LET a random variable x follows the two - parameter

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1 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, Shinkage Bayesian Appoach in Item - Failue Gamma Data In Pesence of Pio Point Guess Value Gyan Pakash Abstact Pesent aticle investigated some popeties of the Bayes Shinkage estimatos fo Two - Paamete Gamma distibution. Based on both the symmetic asymmetic loss functions, popeties in tems of elative efficiency have studied fo Item - failue censoed data. Index Tems Bayes estimato, Bayes Shinkage estimato, Squaed eo loss function SELF, LINEX loss function LLF, Item - failue censoed data. MSC 2010 Codes 62C10, 62C12 I. INTRODUCTION LET a om vaiable x follows the two - paamete Gamma distibution, then the pobability density function is f x;λ, = 1 Γλ x λ 1 λ x ;x > 0, > 0,λ > 0. 1 The paamete λ ae known as the shape scale paamete espectively. Fo λ = 1, two - paamete Gamma distibution is a Negative Exponential distibution. It is well known that, the Bayes estimate of unknown paamete unde SELF is simply the posteio mean. The SELF is often used also because it does not lead to extensive numeical computations but seveal authos like Vaian 1975, Bege 1980, Zellne 1986 othes have ecognized that the inappopiateness of using the symmetic loss function in vaious estimation pediction poblems. The example of a well - known asymmetic loss function is LINEX loss function LLF is poposed fist by Vaian The invaiant vesion of LLF Singh et. al is defined fo any paamete as L = e a a 1; a 0 = ˆ. 2 The sign magnitude of a epesents the diection degee of asymmety espectively. The positive negative value of a is used when oveestimation is moe less seious than undeestimation. L is appoximately squae eo almost symmetic if a nea to zeo. Let us assume hee that x 1,x 2,...,x n ae n items Gyan Pakash is a Assistant Pofesso in the Depatment of Community Medicine, S.N. Medical College, Aga, India. ggyanji@yahoo.com ae put to life test test is teminated when fist n items have failed. This censoing citeion is known as Item - failue censoing citeion. Suppose that x 1,x 2,...x x be the fist odeed obsevations, then the joint pobability density function is obtained as n! 1 f x = n!γλ λ x λ 1 i T ; i=1 { 3 whee T = 1 i=1 x } i +n x. The maximum likelihood ML estimato fo the paamete is ˆ ML = T λ. When pio infomation about paamete is available in fom of a pio point guess value, a pocedue makes uses of this pio infomation by shinking the usual estimatos towads a guess value of paamete with the help of a shinkage facto k0 k 1. Accoding to his belief in guess value, an eimente specifies the values of shinkage facto. Let 0 be the pio point guess value of unknown paamete, then shinkage estimato Thompson 1968 is defined as Ŝ = kˆ+1 k 0. 4 Hee, ˆ be any estimato of the paamete. In pesent pape, we poposed some Bayes shinkage estimatos fo unknown scale paamete unde both symmetic asymmetic loss function. The popeties of the estimatos have studied in tems of elative efficiencies. A simulation has also been caied out fo numeical intepetation. II. BAYES SHRINKAGE ESTIMATORS UNDER CONJUGATE PRIOR The likelihood function fo a given sample x 1,x 2,...,x n of size n fom 1, is given as Lx; = g;t n.hx; 5 whee g;t n = nλ e ˆT/, hx = 1 Γλ ˆT = n i=1 x i. n n i=1 xλ 1 i

2 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, Hence, ˆT is a sufficient statistic fo the consideed undelying model. Hence, thee exists a family of conjugate pio which can be obtained by looking at g;t n. Hence, the inveted Gamma distibution Raiffa & Schlaife 1961 is taken as the conjugate pio density fo paamete is defined fo the paametes α,β as π = βα Γα α+1 1 β ; > 0,α > 0,β > 0. 6 The pio mean pio vaiance ae βα 1 1 ;α > 1 β 2 α 1 2 α 2 1 ;α > 2 espectively. Based on Bayes theoem, the posteio density fo unknown paamete when pio infomation about is consideed as π, is given by = π = f x.π f x.πd 1 λ i=1 xλ 1 i 1 λ i=1 xλ 1 i T T. 1 α+1. 1 π T +β α 1 = α1+1. Γα 1 T +β β α+1 β d ;α 1 = λ+α, > 0. 7 The Bayes estimato ˆ S unde squaed eo loss function SELF fo the paamete is simply the posteio mean obtained as = ˆ S =.π d T +β α 1 T +β α1 d Γα 1 ˆ S = T +β ; λ > 1 α. 8 α 1 1 The guess value of unknown paamete 0 is involved in shinkage estimato 4. Thee ae thee diffeent choices available in liteatue fo selecting pio point guess value fo any given paamete. One may use a pio point guess value ad hock, fom pevious eiments o some eliable souces. Second choice is, consideed a pio distibution having pio mean aound 0, instead of pio point guess value 0. Shike & Nalawade 2003 pesent a new citeion fo obtaining optimum value of paamete in tems of guess value 0 based on Bayes estimation. Following Shike & Nalawade 2003 we have E ˆS = 0 β = α E T β = 0 1 ; 1 = α It is noted hee that 2T 1 distibuted as chi - squae with 2 degee of feedom. Substitutingβ fom 9 in 8, we get an estimato of the fom 4, named as Bayes Shinkage estimato is given by SH = λ 1 T +1 λ 1 ; λ 1 = Now, the Bayes estimato ˆ L unde LLF fo paamete is obtained by simplifying following equality α1+2 = e a T +β aˆ L d α1+2 T +β d T +β aˆ L T +β ˆ L = φ T +β ;φ = 1 a α1+1 = e a 1 a. 11 α 1 +1 On simila line the paamete β, is optimized by Bayes estimato as E ˆL = 0 β = 0 φ E T 1 φ β = φ Hence, the Bayes shinkage estimato unde LLF is LH = λ 2 T +1 λ 2 0 ;λ 2 = φ. 13 The isk unde SELF LLF fo both estimatos ae summaized in following lines Risk unde SELF { SH = 2 +1 λ 2 1 +δδ 2 } + δ λ 1 ;δ = 0

3 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, { LH = 2 +1 λ 2 2 +δδ 2 } + δ λ 2 Risk unde LLF SH = e aδ1 λ1 1 aλ 1 1+a1 δ1 λ 1 { } a LH = α a1 λ 2δ 1 1+a1 δ1 λ 2 The essions of elative efficiencies fo SH with espect to LH unde SELF LLF ae given as RE SELF LH, SH = SH LH RE LLF LH, SH = SH LH. The essions of elative efficiencies ae the functions of, a, λ, δ pio paametes α β. The values of α β ae chosen so as to keep the pio vaiance unity. Thus pe assumed set of pio paametes ae α,β = 3,2,6,10,11,30. Gamma shape paamete is consideed to be fixed at λ = The censoed sample size is consideed hee = the value of shape paamete of LLF is a = 0.25,0.50. By the help of pio paametes α β, geneate fom 6. Based on geneated some pe assumed pio point guess value, a atio of δ obtained unde 10,000 simulation un. Using δ othe pe assumed values, the elative efficiencies have been obtained pesented in Tables espectively. Using Tables 01 02, the shinkage estimato LH pefoms unifomly bette than the shinkage estimato SH unde both loss citeion fo all selected paametic set of values fo small pio paametes. Fo lage pio paametic values the shinkage estimato LH pefoms well only in vicinity of the tue value. Futhe, the elative efficiencies decease as censoed sample size inceases. It is also noted hee that the pio paametes α β incease, simila tend has also been seen. An inceasing tend has seen also hee, when shape paamete of LLF inceases. The efficiency attain maximum at δ = III. BAYES SHRINKAGE ESTIMATORS UNDER QUASI PRIOR The situation whee the life eseaches have no pio infomation about unknown paamete, one may use unifom, quasi o impope pio. We consideed hee a class of quasi pio defined as π 1 = 1 d p ; > 0,p > 0,d > It is noted that fo d p 0 0 π 1 = 1 NonInfomativePio 0 1 π 1 = e 1/ 1 0 π 1 = 1 DiffusePio The pio mean pio vaiance ae given espectively as Γd 2 p d 2 ; d 2 Γd 3 p 2d 4 p d 1 d 3Γd 2 ;d > 3. Using pio density π 1, the posteio density fo is obtained as = π1 = f x.π 1 f x.π 1d 1 λ i=1 xλ 1 i 1 λ i=1 xλ 1 i T +p T T. 1 p d. 1 d p d π1 = T +p d 1 d1+1. Γd 1 ; d 1 = λ+d 1, > On Simila line the Bayes estimatos unde SELF LLF fo paamete ae obtained as ˆ S1 = T +p d 1 1 ˆ L1 = φ 1 T +p ;φ 1 = 1 a ; λ+d > a. 17 λ+d Similaly, the Bayes shinkage estimatos unde SELF LLF ae obtained as by minimizing the Bayes isk SH1 = λ 3 T +1 λ 3 0 ;λ 3 = d LH1 = λ 4 T +1 λ 4 0 ;λ 4 = φ The isk unde SELF LLF fo both estimatos ae summaized in following lines

4 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, Risk unde SELF SH1 = 2 { +1 λ 2 3 +δδ 2 } + δ λ 3 { LH1 = 2 +1 λ 2 4 +δδ 2 } + δ λ 4 Risk unde LLF SH1 = e aδ1 λ3 1 aλ 3 1+a1 δ1 λ 3 { } a LH1 = λ+d +a1 λ 4δ 1 1+a1 δ1 λ 4 [2] H. Raiffa R. Schlaife, Applied statistical decision theoy, Gaduate School of Business administation, Havad Univesity, Boston, MA, [3] D. C. Singh, G. Pakash P. Singh, Shinkage testimatos fo the shape paamete of Paeto distibution using LINEX loss function Communications in Statistics - Theoy Methods, 36, , [4] D. T. Shike K. T. Nalawade, Estimation of the paamete of Binomial distibution in pesence of pio point infomation Jounal of the Indian Statistical Association, 41, 1, , [5] J. R. Thompson, Some shinkage techniques fo estimating the mean, Jounal of the Ameican Statistical Association, 63, , [6] H. R. Vaian, A Bayesian appoach to eal estate assessment. In studies in Bayesian econometics statistics in hono of L. J. Savage, Eds S. E. Feinbege A. Zellne, Amstedam, Noth Holl, , [7] A. Zellne, Bayesian estimation pediction using asymmetic loss function, Jounal of the Ameican Statistical Association, 81, , The essions of elative efficiencies fo SH1 with espect to LH1 unde SELF LLF ae given as SH1 RE SELF LH1, SH1 = LH1 RE LLF LH1, SH1 SH1 =. LH1 The essions of elative efficiencies ae the functions of, a, λ, δ pio paametes d p. The selected quasi families of pio density ae conveted back into the conjugate family of pio when α = d 1 β = p ae substituted. Thus, all the esults ae valid as obtained in pevious section. In pesent case the pio vaiance should not be equated to unity. Hence, we equated pio mean as unity estimate the pio paametes fo simulation. The selected pio paametic set of values ae d,p = 3,1,8.9,2,14.63,4. In pesent section simila set of paametic values selected as consideed ealie apply the simila steps of simulation un. The numeical findings ae pesented hee in Tables Fom, Tables 03 04, the shinkage estimato LH1 pefoms unifomly well than shinkage estimato SH1 unde both isk citeions fo small pio paametic values. When pio paametic values incease the well pefomances of the shinkage estimato LH1 esticted in close vicinity of the tue value. All the popeties have been seen simila efficiency attain maximum at δ = REFERENCES [1] J. O. Bege, Statistical decision theoy Bayesian analysis, 2nd Edition, Spinge - Velag, New Yok, 1985.

5 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, Table 1: Relative Efficiency between Bayes Shinkage Estimatos LH & SH unde SELF RE SELF LH, SH a = 0.25 a = 0.50 α, β δ , , ,

6 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, Table 2: Relative Efficiency between Bayes Shinkage Estimatos LH & SH unde LLF RE LLF LH, SH a = 0.25 a = 0.50 α, β δ , , ,

7 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, Table 3: Relative Efficiency between Bayes Shinkage Estimatos LH1 & SH1 unde SELF RE SELF LH1, SH1 a = 0.25 a = 0.50 d, p δ , , ,

8 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: , VOL. 5, NO. 1, Table 4: Relative Efficiency between Bayes Shinkage Estimatos LH1 & SH1 unde LLF RE LLF LH1, SH1 a = 0.25 a = 0.50 d, p δ , , ,