Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method S. Lertprapa, and M. Tensuwan,,* Department of Mathematcs, Faculty of Scence Burapha Unversty, Chonbur, Thaland Department of Mathematcs, Faculty of Scence Mahdol Unversty, Bangkok, Thaland Centre of Excellence n Mathematcs, CHE S Ayutthaya Road, Bangkok 0400, Thaland * Correspondng author: e-mal: montp.te@mahdol.ac.th Copyrght 0 S. Lertprapa and M. Tensuwan. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract In ths paper we study the problems of estmaton for the populaton varance, n -parameter exponental dstrbuton. In -parameter exponental dstrbuton, () the estmates parameters are ˆ n ( x x ) mmse = and ˆ ( ) = 0 + β () 0 ( n+ )( n+ ) (( ) ) q q x x where q =±, ±. The purpose of ths study s to compare the estmators of based on the Multple Crtera Decson Makng (MCDM) procedure to obtan the best estmator. The results reveal that the best estmator of n -parameter exponental dstrbuton s ˆ ( ). Mathematcs Subject Classfcaton: 6C5, 6G05
08 S. Lertprapa and M. Tensuwan eywords: Exponental dstrbuton, multple crtera decson makng, varance estmator. Introducton The exponental dstrbuton s necessary n lfe testng and relablty theory. In - parameter exponental dstrbuton s formulated as -( x- γ ) f( x) = exp, for x > γ, > 0 where s a scale parameter and γ s a locaton parameter. The locaton parameter s nterpreted as the mnmum tme before whch no falure occurs; the scale parameter s the mean lfe, measured from the locaton parameter (ouroukls, 994). The estmaton of parameters s an nterestng problem n statstcal nference. Let x, x,..., xn be a random sample of sze n from an exponental populaton. Pandey and Sngh (997) obtaned the mnmum mean squared error (MMSE) estmator of varance n -parameter exponental dstrbuton as followng. ˆ MMSE n ( x x() ) = ( n+ )( n+ ), where x() mn{ x }. = Tracy et al (996) proposed a class of shrnkage estmators for the populaton varance gven pror nformaton 0. In -parameter exponental dstrbuton, the estmators are ˆ q (( ) ) n Γ ( n+ q) ( q) = 0 + βq x x () 0 where βq = and q s Γ ( n + 4q ) non-zero real number. In ths artcle the estmators of the populaton varance n -parameter are compared wheren q =±, ±. Based on mean square errors (MSEs), we use the Multple Crtera Decson Makng (MCDM) method for rankng those estmators from the best to the worst. Ths method s brefly descrbed n Secton. Secton contans the MSEs of each estmator and Secton 4 descrbes the man results of ths paper.
Comparson of populaton varance estmators 09. A bref descrpton of MCDM procedure The Multple Crtera Decson Makng (MCDM) s a technque that can be used for assessments and decson makng where the multple crtera are presented (Zeleny, 98). A typcal MCDM problem nvolves a number of alternatves to be selected and a number of crtera or ndcators for assessng these alternatves. Each alternatve has a value for each ndcator and based on these values, the alternatves can be selected. Lertprapa et al (004) presented a comprehensve revew on the MCDM procedure as follows. For a dscrete data matrx X = ( xj ): N, where xj ' s represents the rsk of the th source for the j th category, t s necessary to compare the rows smultaneously wth respect to all the N columns. The MCDM s a novel statstcal procedure to ntegrate the multple rsks ( x, x,, xn), =,,..., for the th alternatve nto a sngle meanngful and overall rsk factor (Flar et al, (999); Matra et al, 00)). The estmators are then compared on the bass of these ntegrated rsk factors. The rsk ntegraton s done by defnng an deal row (IDR) wth the smallest observed value for each column as IDR = (mn x,...,mn x ) = ( u,..., u ) N N and a negatve-deal row (NIDR) wth the largest observed value for each column as NIDR = (max x,...,max x ) = ( v,..., v ). N N For any gven row, now the dstance of each row s computed from the deal row and from the negatve deal row based on a sutably chosen norm. It s computed under L -norm as N L (, IDR) = xj uj wj /( xj ) j= =, N L (, NIDR) = xj vj wj /( xj ) j= = where w j s s approprate weght. The varous rows are now compared based on the overall ndex whch s computed as L (, IDR) L ( Index ) =, =,...,. L(, IDR) + L(, NIDR) Smlarly, under L -norm,
0 S. Lertprapa and M. Tensuwan / N (, ) = ( j j ) j /( j ), j= = L IDR x u w x / N (, ) = ( j j ) j /( j ), j= = L NIDR x v w x and the rows are compared based on L (, IDR) L ( Index ) =, =,...,. L(, IDR) + L(, NIDR) A contnuous verson of ths setup would nvolve ' xj s where the ndex j would vary contnuously. In the context of ths problem, fve estmators of varance n - parameter exponental dstrbuton are compared (see Secton 4). In each estmates obvously = 5, so estmators for varous values of would be redefned as ' xj s s chosen to represent the mean square of errors of those = /, 0< <. In ths case 0 L and L -norm [ ] L (, ) ( ) ( ) ( ) IDR = x u w d [ ] L (, ) ( ) ( ) ( ) NIDR = v x w d L (, IDR) = [ x ( ) u( )] w( ) d / (, ) = [ ( ) ( )] ( ), = = L NIDR v x w d where u( ) mn{ x ( )} and v( ) max{ x ( )}. Lertprapa et al. (004) had shown that based on L -norm the th estmator better th than the j one f x ( ) w( ) d < x ( ) w( ) d j /
Comparson of populaton varance estmators and based on L -norm the th estmator better than the th j one f [ ( ) ( )] ( ) [ j( ) ( )] ( ) < [( ) ( )] ( ) [( ) j( )] ( ) x u w d x u w d. v x w d v x w d Therefore we can rank the estmators by usng these values whch are called reduced-ndex. Based on L -norm, t s computed as * L ( ) ( ) ( ) ndex x w d and based on L -norm, t s computed as L ( ndex ) = * = () [ ( ) ( )] ( ) [( ) ( )] ( ) x u w d. () v x w d There are three choces of weght functon. The frst weght functon s defned by w ( ) =. Refer to Flar et al. (999), the second one denoted by w ( ), s based on the noton of entropy among x ( ), =,,, for varous values of, and the thrd one, denoted by w ( ), s based on the coeffcent of sample varaton of x ( ); =,,, for varous values of. It turns out that φ( ) w ( ) = ( φ( )) d x( ) x( ) where φ ( ) = log log = x ( ) ( ) x = = sd and w =, x( )
S. Lertprapa and M. Tensuwan where and sd = x ( ) x( ) = = x( ) = x ( ). = /. Mean Squared Errors (MSEs) 0 Let =. Refer to Tracy et al (996), the MSEs of ˆ MMSE and ˆ ( p ) are gven as follows. 4 ˆ (n + ) MSE( MMSE ) = ( n+ )( n+ ) In term of 4 n n ( q) = q + q q MSE( ˆ ) ( β ) n β n β ( n )( )( ) ( ) n + n + n β β q q n n. + + ( ˆ 4 MSE ), a common term ( ) s gnored, so the results are n the form ˆ (n + ) MSE( MMSE ) = ( n+ )( n+ ) and ˆ n n MSE( ( q) ) ( β ) q q q n β n β = + n n n n + + ( )( + )( + ) ( ) β β q q n n. We now let as l and MSEs of ˆ MMSE, ˆ, ˆ, ˆ, ˆ as TT, TT, TT, TT 4 and TT 5 respectvely. ( ) ( ) () ()
Comparson of populaton varance estmators 4. Man Results In ths secton, MSEs of each estmator are computed and compared based on the MCDM. The range 0 < l < and n = 0, 5, 0, 5 are consdered for MSEs n four cases as followngs. Case I: n = 0 T T( l) = 7/ TT () l = ( 555009 /56500) l ( 5558/ 78500) l + 889407/ 9065000 TT ( l) = (49 /00) l ( 5/ 500) l + 47/ 5000 TT l 4() = ( 64 /089) l ( 56/6) l+ 7/ TT5 ( l ) = ( 774/ 986) l ( 669 / 997) l+ 407/ 6648 Ther graphcal patterns for n = 0 are presented n Fgure. Case II: n = 5 TT () l = /6 TT ( l) = (0869 / 5565) l (96998 / 57475) + 05900097 /865475 TT l l () l = (59 / 05) (688 / 075) + 440447 / 68475 TT4 () l = (09 / 7984) l (457 /8496) l + /6 TT5 ( l ) = (4747/096) l (59 / 404) l + 656 / 758 Ther graphcal patterns for n = 5 are presented n Fgure. Case III: n = 0 TT ( l) = 4/ TT ( l ) = (88898009 / 400000000) l 589007/ 4000000000l +840450/60000000000 0 TT () l = (4 / 5) l - (4 /5) l + 550/00000 TT l l 4 () l = (96/ 56) (54 / 56) + 4/ TT5 () l = (47009 / 804609) l (5898 / 804609) l + 6/ 8940 l
4 S. Lertprapa and M. Tensuwan Ther graphcal patterns for n = 0 are presented n Fgure. Case IV: n = 5 TT ( l) = 7 /7 TT () l = (88566550/558789065) l 44868580898 / 846976565l + 498546907/ 8485790565 TT ) l l 5 ( l = (68/565) 9098 / 9065 + 4800 / 44406 TT ( ) l l 7 TT 4 l = (599 / 49804) (09 / 4067) + 7 / 5 ( l) = (5540649 / 8066504) l (947448 / 960849) + 467855/ 5840796 Ther graphcal patterns for n = 5 are presented n Fgure 4. l Fgure : Graphcal llustraton of MSEs of ˆ for n = 0.
Comparson of populaton varance estmators 5 Fgure : Graphcal llustraton of MSEs of ˆ for n = 5. Fgure : Graphcal llustraton of MSEs of ˆ for n = 0.
6 S. Lertprapa and M. Tensuwan Fgure 4: Graphcal llustraton of MSEs of ˆ for n = 5. 4. IDR and NIDR For 0< l <, the ntersecton ponts of the fve graphs s happened, there are only sx ponts used to consder the deal row ( a, =,,...,6). Smlarly, there are two ponts for consderng the negatve-deal row ( b, =,). These ponts separate the nterval of l nto several ntervals as 0< a < a < a < a4 < a5 < a6 < ; 0< b < b <. Moreover, the ntersecton ponts are shown n Table whle the deal row (IDR) and negatve-deal row (NIDR) are shown n Table and Table, respectvely. Table The ntersecton ponts of IDR and NIDR n a a a a 4 a 5 a 6 b b 0 0.46 0.59 0.77.860.49.6660 0.448.5657 5 0.06 0.5549 0.75.7.468.740 0.4975.50 0 0.066 0.574 0.6959.754.4674.7806 0.59.4984 5 0.980 0.574 0.68.0.49.8048 0.577.497
Comparson of populaton varance estmators 7 Table IDR for each nterval n 0 < l < a a < l < a a < l < a a < l < a4 a4 < l < a5 a5 < l < a6 a6 < l < u () l u () l u () l u () l u () l u () l u () l 4 5 6 7 0 TT () l TT () l TT () l TT () l TT () l TT () l TT () l 4 5 5 4 5 TT () 4 TT () TT () 5 TT () TT () 5 TT () TT () 4 0 TT () 4 TT () TT () 5 TT () TT () 5 TT () TT () 4 5 TT () 4 TT () TT () 5 TT () TT () 5 TT () TT () 4 n Table NIDR for each nterval 0 < l < b b < l < b b < l < v () l v () l v () l 0 TT () l TT () l TT () l 5 TT () TT () TT () 0 TT () TT () TT () 5 TT () TT () TT () 4. Analyss based on the L and L -norm and For =,,, 4, 5; applyng equatons () and () we get * L ( ndex) = TT() l w() l dl 0
8 S. Lertprapa and M. Tensuwan * ( ) = [ () ()] () / [ () ()] () 0 0 a a = 4 + 0 a a a4 + 5 + a a a5 a6 + [ TT ( l) TT5 ( l)] w( l) dl + a4 a5 b + a6 0 b b b L ndex TT l u l w l dl v l TT l w l dl [ TT( l) TT ( l)] w( l) dl [ TT( l) TT ( l)] w( l) dl [ TT( l) TT ( l)] w( l) dl [ TT( l) TT ( l)] w( l) dl [ TT ( l) TT ( l)] w( l) dl [ TT() l TT()] l wl () dl / [ TT() l TT()] l wl () dl + [ TT( l) TT( l)] wl ( ) dl+ [ TT( l) TT( l)] wl ( ) dl. 4. Comparson of the estmators Reduced ndces are computed based on L and L norm usng w, w and w and we demonstrated them and rank of varance estmators n Table 4. 5. Concluson The MCDM method s used for comparson the estmators of varance populaton n - parameter exponental dstrbuton. In - parameter exponental dstrbuton, based on L and L norm, we conclude that ˆ ( ) s the best estmator whle ˆ (), ˆ (), ˆ ( ) and ˆ MMSE are lower n rank respectvely for n = 0. For n = 5, 0 and 5, based L and L norm, ˆ ( ) s the best estmator whle the worst one s ˆ ( ) whle ˆ (), ˆ (), ˆ MMSE and ˆ ( ) are lower n rank respectvely under the three weghts w, w and w.
Comparson of populaton varance estmators 9 Table 4 Results of analyss based on L and L norms usng w, w and w for n = 0, 5, 0, and 5 n ˆ 0 5 0 5 ˆ MMSE w w w L L L L L L average 0.55 0.549 0.797 0.9848 0.668 0.988 0.6779 5 ˆ ( ) 0.555 0.9598 0.06 0.0058 0.4589 0.948 0.47 4 ˆ ( ) 0.8 0.04 0.49 0.00 0.96 0.074 0.50 ˆ () 0.4055 0.075 0.55 0.74 0.460 0.87 0.07 ˆ 0.657 0.088 0.7 0.00 0.6 0.09 0.697 () ˆ MMSE 0.478 0.0560 0.579 0.8095 0.508 0.9 0.47 4 ˆ ( ) 0.60 0.9977 0.508 0.4 0.5905 0.975 0.6844 5 ˆ ( ) 0.9 0.0056 0.09 0.070 0.08 0.06 0.644 ˆ () 0.965 0.05 0.4697 0.4046 0.476 0.095 0.05 ˆ 0.65 0.09 0.64 0.047 0.494 0.054 0.90 () ˆ MMSE 0.457 0.04 0.4978 0.46 0.4585 0.054 0.90 4 ˆ ( ) 0.64 0.9990 0.5844 0.955 0.64 0.9977 0.7999 5 ˆ ( ) 0.0 0.004 0.4 0.08 0.89 0.0079 0.679 ˆ () 0.96 0.085 0.476 0.498 0.966 0.009 0.66 ˆ 0.645 0.000 0.509 0.0580 0.6 0.09 0.977 () ˆ MMSE 0.4489 0.095 0.4554 0.079 0.46 0.070 0.460 4 ˆ ( ) 0.654 0.999 0.688 0.9946 0.64 0.999 0.8 5 ˆ ( ) 0.6 0.0046 0.9 0.064 0.76 0.0056 0.696 ˆ () 0.999 0.076 0.4047 0.058 0.887 0.074 0.5 ˆ 0.708 0.079 0.684 0.07 0.70 0.009 0.98 () rank
0 S. Lertprapa and M. Tensuwan ACNOWLEDGEMENTS We would lke to thank the Centre of Excellence n Mathematcs, the Commsson on Hgher Educaton, Thaland for the research supported. References [] S. ouroukls, Estmaton n the -parameter exponental dstrbuton wth pror nformaton, IEEE Transactons on Relablty 4(), 994, 446-450. [] B. N. Pandey and J. Sngh, A note on the estmaton of varance n exponental densty, B 9, 997, pp 94 98. [] D.S. Tracy, H.P. Sngh and H.S. Raghuvansh, Some Shrnkage Estmators for the Varance of Exponental Densty, Mcroelectron Relablty 6(5) 996, 65-655. [4] M.Zeleny, Multple crtera Decson Makng, Mc Graw-Hll, New York, USA, 98. [5] S. Lertprapa, M. Tensuwan, and B. Snha, On a comparson of two standard estmates of bnomal proporton based on Multple Crtera decson makng method, Journal of Statstcal Theory and Applcatons, 004, 4 49. [6] J.A. Flar, N.P. Ross and M.L. Wu, Envronmental Assessment Based on Multple Indcators, Techncal Report, Department of Appled Mathematcs, Unversty of South Australa, 999, 4-49. [7] R. Matra, N.P. Ross, and B. Snha, On some aspects of Data ntegraton Technques wth Applcatons, Techncal report, Department of Mathematcs and Statstcs. UMBC, USA, 00. Receved: February, 0