Estimating the Common Mean of k Normal Populations with Known Variance

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1 Iteratoal Joural of Statstcs ad Probablty; Vol 6, No 4; July 07 ISSN E-ISSN Publshed by Caada Ceter of Scece ad Educato Estmatg the Commo Mea of Normal Populatos wth Kow Varace N Sajar Farspour & A Asgharzadeh Departmet of Statstcs,Faculty of mathematcal Sceces, Alzahra Uvercty, Tehra, Ira Departmet of Statstcs,Faculty of Basc Sceces, Uvercty of Mazadara, Babolsar, Ira Correspodece: N Sajar Farspour, Departmet of Statstcs,Faculty of mathematcal Sceces, Alzahra Uvercty, Tehra, Ira E-mal: sajar@alzahraacr Receved: May 3, 07 Accepted: Jue 9, 07 Ole Publshed: Jue 6, 07 do:05539/jspv64p70 URL: Abstract Cosder the problem of estmatg the commo mea of ormal populatos wth ow varaces We study the admsblty of the Best lear Rs Ubased Equvarat (BLRUE) estmator of the commo mea of ormal populatos uder the squared error ad LINEX loss fucto whe the varaces are ow Keywords: Admssblty, asymmetrc loss, best Equvarat estmator, commo mea, LINEX loss, rs ubased Itroducto Suppose we have depedet populatos where the th populato follows from N(θ,σ ), =,, The parameter θ s uow ad σ > 0, =,, are all assumed to be ow Let X j, j =,, be d observatos from the th populato, Defe X as X = X j= j =,,, ad ote that X ~ N(θ, σ ) Combg two or more ubased estmators of a uow parameter θ order to obta a batter ubased estmators ( the sese of smaller rs) s a problem that ofte arses statstcs; for example whe depedet sets of measuremets of the same quatty are avalable The problem of estmatg the commo mea of two or more depedet ormal populatos has receved atteto from several authors the past For some refereces ths regard see Graybll ad Deal(959),Sha ad Mouqadem(989) ad Pal ad Sha(996)for a complete bblography See also Lehma ad Casella (998) pp 95-96, Sajar Farspour (999), Sajar Farspour ad Asgharzadeh (00), for further refereces ad commets I secto, a class of rs ubased estmators whch combes the meas of the samples e, X s, s developed ad secto 3 the rejo of admssblty of the estomators of the from c X + d s derved uder the squared error loss fucto L (δ, θ) = (δ θ) () Whch s a symmetrc loss fucto I secto 4, the admssblty of the estmators of the from j= c X + d are studed uder the Loss fucto I practce, the real loss fucto s ofte ot symmetrc ad overestmato ca lead to more or less severe cosequeces tha uderestmato Vara(975) employed a asymmetrc loss fucto, whch s ow as LINEX loss, ad was extesvely used by Zeller(986), Rojo(987), Sadoogh-Alvad ad Nematollah (989) ad others I ths Regard, our ext loss fucto s L, b e a(δ θ) a where a 0 ad b > 0 The rego of the admssblty ad admssblty of the estmators of the form c X + d uder the loss fucto() are derved secto 5 ad 6 Rs Ubased Equvarat Estmato From decso theoretc approach whe symmetres are preset a problem, It s atural to requre a correspodg symmetry to hold for the estmators The locato parameter estmato problem s a mportat example It s 70

2 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 symmetrc, or, to use the usual termology, equvarat wth respect to traslato of the sample space, that s δ(x + a) = δ(x) + a for all a () Where = (X, X,, X ) A estmator satsfyg () wll called equvarat uder traslato A alteratve mpartalty restrcto whch s applcable to our problem s the codto of ubasedessfollowg Lehma ad Casella (998), a estmator δ of θ s sad to be rs-ubased f t satsfes ad f the loss s as (),() reduces to ow ote that E θ [L(θ, δ)] E θ [L(θ, δ)] for all θ = θ () E θ[e aθ ] = e aθ (3) E θ [e ax a σ ] = e aθ so ax a σ, =,, are all rs ubased estmators of θ Now, cosder a combed estmator of the from θ α = α Where 0 < α <, =,,, are real umbers ad estmator Also ote that ad hece θ RU a aθ α+ E [e (α) = α (X a σ ) α a ( σ )σ (X a σ = We ca verfy that θ α s a traslato-equvarat ] = e aθ ) + a a ( σ )σ Is a rs ubased estmator of θ o the bass of X s uder the LINEX loss fucto The rs fucto of θ RU (α) wth respect to the loss () s easly computed as The rs (4) mmzed uder α R(θ, θ RU ) = ba =, whe α = /σ /σ α σ (4), =,, ad hece the Best Lear Rs Ubased Equvarat (BLURE) estmator of θ uder the LINEX loss s ( /σ /σ )X a /σ (5) Wth the same approach, the BLRUE estmator of θ uder the squared error () s ( /σ )X (6) /σ The estmator (6) s also the uque mmum varace ubased estmator (UMVUE) as well as the best lear ubased estmator (BLUE) (wthout ormalty) for estmatg θ Both estmators (5) ad (6) are specal cases of the more geeral class of lear estmators of the form c X + d To study admssblty of the estmators (5) ad (6), we study admssblty of the class of lear estmators of the form c X + d It should be metoed here that (5) ad (6), the BLURE estmators are see to deped o σ ( =,, )whe σ ( =,, ) are completely uow, they ca be replaced by (X X ) /( ) ( =,, ) 7

3 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 I ths case reasoable estmators of θ are provded by /s ( /s )X a /s (7) Ad ( /s )X (8) /s Obvously the estmators (7) ad (8) are locato equvarat (see()) but ther rss are complcated 3 Admssblty Results uder Loss() Cosder the admssblty of a arbtrary lear fucto fucto X + d wth respect to the squared error loss () s c c ρ(c,, c, d) = E[ c X + d θ] X + d uder the loss () The rs So, we have the followg theorem Theorem 3: The estmator c = c σ + [( c )θ + d] (3) X + d s admssble for θ wheever 0 c, =,,, ad 0 c < or c = /σ Proof: The otato δ(c,, c,d) s used for c X + d /σ, =,,, ad d = 0 ()The case 0 c, =,,, ad 0 c < s cosdered frst If c = 0, =,,, the δ(0,, 0,d) s admssble sce t s the oly estmator wth zero rs at θ = d For fdg the Bayes estmator of θ, cosder the ormal pror wth mea μ ad varace τ The posteror dstrbuto s the ormal wth mea ad varace gve by σ X + μ τ σ + τ ad σ + τ Respectvelyt ca be see that the uque Bayes estmator s σ σ + τ ( ) X + μ τ σ + τ (3) ad that the assocated Bayes rs s fte ad hece admssble It follows that δ(c,, c,d) s admssble wheever 0 c,, =,,, ad 0 c < () If c = σ σ = c (say), =,,, ad d = 0, the rs of δ(c,, c,0) as see from (3) s gve by Note that f c = ad = 0, the we have ρ(c₁,, c, 0) = /σ c σ ρ(c,, c, d) = (33) It ca be show that the rs (33) s mmzed uder c =, whe c = c, ad hece c X s 7

4 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 admssble whe c c To show that δ(c,, c,0) s admssble, the lmtg Bayes method due to Blyth (95)may be used Suppose that δ(c,, c,0) s ot admssble The, there s a estmator δ * such that R(θ, δ ) R(θ, c X ) = /σ For all θ, ad wth strct equalty for at least some θ Now, R(θ, δ) s a cotuous fucto of θ for every δ so that there exsts ε > 0 ad θ 0 < θ such that R(θ, δ ) < ϵ /σ For all θ 0 < θ < θ Let r τ be the average rs of δ wth respect to the pror dstrbuto N(0, τ ), ad let r τ be the Bayes rs of the Bayes estmator (3) wth respect to N(0, τ ) The t follows that Hece r τ = + τ σ r τ σ πτ [ R(θ,δ θ + )]e τ dθ σ = r τ σ σ σ + τ τ( σ )( σ + τ )ε π θ e θ τ dθ θ 0 The tegrad coverges mootocally to as τ ad hece by the Lebesgue mootoe covergece theorem, the tegral coverges to θ θ 0 ad hece the rato coverges to Thus, there exsts τ 0 < such that r τ0 < r τ0, whch cotradcts the fact that r τ0 s the Bayes rs for N(0, τ 0 ) It follows that δ(c,, c,0) s admssble 4 The Iadmssblty Results uder Loss () To see what ca be sad about the other values of c s, =,,, we shall ow prove a admssblty result for lear estmators X + d, whch s qute geeral ad partcular does ot requre the assumpto of ormalty c Theorem 4: The estmator codtos hold () c >, for some =,,, c () c, c +c j >, for some, j =,,, () c < 0, j c j, for some =,,, (v) c, c j < 0, for some =,,, j (v) c 0 Proof : () If c >, for some =,,,, the t follows from (3) that So that c X + d s admssble uder squared error loss wheever oe of the followg ρ(c,, c, d) c σ X + d s domated by X ad hece s admssble > σ = ρ(0,,0,,0,,0) () If c, c +c j >, for some, j =,,,, the c j > ( c ) ad hece 73

5 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 ρ(c,, c, d) c σ + c j σ j j > c σ + ( c ) σ j j But the fucto g(c ) = c σ + ( c)σ j σ s mmzed whe c = j/ j = c j σ / +σ m (say), also j/ j Thus, c g(c m ) = σ σ j/ j σ / +σ j/ j = R(θ, c m X + ( c m )X j ) X + d s domated by c m X + ( c m )X j ad hece s admssble () If c < 0, j c j, for some =,,,, the ( c ) > ( j c j ) ad hece ρ(c,, c, d) > c j σ j j j + [( c ) θ + d] = c j σ j j j + ( c ) d [θ + c ] > c j σ j + ( c j j j j = c j σ j j j + [( c j j ) d [θ + c ] ) θ + ( j cj )d j c = ρ (c,, c, 0, c +,, c, ( j c j )d ) j c ] Thus ths case, c X + d s domated by the estmator c j X j + j ( j c j )d j c (v) If c, j c j < 0, for some =,,,, the ( c ) > (c ) ad hece ρ(c,, c, d) c σ + [( c ) θ + d] = c σ + ( c ) d [θ + c ] > c σ d + (c ) [θ + c ] = c σ + [(c )θ + (c )d c ] 74

6 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 = ρ (0,,0, c, 0,,0, (c )d ) c Thus, c X + d s domated by c X + (c )d c (v) If c 0, the( c ) > ad hece ρ(c,, c, d) [( c )θ + d] = ( c ) d [θ + c ] d > [θ + c ] = ρ (0,,0, d ) c Thus, c X + d s domated by the costat estmator δ = d 5 The Admssblty Results Uder Loss () c Cosder the questo of admssblty of the estmators of the from c X + d uder the loss () Sce the parameter b does ot have ay flueces o our results so wthout loss of geeralty we ca tae b = The rs fucto of the estmator X + d wth respect to the loss () s easly computed as c γ(c,, c, d) = E [e a( c X +d θ) a ( c X + d θ) ] So, we have the followg theorem Theorem 5: The estmator c <, or c = /σ Proof : /σ c a = e ( c σ )+a( c )θ+ad a( c )θ ad (5) X + d s admssble for θ wheever 0 c <, =,, ad 0, =,, ad d = a /σ () The case 0 c <, =,, ad 0 c <, s cosdered frst If c = 0, =,,, the δ(0,, 0, d) s admssble sce t s the oly estmator wth zero rs at θ = d Now cosder the Bayes estmator whe the pror dstrbuto o θ s ormal wth mea μ ad varace τ The, usg (3) Zeller (986), t follows that the uque Bayes estmator s /σ ( ) X /σ +/τ /σ +/τ ad that the assocated Bayes rs s fte ad hece admssble It follows that 0 c <, =,, ad 0 < c < ( a μ τ) (5) c X + d s admssble wheever () If c = /σ /σ (the same c ) ad d = a /σ = d (say ), the the rs of δ(c,, c, d ) as s see from 75

7 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 (5) s gve by γ(c,, c, d ) = a /σ Note that f c =, the we have γ(c,, c, d) = e a ( c σ )+ad ad (53) It ca be show that the rs (53) s mmzed whe c = c ad = d, ad hece ths case δ(c,, c, d) s admssble whe c c ad d d To show that δ(c,, c, d ) s admssble, aga the lmtg Bayes method may be used Suppose that δ(c,, c, d ) s ot admssble, the there exsts a estmator δ such that R(θ, δ ) R(θ, c X + d ) = a /σ for all θ, ad wth strct equalty for at least some θ By the cotuty of R(θ, δ), there exsts ε > 0 ad θ 0 < θ such that R(θ, δ ) < a /σ for all θ 0 < θ < θ Let r τ be the average rs of δ wth respect to the pror dstrbuto N(0, τ ) The t ca be show that Hece r τ = a ε ( /σ + /τ ) a a r /σ τ = r /σ τ πτ + [ a /σ a R(θ, δ )] e θ τ dθ a /σ ( /σ + /τ ) > τ( /σ )( /σ +/τ )ε θ e θ πa τ dθ θ 0 The tegrad coverges mootocally to as τ ad hece by the Lebesgue mootoe covergece theorem, the tegral coverges to θ θ 0 ad hece the rato coverges to Thus, there exsts τ 0 < such that r τ0 < r τ0, whch cotradcts the fact that r τ0 s the Bayes rs for N(0, τ 0 ) It follows that c = c for =,,, ad d = d 6 The Iadmssblty Results Uder Loss () We shall ow prove a admssblty result for lear estmators uder the loss () Theorem 6: The lear estmator codtos hold () c > for some =,, c () c, c + c j >, for some, j =,,, () c < 0 for some =,, Proof: () If c > for some =,,, the c X + d s admssble whe X + d s admssble uder LINEX loss wheever oe of the followg 76

8 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 a γ(c,, c, d) = e ( c σ )+a( c )θ+ad a ( c ) θ ad a σ j ( c j j= ) (sce e x + x) j > a c σ a σ = γ (0,,0,,0,,0, aσ ) So that c X + d s domated by X aσ () If c, c + c j >, for some, j =,,,, the c j > ( c ) ad hece γ(c,, c, d) a c σ > a c σ + a ( c ) σ j j + a c j σ j j But the fucto g(c ) = a c σ + a ( c ) σ j j s mmzed at c = σ j /j σ / +σ j /j (the same c m ), ad g(c m ) = Where a σ σj / j (σ / +σ j /j ) = R(θ, c mx + ( c m )X j + d 0 ) d 0 = a σ σj / j (σ / +σ j /j ) Thus ths case, c X + d s domated by c m X + ( c m )X j + d 0 ) () If c < 0 for some =,,, the t wll be show that c X + d s domated by j c j X j + d where c j = c j c for j ad d = d Now, usg the equalty + ac c ( c ) 3 j c j σ j j sce e a c σ >, we have from (5) γ(c,, c, d) γ(c,, c, 0, c +,, c, d ) a > e a e c j σ j j +a( c j )θ+ad j j= c j σj j +a( c j j )θ+ad j ac θ a j (c j c j )θ a(d d ) e x e y (x y)e y for all x, y, ad otg that c j c j = c (c )c j 0, for all j =,,, ad j,t follows that ( c ) 77

9 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 γ(c,, c, d) γ(c,, c, 0, c +,, c, d ) > [ac θ + a j (c j c j )θ + a(d d )] a e But c j c j = c c j for j ad d = c d ac c j σj j +a( c j j j )θ+ad aθ ac θ a j (c j c j )θ a(d d ) c j σj j j, hece γ(c,, c, d) γ(c,, c, 0, c +,, c, d ) > [ac θ ac ( c j )θ ac d a c c j σ j a e ac θ + ac ( ] j j j c j σj j +a( c j j j )θ+ad aθ j c j )θ + ac d + a c c j σ j = c [ a c j σ j + a( c j )θ + ad aθ] j [e j a c j σj j +a( c j j )θ+ad a j j j θ ] 0 j Sce c < 0 ad (e y ) 0, for all y Remar (6): The BLRUE estmators gve (5) ad (6) are admssble ad mmax Acowledgemet The grat of alzahra uversty s apprecated Refereces Blyth, C R (95) O mmax statstcal decso procedures ad ther admssblty A Math Statst,, -4 Graybll, FA, & Deal, RB (959) Combato of ubased stmators Bometrcs, 5, Lehma, E, L, & Casella, G (998) Theory of pot Estmato Sprger-Verlag, New Yor Pal, N, & Sha, B K (996) Estmato of a commo mea of several ormal populatos: a revew Far East jmath Sc Specal volume Part I, 97-0 Rojo, j (987) O the admssblty of cx + d wth respect to the LINEX loss fucto, commo Statst Theory Meth, 6, Sadoogh-Alvad, S M, & Nematollah, N (989) A ote o the admssblty of cx + d relatve to the LINEX loss fucto, commu Statst Theory Meth 8, Sajar Farspour, N (999) Rs ubased equvarat estmato of a commo ormal mea vector uder LINEX loss fucto Iraa Joural of Scece ad Techology, 3, -6 Sajar Farspour, N, & Asgharzadeh, A (00) O the admssblty of estmators of the commo mea of two ormal populatos uder symmetrc ad asymmetrc loss fuctos South Afrca Statst J 36, Sha, B K, & Mouqadem, O (98) Estmato of the commo mea of two uvarate ormal populatos, commu Statst Theory Meth,, Vara, H R (975) A Bayesa approach to real estate assessmet studes bayesa Ecoometrcs ad statstcs 78

10 Iteratoal Joural of Statstcs ad Probablty Vol 6, No 4; 07 hoor of Leoard j, Savage eds S E Feberg ad A Zeller, Amsterdam: North Hollad Zeller, A (986), Bayesa estmato ad predcto usg asymmetrc loss fucto, Jour Amer statst Assoc, 8, Copyrghts Copyrght for ths artcle s retaed by the author(s), wth frst publcato rghts grated to the joural Ths s a ope-access artcle dstrbuted uder the terms ad codtos of the Creatve Commos Attrbuto lcese ( 79