On estimating the location parameter of the selected exponential population under the LINEX loss function

Transcription

1 On estmatng the locaton parameter of the selected exponental populaton under the LINEX loss functon Mohd. Arshad 1 and Omer Abdalghan Department of Statstcs and Operatons Research Algarh Muslm Unversty, Algarh, Inda. Abstract Suppose that π 1, π 2,..., π k be k( 2 ndependent exponental populatons havng unknown locaton parameters µ 1, µ 2,..., µ k and known scale parameters σ 1,..., σ k. Let µ [k = max{µ 1,..., µ k }. For selectng the populaton assocated wth µ [k, a class of selecton rules (proposed by Arshad and Msra (2016 s consdered. We consder the problem of estmatng the locaton parameter µ S of the selected populaton under the crteron of the LINEX loss functon. We consder three natural estmators δ N,1, δ N,2 and δ N,3 of µ S, based on the maxmum lkelhood estmator, unformly mnmum varance unbased estmator (UMVUE and mnmum rsk equvarant estmator (MREE of µ s respectvely. The unformly mnmum rsk unbased estmator (UMRUE and the generalzed Bayes estmator of µ S are derved. Under the LINEX loss functon, a general result for mprovng a locaton-equvarant estmator of µ S s derved. Usng ths result, estmator better than the natural estmator δ N,1 s obtaned. We also shown that the estmator δ N,1 s domnated by the natural estmator δ N,3. Fnally, we perform a smulaton study to evaluate and compare rsk functons among varous competng estmators of µ S. Keywords: Exponental populatons; Estmaton after selecton; Equvarant estmators; LINEX loss functon; Inadmssble estmators; UMRU estmator. 1 Inroducton In many researches t s often the case that k( 2 populatons are avalable for evaluaton of ther qualty. The qualty of a populaton s defned n terms of an unknown parameter assocated wth t. Usually the parameter of nterest s the populaton mean, so that, the populaton correspond to the largest (or smallest mean s called the best (or worst populaton. The basc goal s to select the best (or worst populaton. All other populatons are dropped-out and the selected populaton s reman for further 1 Correspondng author. E-mal addresses: arshad.tk@gmal.com (M. Arshad, abdalghan.amu@gmal.com (O. Abdalghan. 1

2 nvestgatons. For example, n clncal research, one s nterested n selectng the most effectve treatment from a choce of k avalable treatments. These problems have receved consderable attenton n the lterature and are known as rankng & selecton problems. After one of the populaton s selected, a practcal problem s to estmate the mean (or unknown parameter of the selected populaton. For example, n ndustral applcatons, a company not only wshes to select an electrc generator from a choce of k( 2 avalable generators that produces maxmum power out-put, but also wants an estmate of the average power out-put of the selected generator. In the lterature, the problem descrbed above s termed as estmaton after selecton problem. The man focused of estmaton after selecton problem s to obtan the varous competng estmators of the parameter assocated wth selected populaton and derve the decson theoretc results under varous loss functons. For more detals, see the references: Vellasamy et al. (1988, Vellasamy (1996, 2009, Msra and Sngh (1993, Msra et al. (1998, Parsan and Farspour (1999, Msra and van der Meulen (2001, Vellasamy and Punnen (2002, Gangopadhyay and Kumar (2005, Kumar et al. (2009, Nematollah and Motamed (2012, Nematollah and Jozan (2016 and Nematollah and Pagheh (2017. A plethora on these problems exst under the scenaros of equal nusance parameters and/or sample szes assocated wth the avalable populatons, less work was consdered for the scenaros of unequal nusance parameters and/or sample szes. Some of the contrbuton n ths drecton s due to Rsko (1985, Abughalous and Mescke (1989, Abughalous and Bansal (1994, Msra and Dharyal (1994, Msra and Arshad (2014, Arshad et al. (2015, Arshad and Msra (2015a,2015b, 2016, 2017 and Meena et al. (2017. In some stuatons, where the overestmaton (postve error may be more serous than the underestmaton (negatve error or vce versa, the use of symmetrc loss functon s napproprate. For such cases, Varan(1975 proposed the followng asymmetrc loss functon L(µ, δ = e a(δ µ a(δ µ 1. µ R, δ D, (1.1 where a 0 s locaton parameter of the loss functon, µ denote the unknown parameter and δ s an estmator of µ. The loss functon (1.1 s known as lnear-exponental (LINEX loss functon. In ths paper, we study the problem of estmaton after selecton from exponental populatons wth unequal scale parameters and unequal sample szes, under the LINEX loss functon (1.1, we also provde some generalzaton of the results of Nematollah and Jozan (2016. Let X 1, X 2,..., X n be ndependent random sample of sze n from populaton π, whch are exponentally dstrbuted and havng the pdf { ( 1 f (x = σ e x µ σ, f x µ 0, otherwse, where µ R denotes the unknown locaton parameter and σ denotes the known scale parameter, = 1,..., k. Let µ [1 µ [2 µ [k denote the ordered values of µ 1,..., µ k, and the populaton assocated wth µ [k s called the best populaton. Let X = mn{x 1, X 2,..., X n }, = 1, 2,..., k. Then, X has exponental dstrbuton wth unknown locaton parameter µ and known scale parameter σ n. Wthout loss 2

3 of generalty, we assume that n 1 = n 2 = = n k = 1, so that X Exp(µ, σ, = 1, 2..., k. Note that X = (X 1, X 2,..., X k s a complete and suffcent statstc for µ = (µ 1, µ 2,..., µ k Ω, where Ω denotes the parametrc space. We consder those nference procedures whch depend on observatons only through the complete and suffcent statstc X = (X 1, X 2,, X k. For the goal of selectng the best populaton, Arshad and Msra (2016 consdered the class C = {ψ c : ψ c = (ψ1, c..., δk c, c Rk } of selecton rules, where { 1, f x + c ψ c > max (x j + c (x = j (1.2 0, otherwse, where c = (c 1,..., c k. It follows from Arshad and Msra (2015a that, for k = 2, the selecton rule ψ c = 1, δ2 c, where { { 1, f x1 > x ψ1 c (x = 2 + c 0, f x1 > x 0, f x 1 x 2 + c ; ψ2 c (x = 2 + c 1, f x 1 x 2 + c (1.3, (δ c and ( σ 2 ln c = ( σ 1 ln σ 1 +σ 2 2σ 2 2σ 1 σ 1 +σ 2, f σ 1 < σ 2, f σ 1 σ 2, s generalzed Bayes wth respect to non-nformatve pror and s mnmax under the 0-1 loss functon. Let µ S be the locaton parameter of the populaton selected by the selecton rule ψ c, gven n (1.2. Let C = {x χ : x + c > max (x j + c j } = {x χ : x + c > j x j +c j, j, j = 1,..., k}, = 1,..., k, be the partton of the sample space χ. Then µ S = µ I C (X, where I C ( denotes the ndcator functon of the set C. Note that µ S s a random parameter whch depends on X = (X 1, X 2,..., X k. Our goal s to estmate the locaton parameter µ S under the LINEX loss functon, gven n (1.1. It s easy to verfy that, n the component estmaton problems, X and X σ are the maxmum lkelhood estmator (MLE and the unformly mnmum varance unbased estmator (UMVUE of µ s, respectvely. Also, under the LINEX loss functon, the mnmum rsk unbased estmator (UMRUE and the mnmum rsk equvarant (MRE estmator of µ s X α, where α = 1ln(a aσ a, = 1,..., k. Note that, the UMRUE and the MRE are same under the LINEX loss functon. Now we defned some natural estmators of µ S based on the MLE, UMVUE and MRE (or UMRUE. Natural estmators of µ S are gven by δ N,1 (X = X I C (X, (based on MLE. (1.4 3

4 δ N,2 (X = δ N,3 (X = (X σ I C (X, (based on UMV UE. (1.5 (X α I C (X, (based on MRE, (1.6 where α = α (σ, a = 1 a ln(1 aσ, aσ < 1. The rest of the paper s organzed as follows. In Secton 2, under the crteron of the LINEX loss fucton (1.1, we derve the UMRUE of µ S and the generalzed Bayes estmator wth respect to a non-nformatve pror dstrbuton. In Secton 3, we derve a general result for mprovement of certan locaton equvarant estmators by employng the dea of Brewster and Zdek (1974. A concluson on ths result, an estmator domnatng over the natural estmator δ N,1 s obtaned. We also provde an example to demonstrate how the varous estmators of µ S are computed. In Secton 5, we perform a smulaton study to evaluate and compare rsk functons among varous competng estmators of µ S. 2 UMRUE and Generalzed Bayes Estmator In ths secton, we derve the UMRUE and the generalzed Bayes estmator of µ S under the LINEX loss functon, gven n (1.1. Defnton 1. (Nematollah and Jozan (2016. An estmator δ(x of the locaton parameter µ S s sad to be rsk unbased estmator under the LINEX loss functon (1.1 f E µ [e aδ(x = E µ [e aµ S, µ Ω. The followng lemma s a generalzaton of Theorem 1 of Nematollah and Jozan (2016, whch s useful for fndng the UMRUE of µ S. Lemma 1. Let X 1, X 2,..., X k be k ndependent random varables from Exp(µ, σ, = 1,..., k. Let U 1 (X, U 2 (X,..., U k (X be k real valued functons of X such that (a E µ ( e ax U (X <, µ Ω. (b The ntegeral Then, the functon satsfes U (x 1,..., x 1, y, x +1..., x k e ( σ dy <. x χ. x K (x = e ax [ U (x a x y U (x 1,..., x 1, y, x +1,...e (y x /σ dy E(K (X = E(e aµ U (X, µ Ω. 4

5 Theorem 1. The UMRUE of the locaton parameter µ S of the selected populaton s gven by δ U (X = 1 a ln [ e ax I C (X a σ e ax e ( max(0,y X σ, (2.1 where Y = max j (X j + c j c, =, 2,..., k. Proof. We have [ E(e aµ S = E e aµ I C (X = E [e aµ I C (X. (2.2 Let K (X be an estmator of e aµ I C (X such that E [K (X = E [e aµ I C (X, = 1, 2,..., k. Usng Lemma 1, we get K (X = e ax I C (X ae ax = e ax I C (X ae ax X I C (X 1,..., X 1, y, X +1,..., X k e (y X /σ dy max{x,y } = e ax I C (X aσ e ax e ( max(0,y X e (y X /σ dy σ. It follows from Equaton (2.2 and Defnton 1 that the rsk unbased estmator of µ S s δ U (X = 1 a ln [ K (X = 1 a ln [ e ax I C (X a where Y = max (X j + c j c, =, 2,..., k. j ( σ e ax e max(0,y X σ, Remark 1. Let X [1 X [k denote the ordered values of X 1,..., X k. Let σ 1 = = σ k = σ (say, and c 1 = = c k = 0, t follows from Theorem 1 that the UMRUE of µ S s δ U (X = [e 1 a ln k 1 ax [k aσe ax [k aσ e ax [ e (X [k X [/σ = X [k + [(1 1 a ln k 1 aσ aσ e (1+aσ[X [ X [k. Ths result was derved by Nematollah and Jozan (2016, and also reported n Nematollah and Pagheh (

6 Now we prove that, under the LINEX loss functon (1.1, the natural estmator δ N,3 and the generalzed Bayes estmator of µ S are same. Theorem 2. Under the LINEX loss functon (1.1, the natural estmator δ N,3 s the generalzed Byes estmator wth respect to the non-nformatve pror dstrbuton. Proof. Consder the non-nformatve pror dstrbuton Π µ (µ 1,..., µ k = 1, (µ 1,..., µ k Ω. The posteror probablty densty functon of µ, gven X = x, s gven by { ( k 1 Π σ µ(µ 1,..., µ k x = e x } µ σ, f µ x, = 1,..., k, 0, otherwse. (2.3 The posteror rsk of an estmator δ under the LINEX loss s gven by [ r(δ, x = E Π e a(δ(x µ S a(δ(x µ S 1 X = x. (2.4 The generalzed Bayes estmator δ G (X, whch mnmzes the posteror rsk (2.4, s gven by δ G (x = 1 a ln [ [ E Π e aµ S X = x [ = 1 a ln ( E Π e aµ IC (x Usng the posteror dstrbuton (2.3, we get [ δ G (X = X + 1 a ln(1 aσ I C (X = δ N,3 (X. 3 Improvng a Locaton Equvarant Estmator In ths secton, we wll show that the natural estmator δ N,1 (X s nadmssble under the LINEX loss functon (1.1. The followng lemmas and defnton are useful n dervng a suffcent condton for the nadmssblty of a locaton-equvarant estmator of µ S. Lemma 2. Let X 1,..., X k be ndependent random varables such that X has the exponental dstrbuton wth locaton parameter µ and scale parameter σ, = 1,..., k. Let T j = X j X 1, j = 2,..., k. Then, for fxed t = (t 2,..., t k R k 1, the condtonal dstrbuton of X 1, gven that T = t, s exponental wth locaton parameter ( k 1 µ t = max{µ 1, max (µ 1 j t j } and scale parameter σ 0 = j 1 σ. The proof of the prevous lemma s straghtforward and hence omtted.. 6

7 Let V 1 = { (t 2,..., t k R k 1 : t j < c 1 c j, j = 2,..., k } and { ( } V l = (t 2,..., t k R k 1 : t l > max c 1 c l, max (t j + c j c l 2 j k j l, l = 2,..., k, so that, {V 1,..., V k } forms a partton of R k 1. Defne ϕ(t, µ = 1 a ln [ E ( e a(x 1 µ T = t I V (t Usng lemma 2, we have [ ϕ(t, µ = 1 a ln 1 1 aσ 0 [ 1 ln 1 a = 1 ln 1 a ( e a max{µ 1,max (µ j t j } µ j 1 ( 1 aσ 0 e a max{0,max (µ j µ 1 t j } j 1 1 aσ 0 e a, t R k 1, µ Ω. (3.1 I V (t max{µ 1 µ l, max(µ j µ l t j, t l } j 1 j l, t R k 1 aσ 0 < 1, f t V 1 Lemma 3. Let ϕ(t, µ be the functon as defned n (3.2. Then, for a < 0, 1 ln [ 1 a 1 aσ 0, f t V1 ϕ S = sup ϕ(t, µ = µ Ω 1 ln [ e at l a 1 aσ 0, f t Vl, l = {2, 3,..., k}, for 0 < a < 1 σ 0 1 ln [ 1 a 1 aσ 0, ϕ I = nf ϕ(t, µ = t Ω 1 ln [ e at l a 1 aσ 0, f t V1, f t Vl, l = {2, 3,..., k}. f t V l, l = {2, 3,..., k}. Proof. When a < 0, for fxed t V 1 and µ Ω, ( ϕ(t, µ = 1 ( a ln 1 e a max{0, max (µ µ 1 t j } j 1, t R k 1, µ Ω 1 aσ 0 1 [( 1 a ln, 1 aσ 0 (3.2 and the equalty s attaned when max (µ j µ 1 t j 0. Therefore, ϕ S = supϕ(t, µ = j 1 µ Ω [( 1 ln 1, 1 aσ a 0 > 0. 1 aσ 0 7

8 For fxed t V l, l = {2, 3,..., k} ϕ(t, µ = 1 ( a ln 1 e 1 aσ 0 1 [( e at a ln l, 1 aσ 0 max{µ j µ l, max(µ j µ l t j, t l } a j 1 j l, t Rk 1, µ Ω and the equalty s attaned when max(µ j µ 1 t j t l and µ 1 µ l t l. Therefore, j 1 j l [( ϕ I = supϕ(t, µ = 1 ln e at l a 1 aσ 0. µ Ω [( When 0 < a < 1 σ 0, for fxed t V 1, nf ϕ(t, µ = 1 ln 1 µ Ω a 1 aσ 0. Smlarly, for fxed [( t V l, l = {2, 3,..., k}, nf ϕ(t, µ = 1 ln e at µ Ω a 1 aσ 0. Hence the result follows. Defnton 2. An estmator δ(x 1,..., X k of µ S s sad to be locaton-equvarant f δ(x 1 + p,..., X k + p = δ(x 1,..., X k + p, p R. Clearly, any such estmator wll be of the form δ(x 1,..., X k = X 1 + φ(t 2,..., T k, where T = X X 1, = 2,..., k, and δ : R k 1 R s a real-valued functon. Next, under the LINEX loss functon (1.1, we demonstrate the approach of Brewster and Zdek (1974 to derve a general result provdng a suffcent condton for the nadmssblty of a locaton-equvarant estmator of µ S. Theorem 3. Suppose that δ(x = X 1 + φ(t be locaton-equvarant estmator of µ S, where T = (T 2,..., T k = (X 2 X 1,..., X k X 1, and φ( s a real-valued functon defned on R k 1. ( Suppose that for a < 0 and P µ ({φ(t > ϕ S (T } > 0 T Ω, where ϕ S (T s as defned n Lemma 3. Then, the estmator δ s nadmssble and s domnated by δ 1 (X = X 1 + φ 1 (T, where φ 1 (T = mn{φ(t, ϕ S (T }. ( Suppose that for 0 < a < 1 σ 0 and P µ ({φ(t < ϕ I (T } > 0 µ Ω, where ϕ I (T s as defned n Lemma 3. Then, the estmator δ s nadmssble and s domnated by δ 2 (X = X 1 + φ 2 (T, where φ 2 (T = max{φ(t, ϕ I (t}. Proof. 8

9 ( Consder where, for t R k 1 and µ Ω, R(µ, δ R(µ, δ 1 = E [D µ (T, D µ (t =E [ e a(δ(x µ S a(δ(x µ S 1 T = t E [ e a(δ 1 (X µ S a(δ 1 (X µ S 1 T = t =E [ e a(δ(x µ S e a(δ 1(X µ S T = t ae [ δ(x δ 1 (X T = t =E [ e a(x 1+φ(t µ S e a(x 1+φ 1 (t µ S T = t a [ φ(t φ1 (t = [ e aφ(t e k aφ 1(t E [ e a(x µ T = t IV (t a (φ(t φ 1 (t = [ e aφ(t e aφ 1(t e aϕ(t,µ a (φ(t φ 1 (t. Here, ϕ(t, µ s as gven n (3.1. Now suppose that f φ(t ϕ S (t,.e, φ 1 (t = φ(t, then D µ (t = 0. If φ(t > ϕ S (t.e., φ 1 (t = ϕ S (t, then D µ (t = [ e aφ(t e aϕ S(t e aϕ(t,µ a (φ(t ϕ S (t [ e aφ(t e aϕ S(t e aϕ S(t a (φ(t ϕ S (t = [ e a{φ(t ϕ S(t} 1 a [φ(t ϕ S (t. Usng the nequalty e x > 1 + x, x 0,.e., D µ (t 0. Snce P µ {ϕ(t > ϕ S (t} > 0, µ Ω, we get R(µ, φ R(µ, φ 1 > 0, µ Ω. Hence, the result follows. ( Smlar to the proof of (, hence omtted. The natural estmator δ N,1 of µ S, gven n (1.4, can be wrtten as δ N,1 = X 1 + φ N,1 (T, where φ N,1 (T = T I V (T. Usng the results of Theorem 3, we get the followng corollary. =2 Corollary 1. Under the LINEX loss functon (1.1 and a < 0, the natural estmator δ N,1 s nadmssble and s domnated by the estmator δ I N,1(X = X 1 + φ I N,1(T, (3.3 where φ I N,1 (T = mn{φ N,1(t, ϕ S (t}, and ϕ S (t s as defned n Lemma 3. Remark 2. It can be verfy that, the UMVUE and the natural estmators δ N,2 and δ N,3 are not fulfll the suffcent condton for the nadmssblty of an estmator of µ S. Therefore, the mprovement of the UMVUE and the natural estmators δ N,2 and δ N,3 are not possble by usng Theorem 3. 9

10 Now consder a general class Q 0 = {δ p : p = (p 1,..., p k Ω} of estmators of the form δ p (X = (X + p I C (X, p = (p 1,..., p k Ω. Clearly, the natural estmators δ N,1, δ N,2 and δ N,3 are the members of the class Q 0 for the choces p = (0, 0,..., 0, p = ( σ 1, σ 2,..., σ k and p = ( α 1, α 2,..., α k. Let Q 1 = {δ p Q 0 : p > α = 1, 2,..., k} be a subclass of Q 0. In the followng theorem, we wll derve estmators domnatng over a general estmator δ p Q 1, under the LINEX loss functon (1.1. Theorem 4. Let δ q Q 1 be a gven estmators of µ S, let q = (q 1,..., q k R k such that q ( α, p, = 1,..., k. Then the estmator δ q = (X + q I C (X, q = (q 1,..., q k Ω, domnate the gven estmator δ p under the crteron of the LINEX loss functon (1.1. Proof. Consder the rsk dfference. R(µ, δ p R(µ, δ q = E [ e a(δp(x µ S a(δ p (X µ S E [ e a(δq(x µ S a(δ q (X µ S [ [ = E {e a(x +p µ e a(x +q µ }I C (X ae (p q I C (X = (e ap e aq E [ e a(x µ I C (X a For each = {1,..., k}, and µ Ω, defne (p q E [I C (X. B (µ = E ( e a(x µ I C (X = E [ e aµ e ax I C (X. (3.4 Let M (X be an estmator of B (µ such that E [M (µ = B (µ. Now usng Lemma 1, we get [ M (X =e ax e ax I C (X + a e ay I C (X 1,..., X, y, X +1,..., X k e (y X /σ dy =I C (X + e ( aσ =I C (X + 1 aσ ( aσ =I C (X + 1 aσ X ( 1 aσ ( X σ e 1 aσ σ y dy max{x,y } ( ( 1 aσ X σ e 1 aσ (max{x σ,y } e ( e 1 aσ (max{0,y σ X }. (3.5 10

11 Usng Equatons (3.4 and (3.5, we get R(µ, δ p R(µ, δ q = = (e ap e aq E [M (X a E ( [ (e ap e aq I C (X + (p q E [I C (X. ( aσ 1 aσ ( e 1 aσ (max{0,y σ X } a(p q I C (X = E [ψ (X, (say. (3.6 For each fxed {1, 2,..., k}, we have ψ (X = (e ap e aq I C (X a(p q I C (X + aσ (e ap e aq ( e 1 aσ (max{0,y σ X } (1 aσ = (e ap e aq I C (X a(p q I C (X + aσ (e ap e aq I C (X (1 aσ + aσ (e ap e aq ( e 1 aσ (Y σ X I C c (1 aσ (X [ (e ap e aq = a(p q I C (X + aσ (e ap e aq ( e 1 aσ (Y σ X I C c (1 aσ (1 aσ (X [ e aq ( = e ap aq 1 a(p q I C (X + aσ (e ap e aq ( e 1 aσ (Y σ X I C c (1 aσ (1 aσ (X, where C c denotes the complement of the set C. Snce (1 aσ > 0, and e ap aq 1 > (ap aq, t follows that ( e aq ψ (X > 1 a(p q I C (X + aσ (e ap e aq ( e 1 aσ (Y σ X I C c 1 aσ (1 aσ (X Here = ψ (X, (say. Snce the sgn of ψ ( When a < 0. ( e aq ψ 1 aσ 1 a(p q, f X C (X = ( aσ (e ap e aq e 1 aσ (Y σ X, f X C c. 1 aσ depends on the sgn of a, the followng two cases arse: We know that q > α = 1 ( e aq a ln(1 aσ = 1 < 0 (snce a < 0 and p > q 1 aσ ( e aq = a(p q 1 > 0 1 aσ = ψ (X > 0 X C. 11

12 Further, we have, p > q = e ap < e aq = (eap e aq (1 aσ < 0 ( 1 aσ σ = aσ (e ap e aq e 1 aσ = ψ (X > 0, X C c. (Y X > 0 Now, we can conclude that, for fxed {1, 2,..., k}, ψ (X > ψ (X > 0, X χ. It follows from Equaton (3.6 and ψ (X > 0, X χ, = 1, 2,..., k, that R(µ, δ p > R(µ, δ q. ( When 0 < a < 1 σ 0. The proof s smlar to case (, hence omtted. Remark 3. Snce the natural estmator δ N,1 s a member of the class Q 1 (δ p wth p = 0, = 1,..., k, t follows from Theorem 4 that, for all a < 1 σ 0, the natural estmator δ N,1 s nadmssble and s domnated by the natural estmator δ N,3 (δ p wth p = α, = 1, 2,..., k. Remark 4. For 0 < a < 1 σ 0, t s verfy from Theorem 4 that the natural estmator δ N,2 s domnated by the natural estmator δ N,3. But, for a < 0, Theorem 4 fals to fnd an estmator domnatng the natural estmator δ N,2. In Secton 4, we also provde the rsks values of these estmators, whch suggest that the natural estmator δ N,3 performs better than other estmators except few cases. In the followng example, we demonstrate how the varous estmators of the locaton parameter µ S of the selected populaton are computed. Example 1. The followng datasets represent the mean of the daly power output (kw for daly average wnd speed greater than 3(m/s for two dfferent regons n the Unted States under the same condtons. The datasets are compatble wth hypothess that the two populatons are exponentally dstrbuted. Furthermore, these values are mportant because they provde nformaton about the value of the power output of wnd turbnes as well as to mantan the same power densty. Detaled nformaton can be found n the applcaton (Wnd Energy Resource Atlas of the Unted States at Populaton Observatons Regon A (π Regon B (π Let π 1 represent the populaton under Regon A and π 2 represent the populaton under Regon B. We assume that an equal number of samples are randomly taken from 12

13 both the regons. We have ftted the avalable datasets and found the followng: For the populaton π 1, the Kolmogorov-Smrnov (K-S dstance between the actual data and the ftted exponental (13.74, dstrbuton s and the correspondng p-value s Smlarly, for the populaton π 2, the K-S dstance between the actual data and the ftted exponental (10.65, dstrbuton s and the correspondng p-value s Therefore, the avalable datasets have provded suffcent evdent to ndcate that the two populatons are exponentally dstrbuted. The qualty of the populaton s measured n terms of average power out-put,.e., the populaton π 1 Exp(µ 1, σ 1 s better than the populaton π 2 Exp(µ 2, σ 2 f µ 1 > µ 2, and the populaton π 2 s better than the populaton π 1 f µ 1 µ 2. For the goal of selectng the better populaton, we use the mnmax selecton rule ψ c, gven n (1.3. From the data, we have X 1 = 13.74, X 2 = 10.65, σ 1 = , σ 2 = , and c = We wll construct the computaton of the estmators for dfferent values of a. The Regon A has been selected as the best regon by usng the mnmax selecton rule defned n (1.3. Therefore, the estmators of µ S defned n (1.4, 1.5, 1.6, 2.1, 3.3 are δ N,1 = 13.74, δn,1 I = (a = 0.1, (a = 0.1, δ N,2 = , δ N,3 = (a = 0.1, (a = 0.1, δ U = (a = 0.1, (a = Rsk Comparsons of Estmators. In ths secton, we perform a smulaton study to assess the performances of varous estmators of µ S under the LINEX loss functon (1.1. For numercal comparson of varous competng estmators, we take k = 2 and use the mnmax selecton rule ψ c, gven n (1.3, for selectng the best exponental populaton. Clearly ψ c s a functon of (σ 1, σ 2,.e., ψ c vares for dfferent confguratons of (σ 1, σ 2. It can be seen that the rsks of the competng estmators of µ S,.e, the UMRU estmator δ U, the natural estmators δ N,1, δ N,2 and δ N,3, and the estmator δn,1 I (whch mproves on the natural estmator δ N,1 depend on unknown parameter (µ 1, µ 2 only through the dfference µ = µ 2 µ 1. We have compared the rsk functons of the fve competng estmators of µ S for varous values of µ and for dfferent confguratons of a and (σ 1, σ 2. For notatonal convenence, let R 1 (µ = R(µ, δ N,1, R 2 (µ = R(µ, δ N,2, R 3 (µ = R(µ, δ N,3, R 4 (µ = R(µ, δ U and R 5 (µ = R(µ, δn,1 I denote the rsk functons of the varous estmators. Rsk values are provded n Tables 1, 2, 3 and 4 for µ { 4, 3.6, 3.2, 2.8, 2.4, 2, 1.6, 1.2, 0.8, 0.4, 0, 0.4, 0.8, 1.2, 1.6, 2, 2.4, 2.8, 3.2, 3.6, 4}, (σ 1, σ 2 {(2, 1, (1, 2} and a {0.1, 0.1}. The followng observatons are deduced from Tables 1, 2, 3 and The natural estmator δ N,1 s domnated by all other competng estmators for all confguratons of a and (σ 1, σ The UMRUE δ U s domnated by the natural estmator δ N,3 for all confguratons of a and (σ 1, σ Improved estmator δ I N,1 performs better than the natural estmator δ N,1 and 13

14 worst than the natural estmator δ N,3 for all confguratons of a and (σ 1, σ The natural estmator δ N,2 s domnated by the natural estmator δ N,3 for 0 < a < 1 σ 0 and all confguratons (σ 1, σ For a < 0 and all confguratons (σ 1, σ 2, the natural estmator δ N,2 performs better than the natural estmator δ N,3 except for few values of µ. The above observatons suggest that, n practcal applcatons, the use of natural estmator δ N,3 s recommended for 0 < a < 1 σ 0 and the natural estmator δ N,2 s recommended for a < 0. References Abughalous, M. M. and Bansal, N. K. (1994. On the problem of selectng the best populaton n lfe testng models. Communcatons n StatstcsTheory and Methods, 23, Abughalous, M. M. and Mescke, K. J. (1989. On selectng the largest success probablty wth unequal sample szes. Journal of Statstcal Plannng and Inference, 21, Arshad, M. and Msra, N. (2015a. Selectng the exponental populaton havng the larger guarantee tme wth unequal sample szes. Communcatons n StatstcsTheory and Methods, 44, Arshad,M. and Msra, N. (2015b. Estmaton after selecton from unform populatons wth unequal sample szes. Amercan Journal of Mathematcal and Management Scences, 34, Arshad, M. and Msra, N. (2016. Estmaton after selecton from exponental populatons wth unequal scale parameters. Statstcal Papers, 57, Arshad, M. and Msra, N. (2017. On estmatng the scale parameter of the selected unform populaton under the entropy loss functon. Brazlan Journal of Probablty and Statstcs, 31, Arshad, M., Msra, N. and Vellasamy, P. (2015. Estmaton after selecton from gamma populatons wth unequal known shape parameters. Journal of Statstcal Theory and Practce, 9, Brewster, J. F. and Zdek, Z. V. (1974. Improvng on equvarent estmators. The Annals of Statstcs, 2, Gangopadhyay, A. K., and Kumar, S. (2005. Estmatng average worth of the selected subset from two-parameter exponental populatons. Communcaton s Statstcs- Theory and Methods, 34,

15 Kumar, S., Mahapatra, A. K., and Vellasamy, P.(2009. Relablty estmaton of the selected exponental populatons. Statstcal Probablty Letters, 79, Meena, K. R., Arshad, M. and Gangopadhyay, A. K. (2017. Estmatng the parameter of selected unform populaton under the squared log error loss functon. Communcaton s Statstcs-Theory and Methods, (just-accepted. Msra, N., Anand, R., and Sngh, H. (1998. Estmaton after subset selecton from exponental populatons: locaton parameter case. Amercan Journal Mathematcal Managemment Scences, 18, Msra, N. and Arshad, M. (2014. Selectng the best of two gamma populatons havng unequal shape parameters. Statstcal Methodology, 18, Msra, N. and Dharyal, I. D. (1994. Non-mnmaxty of natural decson rules under heteroscedastcty. Statstcs and Decsons, 12, Msra, N., and Sngh, G. N. (1993. On the UMVUE for estmatng the parameter of the selected exponental populaton. Journal of Indan Statstcal Assocaton, 31, Msra, N., and van der Meulen, E. C. (2001. On estmaton followng selecton from nonregular dstrbutons. 30, Nematollah, N. and Jozan, M. J. (2016. On rsk unbased estmaton after selecton. Brazlan Journal of Probablty and Statstcs, 30, Nematollah, N. and Motamed-Sharat, F. (2012. Estmaton of the parameter of the selected unform populaton under the entropy loss functon. Journal of Statstcal Plannng and Inference, 142, Nematollah, N. and Pagheh, A. (2017. Estmaton of the locaton parameter and the average worth of the selected subset of two parameter exponental populatons under LINEX loss functon. Communcatons n Statstcs-Theory and Methods, 46, Parsan, A. and Farspour, N. S. (1999. Estmaton of the mean of the selected populaton under asymmetrc loss functon. Metrka, 50, Rsko, K. J. (1985. Selectng the better bnomal populaton wth unequal sample szes. Communcatons n StatstcsTheory and Methods, 14, Varan, H. R. (1975. A Bayesan approach to real estate assessment. Studes n Bayesan Econometrc and Statstcs n honor of Leonard J. Savage, Vellasamy, P. (1996. A note on the estmaton of the selected scale parameters. Journal of Statstcal Plannng and Inference, 55, Vellasamy, P. (2009. A note on unbased estmaton followng selecton. Statstcal Methodology, 6,

16 Vellasamy, P. and Punnen, A. P. (2002. Improved estmators for the selected locaton parameters. Statstcal Papers, 43, Vellasamy, P., Kumar, S. and Sharma, D. (1988. Estmatng the mean of the selected unform populaton. Communcatons n StatstcsTheory and Methods, 17,

17 Table 1: Comparson of rsk functons for a = 0.1 µ (σ 1, σ 2 = (2, 1; c c (σ 1, σ 2 = R 1 (µ R 2 (µ R 3 (µ R 4 (µ R 5 (µ

18 Table 2: Comparson of rsk functons for a = 0.1 µ (σ 1, σ 2 = (2, 1; c c (σ 1, σ 2 = R 1 (µ R 2 (µ R 3 (µ R 4 (µ R 5 (µ

19 Table 3: Comparson of rsk functons for a = 0.1 µ (σ 1, σ 2 = (1, 2; c c (σ 1, σ 2 = R 1 (µ R 2 (µ R 3 (µ R 4 (µ R 5 (µ

20 Table 4: Comparson of rsk functons for a = 0.1 µ (σ 1, σ 2 = (1, 2; c c (σ 1, σ 2 = R 1 (µ R 2 (µ R 3 (µ R 4 (µ R 5 (µ