zhao Z. W., ad Sog J. F., 008. Mateace-decso-oreted modelg ad emulatg of battlefeld jury campag macrocosm. Joural of System Smulato, 0 (0): 5669-567. [5] Log B., 05. Bayesa estmato of parameter o Эрлангa dstrbuto uder dfferet pror dstrbuto. Mathematcs Practce & heory, (4): 86-9. [6] Jao J., Vekat K., Ha Y., Wessma, 05. Mmax estmato of fuctoals of dscrete dstrbutos. Iformato heory I rasactos o, 6 (5): 835-885. [7] Gao C., Ma Z., Re Z., Zhou H. H, 04. Mmax estmato sparse caocal correlato aalyss. Aals of Statstcs, 43 (5): 905-9. Fgure 4. Performace of estmators wth =50. From Fgure to Fgure 4, we kow that o of these estmators s uformly better that other estmators. he practce, we recommed to select the estmator accordg to the pror parameter value d whe assumg the quas-pror as the pror dstrbuto. 6. Cocluso hs paper derved Bayes estmators of the parameter of Э Рланга dstrbuto uder weghted squared error loss, squared log error loss ad etropy loss fuctos. Mote Carlo smulatos show that the rsk fuctos of these estmators, defed uder squared error loss fucto, are all decrease as sample sze creases. he rsk fuctos more ad more close to each other aehe the sample sze s large, such as >50. Ackowledgemet hs study s partally supported by Natural Scece Foudato of Hua Provce (No. 05JJ3030) ad Foudato of Hua ducatoal Commttee (No. 5C08). he author also gratefully ackowledge the [8] Koga M. M., 04. LMI-based mmax estmato ad flterg uder ukow covaraces. Iteratoal Joural of Cotrol, 87 (6): 6-6. [9] chraka.., Zhuk S., 05. A macroscopc traffc dataassmlato framework based o the Fourer Galerk method ad Mmax estmato [J]. I rasactos o Itellget rasportato Systems, 6 (): 45-464. [0] Roy, M. K., Podder C. K. ad Bhuya K. J., 00. Mmax estmato of the scale parameter of the Webull dstrbuto for quadratc ad MLINX loss fuctos, Jahagragar Uversty Joural of Scece, 5: 77-85. [] Podder, C. K., Roy M. K., Bhuya K. J. ad Karm A., 004. Mmax estmato of the parameter of the Pareto dstrbuto for quadratc ad MLINX loss fuctos, Pak. J. Statst., 0 (): 37-49. [] Dey, S., 008. Mmax estmato of the parameter of the Raylegh dstrbuto uder quadratc loss fucto, Data Scece Joural, 7 (): 3-30 [3] Shadrokh, A. ad Pazra H., 00. Mmax estmato o the Mmax dstrbuto, Iteratoal Joural of Statstcs ad Systems, 5 (): 99-8. [4] Rasheed H. A., Al-Shareef. F, 05. Mmax estmato of the scale parameter of Laplace dstrbuto uder squared-log error loss fucto. Mathematcal heory & Modelg, 5 (): 83-93.
Scece Joural of Appled Mathematcs ad Statstcs 06; 4(5): 9-35 35 [5] L L. P., 04. Mmax estmato of the parameter of expoetal dstrbuto based o record values. Iteratoal Joural of Iformato echology & Computer Scece, 6 (6): 47-53. [6] L L. P., 06, Mmax estmato of the parameter of Maxwell dstrbuto uder dfferet loss fuctos, Amerca Joural of heoretcal ad Appled Statstcs, 5 (4): 0-07. [7] L X., Sh Y., We J., Cha J., 007. mprcal Bayes estmators of relablty performaces usg LINX loss uder progressvely ype-ii cesored samples [J]. Mathematcs & Computers Smulato, 007, 73 (5): 30-36. [8] Zeller, A., 986. Bayesa estmato ad predcto usg asymmetrc loss fucto. Joural of Amerca statstcal Assocato, 8 (394): 446-45. [0] Dey, D. K., Ghosh M. ad Srvasa C., 987. Smultaeous estmato of parameters uder etropy loss, J. Statst. Pla. ad Ifer., 5 (3): 347-363. [] Kapoura A. ad Nematollahb N., 0. Robust Bayesa predcto ad estmato uder a squared log error loss fucto. Statstcs & Probablty Letters, 8 (): 77-74. [] Sgh S. K., Sgh U. ad Kumar D., 0. Bayesa estmato of the expoetated Gamma parameter ad relablty fucto uder asymmetrc loss fucto. RVSA, 9 (3): 47-60. [3] Nematollah N., Motamed-Sharat F., 009. stmato of the scale parameter of the selected Gamma populato uder the etropy loss fucto. Commucato Statstcs- heory ad Methods, 38 (7): 08-. [9] Brow L., 968. Iadmssblty of the usual estmators of scale parameters problems wth ukow locato ad scale parameters. Aals of Mathematcal Statstcs, 39 (): 9-48.