Prelimiary test ad Bayes Estimatio of a Locatio Parameter Uder Bliex Loss J. COETSEE, A. BEKKER, AND S. MILLARD Departmet of Statistics, Uiversity of Pretoria, Pretoria, South Africa I this article, the prelimiary test estimator is cosidered uder the BLINEX loss fuc-tio. The problem uder cosideratio is the estimatio of the locatio parameter from a ormal distributio. The risk uder the ull hypothesis for the prelimiary test esti-mator, the exact risk fuctio for restricted maximum likelihood ad approximated risk fuctio for the urestricted maximum likelihood estimator, are derived uder BLINEX loss ad the differet risk structures are compared to oe aother both aalytically ad computatioally. As a motivatio o the use of BLINEX rather tha LINEX, the risk for the prelimiary test estimator uder BLINEX loss is compared to the risk of the pre-limiary test estimator uder LINEX loss ad it is show that the LINEX expected loss is higher tha BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived uder BLINEX loss, ad a feasible Bayes prelimiary test estimator is defied ad compared to the classical prelimiary test estimator. Keywords Prelimiary test estimator; LINEX loss fuctio; BLINEX loss fuctio; Feasible Bayes estimator; Feasible Bayes prelimiary test estimator. Mathematics Subject Classificatio 6F15; 6F10. 1. Itroductio I this article, the performace of the prelimiary test estimator PTE), restricted maximum likelihood estimator RMLE), ad urestricted maximum likelihood estimator UMLE) are all cosidered uder the BLINEX loss fuctio ad the risk fuctios are compared to oe aother i order to determie which estimator performs better compared to the rest. At the ed of this article, all three estimators metioed above are cosidered uder both the BLINEX ad LINEX loss fuctio i order to determie which of the two loss fuctios performs the best i the sese of havig smaller risk. A umber of articles have already bee published o the prelimiary test estimators usig differet proposed loss structures by Bacroft 1944, 1964), Giles ad Giles 1996), Giles 00), Ohtai et al. 1997), Kibria ad Saleh 1993, 006), Saleh 006), ad Arashi et al. 008), to ame a few. Furthermore, two feasible Bayes estimators are derived uder BLINEX loss ad a feasible Address correspodece to J. Coetsee, Departmet of Statistics, Faculty of Natural ad Agricultural Scieces, Uiversity of Pretoria, Pretoria, South Africa; E-mail: judy.coetsee@up.ac.za 1
Bayes prelimiary test estimator FBPTE) is defied ad it s performace is compared to the performace of the classical prelimiary test estimator. I a article by We ad Levy 001), a ew parametric family of bouded ad asymmetric loss fuctios, called the BLINEX loss fuctio, was developed ad the mathematical properties of the BLINEX loss fuctio were discussed. I Parsia ad Kirmai 00), Porosiński ad Kamińska 009), Arashi 010), ad Arashi ad Tabatabaey 010), the properties of the LINEX fuctio, which was first proposed by Varia 1975), was preseted together with results of estimatio uder LINEX loss for a umber of probability distributios. The BLINEX fuctio is the bouded alterative to the LINEX loss fuctio, therefore it does t have the same limitatios as the LINEX loss fuctio, for example, the LINEX loss fuctio exhibits a huge icrease i expected loss which limits the applicatio of the loss fuctio i practice. The BLINEX fuctio is therefore both bouded ad asymmetric, which allowed the same flexibility as exhibited by the LINEX loss fuctio but it also has the added advatage of beig bouded. The relatioship betwee the BLINEX ad LINEX loss fuctios will be cosidered ext. Assume that τ is a estimator of the ukow parameter θ the the LINEX loss fuctio is defied as follows: L τ,θ) = d e aτ θ) a τ θ) 1 ), where a is the shape parameter ad d is the scale parameter. The BLINEX loss fuctio is derived from the LINEX loss fuctio; let L τ,θ) = d e aτ θ) a τ θ) 1 ) 1 + λd e aτ θ) a τ θ) 1 ) = 1 1 1 λ 1 + b e aτ θ) a τ θ) 1 ) 1) with b = λd. The BLINEX loss fuctio depeds o the selected values of three parameters, amely a, b, ad λ, where each of these values plays a sigificat role i the shape of the loss fuctio. The boudig parameter, λ, specifies the rage of the loss which is bouded betwee 0 ad 1. The sig of the costat a will determie the directio of error pealisatio, λ where egative values will pealise egative errors ad a positive value will pealise positive errors. This costat also iflueces the flatess of the curve, for small positive values of a, the curve will be flatter ad large positive values will result i a steeper curve. The costat b is a asymmetry parameter ad it ca be see that for smaller values of b the loss fuctio is more asymmetric ad for larger values of b the fuctio is more symmetric. The ifluece of the differet parameters as discussed above is depicted i Figure 1 for differet parameter values. The problem uder study is to cosider the estimatio of a locatio parameter μ from a ormal populatio with ukow variace i order to evaluate the fiite-sample properties of the prelimiary test estimator i terms of it s risk. Suppose that X 1, X,...,X is a radom sample from Nμ, σ ) where both μ R ad σ R + are ukow. The outcome of a prelimiary test o μ will determie which of the estimators will be used to estimate the populatio mea μ. The urestricted maximum likelihood estimator, X = X i, will be regarded as the estimator of μ whe othig is a priori kow about the parameter μ, whereas the restricted maximum likelihood estimator will be chose whe
Figure 1. BLINEX loss fuctio 1 with i) λ = 0.8,a = b = 1, ii) - - - λ = b = 1,a = 5, iii)... λ = b = 1,a = 5, ad iv).-.λ = a = 1,b = 5. some costraits are imposed to the model as a ull hypothesis. The stadard t test for the populatio mea will be the appropriate test sice the scale parameter σ is ukow. Now cosider the case where we are dealig with the followig hypotheses: { H0 : μ = μ 0 H A : μ μ 0. The it is well kow that the test statistic is give as t = X μ 0 s t 1) with X Nμ, σ ). Alteratively, we ca use the followig statistic t = X μ 0) = F F 1, 1). I more geeral terms, the samplig distributio of the test statistic will be the o cetral F variate, F F 1, 1,λ ) with λ = δ kow as the ocetrality parameter σ ad the estimatio error is defied as δ = μ 0 μ. It ca easily be see that whe δ = 0, the samplig distributio uder the ull hypothesis will be the cetral F 1, 1, 0) distributio. Based o this proposal the urestricted maximum likelihood estimator UMLE) is give by μ 1 = X ad the restricted maximum likelihood estimator RMLE) is equal to μ 0. The the prelimiary test estimator of μ is defied as where ad I R F ) = 1 I NR F ) = 1 μ p = I R F ) μ 1 + I NR F ) μ 0 whe the ull hypothesis is rejected = 0 otherwise whe the ull hypothesis is ot rejected = 0 otherwise. The ull hypothesis will ot be rejected whe F c α, where c α is a critical value defied as F α,1, 1,λ =0, ad α is the level of sigificace. s 3
The outlie of this article is as follows. I Sec., the risk fuctio of the RMLE, UMLE ad PTE are derived uder BLINEX loss. These differet risk fuctios will be computatioally compared. I Sec. 3, two feasible Bayes estimators are derived uder BLINEX loss. A feasible Bayes prelimiary test estimator FBPTE) is the defied ad the risk fuctio of the PTE is compared to that of the FBPTE. Sectio 4 is a discussio of the results obtaied i this article ad illustratios are provided for the motivatio to use BLINEX loss rather tha LINEX loss fuctios.. Risk Fuctios.1 Risk Fuctio of RMLE The risk fuctio of RMLE is defied ad followed by a illustratio of how the risk fuctio behaves for differet choices of the parameters i the fuctio. The risk fuctio of the RMLE is defied as follows: 1 1 R μ 0 ) = E 1 λ 1 + b e aδ aδ 1 ) = 1 1 1 λ 1 + b e aδ aδ 1 ) with δ = μ 0 μ. I the case where δ = 0, Rμ 0 ) = 0. By examiig the risk fuctio it ca be see that whe both a ad δ are egative values or whe both a ad δ are positive, with δ either tedig to or, the risk of RMLE, Rμ 0 )) teds to 1 λ. Whe δ teds to zero, the term beaδ aδ 1) teds to zero ad therefore Rμ 0 ) teds to 0.. Risk Fuctio of UMLE By defiitio, the risk fuctio of UMLE ca be calculated as below: 1 1 R μ 1 ) = 1 λ 1 + b e aμ 1 μ) a μ 1 μ) 1 p μ 1 ) dμ 1, ) where pμ 1 ) is the desity of N μ, σ ). Therefore, by applyig the biomial expasio to 1 Z) 1 where Z = baμ 1 μ) e aμ 1 μ) + 1 R μ 1 ) = 1 λ 1 λ ) 1 πσ e μ 1 μ) σ ) 1 z dμ 1. The risk fuctio of UMLE ca be expaded to: = b e a σ 1 + a b σ e a σ 1 + b e a σ e a σ 1 λ λ λ 4
+ 3a b 3 σ λ + 3b3 λ e a σ + e a σ e a σ e a σ a σ ) 1 + 6a b 3 σ λ e a σ e a σ + b3 e 9a σ 1 +... 3) λ The risk fuctio of the UMLE is ot a fuctio of the estimatio error, δ, ad is therefore also costat with regard to the o cetrality parameter λ...1 Mote Carlo Simulatios of Risk of UMLE. The risk fuctio of UMLE was also evaluated as a fuctio of λ ad it was foud that the risk fuctio is a decreasig fuctio of both ad λ. Table 1 gives the simulatio results where the fuctio was evaluated for differet values of a,b,σ, ad ad usig p = 1000 iteratios. The results i the table just cofirm some of the expected characteristics of the risk fuctio, amely that the risk of UMLE is a decreasig fuctio of both λ ad, butthe risk is a icreasig fuctio of the variace for a specific choice of parameter..3 Risk of PTE I this sectio the ull risk risk uder the ull hypothesis) of the prelimiary test estimator is derived uder BLINEX loss. The followig lemma is due to Judge ad Bock 1978) ad work doe by Clarke 1986). Lemma.1. Let w be a o cetral chi-square variate with g degrees of freedom ad o cetrality parameter θ, letφ ), be ay real-valued fuctio ad let be ay real value such that > g, the Ew φw)) = e θ θ m Ɣ 1 g + + m) m! Ɣ 1 g + m) E φ χg++m) ). m=0 Theorem.1. Igorig the terms of order 3, the risk fuctio of the prelimiary test estimator usig the BLINEX loss fuctio uder H 0 is give by Rμ p ) = ab σ ) 1 λ 1 + e λ λ m Ɣ 1 + m) eaδ ) m! Ɣ 1 1 + m)p F 1 <cα + 3b a r λ r! r=1 e λ λ i i! i=0 ) σ a b λ ) r σ π m=0 π Ɣ ) r+1 + δ r P F <c α r Ɣ 1 1 + r j + i) Ɣ 1 1 + i) P F 3 <cα ) Ɣ ) 3 + δp F <c α 1 σ δ j 1 j) δ j ) r j ) σ r j) 5
Table 1 Risk of UMLE for specific choices of a,b,σ, ad λ = 1 a = 1, b = 0.1,λ= 1 a = 0.1, b = 1,λ= 1 a = 0.1, b = 0.1,λ= 1 Sample size σ = 1 σ = σ = 5 σ = 1 σ = σ = 5 σ = 1 σ = σ = 5 = 10 0.00469 0.01903 0.1077 0.00046 0.00185 0.0113 0.00005 0.00019 0.00117 = 30 0.0016 0.00647 0.04009 0.00016 0.00065 0.00401 0.0000 0.00007 0.00041 = 50 0.00105 0.0041 0.063 0.00011 0.0004 0.0061 0.00001 0.00004 0.0006 = 100 0.00051 0.0003 0.0181 0.00005 0.0000 0.00130 0.00000 0.0000 0.00013 = 1000 0.00005 0.000 0.00135 0.00001 0.0000 0.00013 0.00000 0.00000 0.00001 6
e λ λ k k=0 k! b a) r λ r! r r=1 δ j ) r j Ɣ 1 j + k) Ɣ 1 1 + k) P F 4 <cα ) r σ Ɣ ) r+1 ) σ π r j) i=0 P F 3 <cα ) + O Z 3) + δ r P F <c α e λ λ i i! Ɣ 1 1 + r j + i) Ɣ 1 1 + i) with F 1 = 1)χ +m,λ ) +m)χ 1),F = 1)χ 1,,λ ),F χ 1) 3 = 1)χ 1+r j+i,λ ) 1+r j+i)χ 1) ad F 4 = 1)χ j+k,λ ). j+k)χ 1) Proof. By applyig the biomial expasio for Z = baμ p μ) e aμp μ) + 1 the risk fuctio is give by Rμ p ) = ELμ p,μ) 4) 1 λ E Z) + E Z ) + O Z 3) where a r E μ p μ ) r E Z) = b. r! r= Usig the fact that μp μ ) r = μ1 μ) r I R F ) δ r I NR F ) = μ 1 μ) r + δ r μ 1 μ) r I NR F ) for r = 1,, 3,... r = μ 1 μ) r + δ r δ j ) ) r μ 1 μ 0 ) r j I NR F ) j r = μ 1 μ) r + δ r δ j ) ) r σ χ1,λ r j) ) I NR F ), j where χ1,λ ) is a o cetral Chi-squared variate with 1 degree of freedom ad a o cetrality parameter λ = δ. σ Thus, E μ p μ ) r = E μ1 μ) r r + δ r E I NR F ) δ j ) ) r σ r j) j E I NR F ) χ1,λ ) r j) ). 7
Usig Lemma.1, we have: E I NR F ) χ1,λ ) r j) ) = r δ j ) r j ) σ r j) i=0 e λ λ i Ɣ 1 1 + r j + i) Ɣ 1 1 + i) P F 3 <cα i! E Z) = b r= a r E μ 1 μ) r + δ r P F <cα r! e λ λ i i=0 i! r δ j ) r j Ɣ 1 1 + r j + i) Ɣ 1 1 + i) P F 3 <cα ). Cosider the followig expressios see Zeller, 1971, pp. 364-365): We fially get EZ) = b E μ 1 μ) r = σ ) r ) Ɣ r + 1 π E μ 1 μ) r 1 = 0 for r = 1,, 3,... r= a r r! e λ λ i i=0 Furthermore, we have Sice i! ) r σ π Ɣ ) r+1 + δ r P F <c α for r = 1,, 3,... r δ j ) r j ) σ ) σ r j) r j) Ɣ 1 1 + r j + i) Ɣ 1 1 + i) P F 3 <cα ). 5) E Z ) = a be μ p μ ) + abe μp μ ) μp abe μ ) e aμ p μ) E e aμ p μ) + be e aμ p μ) + b. μp μ ) = μ 1 μ) I R F ) + μ 0 μ) I NR F ) = μ 1 μ) + δ μ 1 μ) I NR F ) = μ 1 μ) + μ 1 μ 0 ) I NR F ) σ ) 1 χ ) 1 = μ 1 μ) 1,λ I NR F ) 8
ad uder H 0 we fid μp E μ ) e aμ p μ) σ χ = E μ 1 μ) E 1,λ σ ) 1 χ = E 1,λ σ ) 1 χ = e aμ0 μ) E 1,λ σ ) 1 m=0 = e aδ) e λ λ m m! otherwise We the fid E Z ) uder H 0 : E Z ) ) σ = a b π ) 1 E μp μ ) e aμ p μ) = 0. Ɣ ) 3 + δp F <cα ) 1 σ δ j ) 1 j e λ λ k k! k=0 m=0 ) 1 I NR F ) e aμ p μ) ) 1 I NR F ) e aμ 1 μ)i R F )+aμ 0 μ)i NR F ) ) 1 I NR F ) Ɣ 1 + m) Ɣ 1 1 + m)p F 1 <cα, Ɣ 1 j + k) Ɣ 1 1 + k) P F 4 <cα σ ) 1 e λ λ m Ɣ 1 + m) ab m! Ɣ 1 m=0 1 + m)p F 1 <cα + ab e aδ) σ ) 1 e λ λ m Ɣ 1 + m) m! Ɣ 1 1 + m)p F 1 <cα b r=1 a r r! ) r σ i=0 π σ r j) e λ λ i i! + b r=1 a) r r! σ ) r Ɣ ) r+1 π σ r j) e λ λ i i! i=0 + δ r P F <c α Ɣ 1 1 + r j + i) P 1 + i) Ɣ ) r+1 Ɣ 1 + δ r P F <c α r δ j ) ) r j ) F 3 <cα r δ j ) ) r j Ɣ 1 1 + r j + i) Ɣ 1 1 + i) P F 3 <cα ). 6) Substitutig 5) ad 6) ito 4) the result follows. 9
.4 Compariso Results The risk of UMLE ad PTE are aalytically compared to oe aother ad result is give i the followig theorem. Theorem.. The PTE performs better tha the UMLE i the sese of havig smaller risk if the followig iequality holds: δ 1 1 a l b E 1 1 x μ1 + ae μ p μ ) + 1 E e P F >c) /P F <c). aμ 1 μ)i RF Proof. It is eough to show that Rμ p ) Rμ 1 ). Therefore, we should have E1 1 x μp E1 1 x μ1., where x μp = 1 + be aμp μ) aμ p μ) 1 ad x μ1 = 1 + be aμ1 μ) aμ 1 μ) 1, kowig that E1 1 x μ1 is ot a fuctio of δ. Applyig the iequality, 1 1 x 1, x>0to the LHS of the above expressio it is x sufficiet to prove that Ex μp 1 E1 1 x μ1. Simplifyig the relevat term it yields E e aμ p μ) 1 b E 1 1 x μ1 + ae μ p μ ) + 1. Settig 1 b E1 1 x μ1 + aeμ p μ) + 1 = Q we the have E e aμ p μ) Q. 7) The LHS of the above iquality ca be writte as E e aμ p μ) = E e aμ 1 μ)i RF e aδi NRF. 8) By applyig the Cauchy-Schwarz iequality ad usig 7) ad 8), we obtai E e aμ 1 μ)i RF e aδi NRF E e aμ 1 μ)i RF E e aδi NRF. The, it is eough to show that E e aμ 1 μ)i RF E e aδi NRF Q. From the fact that E e aμ 1 μ)i RF > 0, we have E e aδi NRF Q E. 9) e aμ 1 μ)i RF However, the LHS of the above iequality ca be writte as E e aδi NRF aδ) k = E I NR F k! k=0 aδ) k = P F >c) + P F <c) k! k=0 = P F >c) + P F <c) e aδ 10) 10
Figure. a = b = λ = 1,σ = 1,= 10. By makig use of the expressio i 9) ad 10) we have P F >c) + P F <c) e aδ Q E e aμ 1 μ)i RF. By simplifyig ad takig logs o both sides of the iequality the result follows. The risk of RMLE, UMLE, ad PTE calculated uder BLINEX loss are computatioally compared to oe aother i order to determie the performace of the estimators relative to oe aother. The sketches see Figs. -10) i this sectio will be discussed i two ways, first by keepig σ costat ad allowig the sample size,, to icrease ad secodly by keepig costat ad allowig σ to icrease. Figure 3. a = b = λ = 1,σ =,= 10. 11
Figure 4. a = b = λ = 1,σ = 5,= 10. For the case where σ = 1, it ca be see that as the sample size icreases for this choice of σ, the risk of PTE teds towards the risk of UMLE as δ ±. It is well kow that X UMLE) is the most efficiet estimator as the sample size icreases, sice Var X ) = σ. It is clear that as the sample size icreases the Var X ) will ted to zero. Therefore as the sample size icreases, PTE becomes just as efficiet as the UMLE. Similar results are obtaied for differet values of σ. Whe σ icreases for a specific sample size, it ca be see that i the iterval ear the origi, the domiatio iterval of PTE relative to the UMLE becomes larger. Therefore, whe workig with a specific sample size ad if the variace of the sample is relatively high, the PTE is a more efficiet estimator compared to the UMLE. Figure 5. a = b = λ = 1,σ = 1,= 30. 1
Figure 6. a = b = λ = 1,σ =,= 30. Figure 7. a = b = λ = 1,σ = 5,= 30. Figure 8. a = b = λ = 1,σ = 1,= 100. 13
Figure 9. a = b = λ = 1,σ =,= 100. 3. Bayesia Estimatio uder the BLINEX Loss Fuctio 3.1 Feasible Bayes Estimator The Bayes estimator uder BLINEX loss is derived i this sectio ad due to the complex ature of BLINEX loss fuctio it was ecessary to make use of alterative methods i order to propose feasible estimators. Suppose that X 1, X,..., X is a sample from Nμ, σ ) where μ R ad σ R + is assumed to be kow. For the case of σ ukow, oe may look at the work of Arashi 010), the results obtaied i his paper are very complex. It is wellkow that the cojugate prior for the ukow parameter μ is N μ,σ ) ad the posterior of μ x is N μ,σ ) where μ = μ 1σ +μ σ )) ad σ = σ σ. σ +σ σ +σ Figure 10. a = b = λ = 1,σ = 5,= 100. 14
To derive the Bayes estimator, the expected loss is equal to EL = L μ B,μ) p μ x) dμ where x = x 1,x,..., x ) The uder the pre-specified assumptios we have ELμ B ) = 1 λ 1 λ E 1 1 + b e aμ B μ) aμ B μ) 1 ). 11) The the Bayes estimator of μ deoted by μ B is the oe that miimises the posterior expectatio of the loss fuctio, i.e., μ B = arg mi μ B ELμ B ). Note that ELμ B ) μ B = b λ E ae aμb μ) a 1 + b e aμ B μ) aμ B μ) 1 ) = 0. Take f a μ B μ) = ae aμ B μ) a. It ca be see that f a μ B μ) 0 whe μ B μ ad f a μ B μ) < 0 whe μ B <μ.also ote that for a radom variable X 0,EX) = 0 implies X = 0 a.e. Therefore, subject to the costrait μ B μ the Bayes estimator is equal to μ B = μ. 1) Now uder the restrictio μ B <μ,cosider the followig optimisatio problem arg max μ B ELμ B ) arg max μb 1 E 1 + b e aμ B μ) aμ B μ) 1 ). 13) 1 Cosider that gμ B μ) = 1+be aμ B μ) aμ B μ) 1) is a icreasig fuctio sice g μ B μ) > 0 whe μ B <μ.also from the fact that e aμb μ) >aμ B μ) + 1 whe μ B <μ, it ca be see that gμ B μ) > 0. There is o uique solutio for 13), therefore 1) is a valid result. The problem with the above solutio 1) is that μ B is ot a valid estimator, because μ is ukow. I this case a feasible estimator, deoted by μ FB, ca be obtaied by substitutig a proper estimator of μ ito the solutio to get μ 1) FB = μ. Aother possible solutio is to use two sets of available iformatio cotaied i the prior distributio, for the ukow parameter μ ad the desity fuctio.x μ, which cotais the iformatio of the origial sample. 15
I this case, defie λ L μ, σ ).p μ) μ μ ) e 1 σ x i μ) σ. The takig the derivative with respect to μ, equatig to zero ad substitutig the fial result ito 1) we obtai the feasible estimator i=1 x i + μ σ μ ) FB = σ σ σ ) ). 14) + Note that λ is just a way to produce a estimator of μ ad it totally idepedet of the loss fuctio. The result of λ was substituted ito 11), that was iitiated from BLINEX loss ad the sketches were geerated usig simulatio exercises based o 11). This feasible Bayes estimator is exactly the same as the Bayes estimator uder squared error loss. The properties of μ ) FB are give below: E μ ) FB = μ + μ σ σ ) σ + σ σ if μ = μ the E μ ) μ FB = = μ, therefore uder these circumstaces μ ) σ σ + FB ubiased estimator of μ; if σ = σ the Var Var μ ) FB 0. σ + ) Var μ ) FB = μ ) FB = σ σ +1) σ ) = + ; σ σ σ + σ ) ; is a σ, it ca therefore be see that whe, 3. A Feasible Bayes Prelimiary Test estimator Facig the same problem as i the itroductio ad usig μ ) FB as the urestricted estimator, the feasible Bayes prelimiary test estimator is defied by μ FBP = I R F ) μ ) FB + I NR F ) μ 0. The performace of FBPTE will be explored uder BLINEX loss i the followig sectio. 3..1 Comparig PTE with FBPTE uder BLINEX loss fuctio. The risk fuctios for the PTE ad FBPTE were calculated for specific choices of the parameters ivolved ad are compared to oe aother. I all the sketches the risk fuctio of the PTE is preseted by the dashed lie curve ad the risk fuctio of the FBPTE is give by the solid lie curve. Both risk fuctios of the estimators are fuctios of δ, where 1 δ 1. I the calculatio of the FBPTE oly a fixed value for the prior mea, μ = 13 ad prior stadard deviatio, 16
Figure 11. Compariso of PTE ad FBPTE for differet parameter values. σ = 1, were cosidered ad the performace of the estimators must still be evaluated for differet choices of parameters. For these choices it ca be see that depedig o the value of δ, σ ad, the oe estimator outperforms the other oe. 4. Discussio I this article, the PTE ad its exact risk fuctio is derived uder BLINEX loss, It is the compared i terms of it s risk fuctio, to the risk of the RMLE ad UMLE. For specific choices of parameters ad sample sizes it ca be see that the PTE is just as efficiet as UMLE, these results were obtaied whe δ ± as the sample size icreases ad by keepig σ costat. Whe the sample size remais costat ad σ icreases, the iterval where PTE domiates UMLE icreases. I other words, whe σ icreases we rather prefer to use the PTE i our problems. The FBPTE was also defied i Sec. 3. ad the PTE was computatioally compared to the FBPTE. For specific choices of parameters ad sample sizes it ca be see that both estimators perform well uder specific coditios ad for specific values of δ. Sice the feasible Bayes estimator, give by 14), is othig more tha the Bayes estimator uder squared error loss, we ca ot expect that the risk of FBPTE has a special form. It is still uclear how oe ca fid a Bayes estimator uder BLINEX loss with stable risk performace, which leaves this problem ope for further research. By examiig the results obtaied, we foud that the PTE performs better tha FBPTE for some specific parameter choices ad vice versa. The BLINEX ad LINEX risk fuctios are illustrated i the followig sketches see Figs. 1a c) for specific values of the boudig parameter, λ, the asymmetry parameter b, ad the shape parameter a which iflueces the directio of the pealisatio of the delta. I additio to these parameters, differet samples sizes were selected ad the populatio variace was also adjusted to determie the effect o the risk fuctios. I each sketch the 17
Figure 1. a. Risk fuctios uder BLINEX ad LINEX loss for differet choices of the scale parameter ad = 10. b. Risk fuctios uder BLINEX ad LINEX loss for differet choices of the scale parameter ad = 30. c. Risk fuctios uder BLINEX ad LINEX loss fordifferet choices of the scale parameter ad = 100. 18
Figure 1. cotiued) correspodig risk fuctios for BLINEX ad LINEX loss are compared to oe aother. The BLINEX risk fuctio is represeted by the dashed lie curve ad the LINEX risk fuctio is represeted by the solid lie curve. All the risk fuctios are represeted as a fuctio of δ, where 1 δ 1. I the first colum of the pael the risk fuctios of the PTE uder BLINEX ad LINEX loss are give. I the secod colum the risk fuctios of the RMLE uder BLINEX ad LINEX loss are give ad i the last colum the risk fuctios of the UMLE uder BLINEX ad LINEX loss are give. Eve though oly specific choices of the parameters a,b,λ,σ, ad were ivestigated, it ca clearly be see that the BLINEX risk fuctios represet lower risk tha that of the correspodig LINEX risk fuctio across all estimators. These results ca be used to motivate why the BLINEX loss fuctio should be used, rather tha the LINEX loss fuctio. The risk fuctios of RMLE remai the same for all choices of σ ad, sice the risk fuctio is ot a fuctio of these two parameters. The risk fuctios of UMLE are ot a fuctio of δ, therefore the risk is costat over the domai of δ. As show i We ad Levy 001), the BLINEX loss fuctio has the same flexibility as the LINEX, together with the added advatage of beig a bouded fuctio, ow it is also show that uder BLINEX loss all the estimators performs better whe compared to LINEX loss. All the simulatios i this article were doe i SAS 9.. Ackowledgmets This authors are grateful for all the valuable commets ad recommedatios made by the aoymous reviewers. 19
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