ASYMPTOTIC BEST LINEAR UNBIASED ESTIMATION FOR THE LOG-GAMMA DISTRIBUTION

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1 Sakhyā : The Idia Joural of Statistics 994, Volume 56, Series B, Pt. 3, pp ASYMPTOTIC BEST LINEAR UNBIASED ESTIMATION FOR THE LOG-GAMMA DISTRIBUTION By N. BALAKRISHNAN McMaster Uiversity, Hamilto P.S. CHAN The Chiese Uiversity of Hog Kog SUMMARY. I this paper, we discuss Beett s determiatio of optimal asymptotic weights for the BLUEs of the locatio scale parameters based o geeral Type-II cesored samples the apply this method to the log-gamma distributio.. Itroductio Let Y be a log-gamma rom variable with probability desity fuctio g(y) = (y µ)/σ σγ(κ) e e +κ(y µ)/σ, <y<, <µ<,σ >0,κ>0,...(.) where µ is the locatio parameter, σ is the scale parameter, κ is the shape parameter, or, equivaletly, ( g R (y) = κκ /2 κ bγ(κ) exp y u b ) κe (y u)/b κ, <y<,...(.2) where u = µ + σ log k, b = σ κ....(.3) Lawless (980, 982) has illustrated the usefuless of the log-gamma model i (.) as a life-test model discussed the maximum likelihood estimatio of the parameters; see also Pretice (974). Youg Bakir (987) have discussed the log-gamma regressio model. Oe may also refer to Lawless (980) DiCiccio (987) for some valuable work o the iferece for a related geeralized gamma distributio. Paper received. October 992; revised August 993. AMS (990) subject classificatio. 62F0. Keywords phrases. Log-gamma distributio, asymptotic best liear ubiased estimators (ABLUEs), type-ii cesored samples, life-test data.

2 best liear ubiased estimatio 35 Recetly, Balakrisha Cha (994) have studied order statistics from the log-gamma distributio determied their meas, variaces covariaces, used these quatities to determie the best liear ubiased estimators (BLUEs) of µ σ i (.) based o complete as well as Type-II cesored samples. They have also discussed the liear estimatio of µ σ based o k optimally selected order statistics. I this paper, we discuss asymptotic approximatios to the BLUEs of µ σ based o Type-II cesored samples. 2. Asymptotic best liear ubiased estimatio I this sectio, we shall cosider a Type II cesored sample Y r+: Y r+2: Y :...(2.) from a populatio with c.d.f F ((y µ)/σ) p.d.f f((y µ)/σ)/σ briefly describe Beett s determiatio of optimal asymptotic weights for the BLUEs of µ σ. Let us deote X i: = (Y i: µ)/σ, α i: = E(X i: ), β i,j: =Cov (X i:,x j: ). The the best liear ubiased estimators of µ σ for the cesored sample i (2.) are (see Lloyd, 952; David, 98, p. 29; Balakrisha Cohe, 99, p. 80-8) give by { α µ Ωα Ω α Ωα } Ω = (α Ωα)( Ω) (α Ω) 2 Y...(2.2) where σ = { Ωα Ω Ωα } Ω (α Ωα)( Ω) (α Ω) 2 Y...(2.3) Y = (Y r+:,y r+2:,...,y : ), α = (α r+:,α r+2:,...,α : ), = (,...,) r s, Ω =(ω i,j )=(β i,j: ), r + i, j s. The variaces covariace of µ σ i (2.2) (2.3) are (see Lloyd, 952; David, 98, p. 30; Balakrisha Cohe, 99, p. 8) give by { var(µ )=σ 2 α } Ωα (α Ωα)( Ω) α Ω) 2,...(2.4) { var(σ )=σ 2 } Ω (α Ωα)( Ω) α Ω) 2,...(2.5)

3 36. balakrisha p.s. cha { cov(µ,σ )= σ 2 α } Ω (α Ωα)( Ω) (α Ω) 2....(2.6) As, i, j such that i/ p i, j/ p j, 0 <p i < p j <, we have the expected value of X i: the covariace of X i: X j: up to order / as (see David, 98, p. 80; Arold Balakrisha, 989, p ) α i: = F (p i )=G i...(2.7) β i,j: = p iq j,i j, +2f(G i )f(g j )...(2.8) where q i = p i, G i = F (p i ) for r + i s. The Ω may be algebraically worked out usig (2.8), is give by { ω i,i = f 2 (G i ) p i+ p i + ω i,i = ω i,i = f(gi)f(gi ) p i p i, ω i,j = 0 otherwise, p i p i },...(2.9) where p r = 0 p + =. Usig (2.9) lettig r/ λ ()/ λ 2 as, it ca be easily show that Ω = α Ω = α Ωα = i=r+ {f(g i+ ) f(g i )} 2 p i+ p i + f 2 (G r+ ) p r+ + f 2 (G ) p = λ 2 λ Ψ 2 (v)du + f 2 (G(λ )) λ + f 2 (G(λ 2)) λ 2 = I (say) {f(g i+ ) f(g i )}{G i+ f(g i+ ) G i f(g i )} p i+ p i i=r+ + Gr+f 2 (G r+) p r+ + Gf 2 (G ) p = λ 2 λ Ψ(v)( + vψ(v))du + G(λ )f 2 (G(λ )) λ + G(λ 2)f 2 (G(λ 2 )) λ 2 = I 2 {G i+ f(g i+ ) G i f(g i )} 2 i=r+ + G2 r+ f 2 (G r+) p r+ p i+ p i + G2 f 2 (G ) p = λ 2 λ ( + vψ(v)) 2 du + G2 (λ )f 2 (G(λ )) λ + G2 (λ 2 )f 2 (G(λ 2 )) λ 2 = I 22 ;...(2.0)...(2.)...(2.2)

4 best liear ubiased estimatio 37 i the above formulae, v = G(u) =F (u), G (u) =df (u)/du =/f(g(u)), Ψ(v) =f (v)/f(v). The fact that d{+vf(v)}/du =+vψ(v) is also used i the derivatio of these formulae. From Eqs. (2.2) (2.3), we ca the write the BLUEs of µ σ as where, for i = r +,..., s, γ i = (I I 22 I2 2 ) I 22 µ = σ = j=r+ i=r+ i=r+ γ i Y i:...(2.3) δ i Y i:...(2.4) ω i,j I 2 j=r+ α j: ω i,j...(2.5) δ i = (I I 22 I2 2 ) I j=r+ α j: ω i,j I 2 j=r+ ω i,j....(2.6) Usig (2.9), the quatities i γ i δ i ca be further simplified as j=r+ ω i,j f(g(ξ i )) d2 f(g(ξ i)) dξ dξi 2 i = φ (ξ i )dξ i (say) φ (ξ i )/...(2.7) with i =[ξ i ]+, [m] idicatig the itegral part of m, where j=r+ j=r+ Ω r+,j φ (ξ r+ )dξ r+ + f 2 (G r+) p r+ f (G r+ ) φ (ξ r+ ) + φ,r+ ω,j φ (ξ )dξ + f 2 (G ) p + f (G ) φ(ξ) + φ,, φ,r+ = f 2 (G r+ ) p r+ f (G r+ ), φ, = f 2 (G ) p + f (G )....(2.8)...(2.9)

5 38. balakrisha p.s. cha Moreover, j=r+ α j: ω i,j f(g(ξ i )) d2 {G(ξ i )f(g(ξ i ))} dξ dξi 2 i = φ 2 (ξ i )dξ i (say) φ 2 (ξ i )/,...(2.20) where j=r+ j=r+ α j: ω r+,j φ 2 (ξ r+ )dξ r+ + G r+f 2 (G r+ ) p r+ {f(g r+ )+G r+ f (G r+ )} + φ 2,r+ φ 2(ξ r+ ) α j: ω,j φ 2 (ξ )dξ + Gf 2 (G ) p +{f(g )+G f (G )} + φ 2,, φ 2(ξ ) φ 2,r+ = G r+f 2 (G r+ ) p r+ {f(g r+ )+G r+ f (G r+ )},...(2.2)...(2.22) φ 2, = G f 2 (G ) + {f(g )+G f (G )}. p I the above derivatios, we may ote that the fuctios φ φ 2 are of the form (with v = G(u) =F (u) Ψ(v) =f (v)/f(v), as before) φ (u) = f(v) d2 f(v) du 2 = Ψ (v)...(2.23) φ 2 (u) = f(v) d2 {vf(v)} du 2 = {Ψ(v)+vΨ (v)}....(2.24) Hece, from Eqs. (2.5) (2.6), we may determie asymptotically the coefficiets γ i δ i (except for the two extreme coefficiets, viz., for i = r + s) by settig u = p i = i/( + ) i the followig cotiuous weight fuctios γ(u) = φ (u)i 22 φ 2 (u)i 2 (I I 22 I 2 2 )...(2.25) δ(u) = φ 2(u)I φ (u)i 2 (I I 22 I 2 2 )....(2.26)

6 best liear ubiased estimatio 39 For the two extreme cases we may determie asymptotically the coefficiets by the formulas γ r+ = γ(ξ r+ )+ φ,r+i 22 φ 2,r+I 2 I I 22 I2 2...(2.27) γ = γ(ξ )+ φ,i 22 φ 2,I 2 I I 22 I (2.28) δ r+ = δ(ξ r+ )+ φ 2,r+I φ,r+i 2 I I 22 I2 2...(2.29) δ = δ(ξ )+ φ 2,I φ,i 2 I I 22 I (2.30) By usig (2.4) - (2.6) (2.0) - (2.2), we ca easily obtai the asymptotic variaces covariace of the estimators µ σ i (2.3) (2.4) to be var(µ ) I 22 σ 2 = I I 22 I2 2 var(σ ) I σ 2 = I I 22 I2 2...(2.3)...(2.32) cov(µ,σ ) I 2 σ 2 = I I 22 I (2.33) 3. Derivatio for the log-gamma distributio I this sectio, we derive exact explicit expressios for the various fuctios itroduced i the last sectio; these formulae make the computatio of the asymptotic BLUEs for this case a lot simpler. For this purpose, we establish the followig three lemmas which will be repeatedly used i the algebraic maipulatios to follow i this sectio. Lemma. For κ > 0, let F κ (λ) = λ e κy ey Γ(κ) dy. The F κ+ (λ) =F κ (λ) eκλ eλ Γ(κ +). Lemma 2. For κ > 0, let G κ (λ) = λ ye κy ey dy. Γ(κ)

7 320. balakrisha p.s. cha The G κ+ (λ) =G κ (λ)+ κ F κ(λ) λeκλ eλ Γ(κ +), where F κ (λ) is as defied i Lemma. Lemma 3. For κ > 0, let H κ (λ) = λ y 2 e κy ey dy. Γ(κ) The H κ+ (λ) =H κ (λ)+ 2 κ G κ(λ) λ2 e κλ eλ Γ(κ +), where G κ (λ) is as defied i Lemma 2. Now, for the stard log-gamma distributio with desity fuctio f(x), with kow shape parameter κ, wehave Hece, the fuctio Ψ(v) becomes f (x) =f(x)(κ e x )....(3.) Ψ(v) = f (v) f(v) = κ ev...(3.2) Let us deote ξ λ ξ λ2 by Fκ (λ ) Fκ (λ 2 ), respectively, where Fκ ( ) is iverse c.d.f of the log-gamma distributio with shape parameter κ. Sice v = Fκ (u) dv =(/f(v))du, applyig Lemma, the fiite itegral i Eq. (2.3) ca be writte as λ Ψ 2 (v)du = ξ λ2 ξ λ (κ 2 2κe v + e 2v )f(v)dv = κ 2 [λ 2 λ ] 2κ 2 [F κ+ (ξ λ2 ) F κ+ (ξ λ )] +κ(κ + )[F κ+2 (ξ λ2 ) F κ+2 (ξ λ )]...(3.3) = κ(λ 2 λ )+ κ Γ(κ) [eκξ λ 2 e κξ λ ] Γ(κ) [e(κ+)ξ λ 2 e (κ+)ξ λ ]. Similarly, usig Lemmas 2, the fiite itegral i (2.4) ca be writte as λ Ψ(v)[ + vψ(v)]du = κ(λ 2 λ ) κ[f κ+ (ξ λ2 ) F κ+ (ξ λ )] +κ 2 [G κ (ξ λ2 ) G κ (ξ λ )] 2κ 2 [G κ+ (ξ λ2 ) G κ+ (ξ λ )] + κ(κ + )[G κ+2 (ξ λ2 ) G κ+2 (ξ λ )] = (λ 2 λ )+κ[g κ (ξ λ2 ) G κ (ξ λ )] + κ(κ ) Γ(κ) [ξ λ 2 e κξ λ 2 ξ λ e κξ λ ] Γ(κ) [ξ λ 2 e (κ+)ξ λ 2 ξ λ e (κ+)ξ λ ]....(3.4)

8 best liear ubiased estimatio 32 Fially, usig Lemmas, 2 3, the fiite itegral i (2.5) ca be writte as λ [ + vψ(v)] 2 du = (λ 2 λ )+2κ[G κ (ξ λ2 ) G κ (ξ λ )] 2κ[G κ+ (ξ λ2 ) G κ+ (ξ λ )] + κ 2 [H κ (ξ λ2 ) H κ (ξ λ )] 2κ 2 [H κ+ (ξ λ2 ) H κ+ (ξ λ )] +κ(κ + )[H κ+2 (ξ λ2 ) H κ+2 (ξ λ )] = (λ 2 λ )+2[G κ (ξ λ2 ) G κ (ξ λ )] +κ[h κ (ξ λ2 ) H κ (ξ λ )] + κ Γ(κ) [ξ2 λ 2 e (κ)ξ λ 2 ] ξλ 2 e (κ)ξ λ ] Γ(κ) [ξ2 λ 2 e (κ+)ξ λ 2 ξλ 2 e (κ+)ξ λ ]....(3.5) Hece, we ca evaluate I,I 2 I 22 if we kow the values of G κ (ξ λ ) H κ (ξ λ ). Note that [G κ (ξ λ2 ) G κ (ξ λ ]= λ ye κy e y dy Γ(κ) ye κy e y [H κ (ξ λ2 ) H κ (ξ λ ]= dy λ Γ(κ) ca be easily computed by employig stard umerical methods sice both itegrs are cotiuous i ay ope iterval. Ackowledgemets. The first author would like to thak the Natural Scieces Egieerig Research Coucil of Caada for fudig this research. The authors would also like to thak Prof. Bimal K. Siha a referee for suggestig some chages which led to a improvemet i the presetatio of this paper. Refereces Arold, B. C. Balakrisha, N. (989). Relatios, Bouds Approximatios For Order Statistics. Lecture Notes i Statistics 53, Spriger-Verlag, New York. Balakrisha, N. Cohe, A. C. (99). Order Statistics Iferece: Estimatio Methods, Academic Press, Sa Diego. Balakrisha, N. Cha P. S. (994). Log-gamma order statistics liear estimatio of parameters. Computatioal Statistics & Data Aalysis (Revised). David, H. A. (98). Order Statistics, Secod editio, Joh Wiley & Sos, New York. DiCiccio, T. J. (987). Approximate iferece for the geeralized gamma distributio. Techometrics 29, Lawless, J. F. (980). Iferece i the geeralized gamma log-gamma distributio. Techometrics 22, (982). Statistical Models & Methods For Lifetime Data, Joh Wiley & Sos, New York.

9 322. balakrisha p.s. cha LLoyd, E. H. (952). Least-squares estimators of locatio scale parameters usig order statistics. Biometrika 39, Pretice, R. L. (974). A log gamma model its maximum likelihood estimatio. Biometrika, 6, Youg, D. H. Bakir, S. T. (987). Bias correctio for a geeralized log-gamma regressio model. Techometrics 29, Departmet of Mathematics Statistics McMaster Uiversity Hamilto, Otario Caada l8s 4K Departmet of Statistics The Chiese Uiversity of Hog Kog Shati Hog Kog