Estimators when the Correlation Coefficient. is Negative

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1 It J Cotemp Math Sceces, Vol 5, 00, o 3, Estmators whe the Correlato Coeffcet s Negatve Sad Al Al-Hadhram College of Appled Sceces, Nzwa, Oma abur97@ahoocouk Abstract Rato estmators for the mea of Y use avalable formato o a aular varable X to mprove the estmato Whe the correlato coeffcet betwee X ad Y s egatve, the the product estmator s used I ths paper two geeral tpes of estmators based o aular varable are suggested ad vestgated to reach the optmalt of these estmators Kewords: Rato estmator, product estmator; Mea square error, Aular varable INTRODUCTION Assumg that the populato mea μ of a aular varable X s kow, the product estmators for the populato mea are = ( ) / μ = μ = Where = (/ ) X, = (/ ) Y Murth( 963) showed that the = = Bas ( ) = Bas ( ) The he combed the two estmators ad proposed the followg ubased estmator 3 = Two product-tpe estmators for the populato mea were suggested Upadhaa & Sgh (999) ( + V ) 4 = μ + V ( )

2 46 S A Al-Hadhram 5 = ( + K ) ( μ + K ) where V ad K are the coeffcet of varato ad the coeffcet of kurtoss of X, respectvel Upadhaa & Sgh (999) also suggested the two product- tpe estmators ( K + V ) 6 = ( μ K + V ) ad [ V + K ] 7 = [ μv + K ] I ths artcle we provde two geeral forms of estmators ad compared them wth the tradtoal product estmators FIRST FORM OF PRODUCT TYPE ESTIMATOR The estmators 4, 5, 6 ad 7 metoed above are actuall specal cases of the geeral form α ˆ μ = () αμ wth α ad γ are costats wth α 0 or γ 0 The epectato of ths estmator s α γ E( ˆ μ ) = E(XY) + E(Y) αμ αμ ασ XY = + μ ( αμ ) αμ ασ XY E ( ˆ μ ) = μ + ( αμ The estmator s based ad the amout of bas s Bas ( ˆ μ ) = ασ XY / ( αμ ) () If / μ > α /( αμ, the we get Bas ( ˆ μ ) < Bas ( ) It s clear that f α = 0, the the estmator () s reduced to mea per ut,, whch s ubased The bas ca be also reduced b creasg sample sze or/ad choose αμ to be large used Let To stud the MSE of the estmator, oe degree of Talor epaso s ( ) ˆ μ μ + ( μ ) + D μ

3 Estmators whe the correlato coeffcet s egatve 47 where D = αμ /( αμ The a appromated MSE of the estmator s MSE V ar D V ar DCov [ ] ( ) ( ˆ μ ) (Y) + (X) + X,Y = Var(Y) + DVar(X) D + I order to get α ad γ that mmze the MSE, we take the partal dervatves wth respect to α ad γ These gve the followg two equatos ( γμ) αμ MSE ( ˆ μ) = Var (X) + α ( αμ ( αμ αμ αμ MSE ( ˆ μ) = Var (X) + γ ( αμ ( αμ MSE ( ˆ μ) MSE ( ˆ μ) Settg = 0 ad = 0 wll gve γ = ( α) μ / + μ α γ Whch ca be wrtte as γ = ( αμ) [ V X /( ρvy ) + ] where V X ad VY are the coeffcet of varatos ad ρ s the correlato coeffcet ad α 0, ad the mmum MSE s MSEm ( ˆ μ) Var (Y) Var (X) = Var(Y) ρ The costat γ cludes the parameter of terest, μ, whch s ukow However, we ma use a good estmate for μ Let cosder the choce ˆ μ = = (/ ) = So ˆ γ = ( α) + μ Substtute ths the ma estmator () gves ˆ μ = + ( μ ) (3) whch s the regresso estmator The appromated MSE of the estmator (3) s MSE ( ˆ μ ) Var (Y) Var (X) ( σy / )( ρ ) = Let ow use the trasformato u = α whereα ad γ are a arbtrar costats wthα 0 or γ 0 Ad let also u = (/ ) u ad U = αμ The, we defe a ew estmator ) ) ˆ μ = ) ) where = (u / U) ad = (/ U ) u = = (4)

4 48 S A Al-Hadhram Lemma(): The estmator [ ] Proof ) ) ˆ μ = /( ) s ubased estmator for ˆ μ = u / U ( / U ) ( ) α = = ( / ) ( / ) ( ) U α α γ = = = ( α / ) ( ) γ ( ) U + = = = = {( α / ) + ( ) γ j ( ) U } Takg the epectato we get E( ˆ μ) = {( α / ) μ μ μ } U = μ μ 3 THE SECOND FORM OF PRODUCT TYPE ESTIMATOR Let cosder estmators the form + ( μ ) ˆ μ = ( α (5) αμ wth = σxy / σx, α ad γ are costats whereα 0 or γ 0 + ( μ ) Lemma( ): The estmator ˆ μ = ( α s ubased estmator for αμ μ Proof Let us wrte ˆSRS μ as ( α ) ( μ )( α ˆ μ = + αμ αμ α ˆ μ = + αμ μ α γ αμ αμ The the epectato of ˆμ s

5 Estmators whe the correlato coeffcet s egatve 49 E ˆ E E ( μ) = α (XY) μ + αμ α (X ) αμ αμ = α( Cov + μ μ ) μ + αμ α + μ αμ αμ (X,Y) ( Var (X) ) After smplfcato we get E( ˆ μ) = α Cov(X,Y) Var( X ) μ ( αμ γ) αμ γ Sce = Cov ( X, Y )/ V ( X ), we get E ( ˆ μ ) = μ To vestgate the MSE of ˆSRS μ, we used frst order of appromato from Talor epaso Ths gves αμ ˆ μ μ + ( μ) + ( )( μ) αμ wth appromated MSE s MSE ( ˆSRS ) Var (Y) αμ αμ μ + Var (X) + Cov (X, Y) αμ αμ To compare the MSE of ths form of estmators wth the tradtoal product tpe, αμ = ( )/ μ, let state R = μ / μ, ad D = αμ The we got MSE ( μ ) < MSE (Y) f ad ol f ˆ D Var + DCov < R Var + RCov (X) (X, Y) (X) (X, Y) whch ca be wrtte as Var(X) D R < Co v(x, Y) [ R D] Provded that R D 0, we got Var(X) [ D + R] < Co v(x, Y) Therefore, [ D + R] < / where = σxy / σx That s MSE ( ˆ μ ) < MSE (Y) f [ D + R] < / I order to get the costats α ad γ that mmze the MSE, we take the partal dervatves of the MSE wth respect to α ad γ These gve the followg two equatos ( γμ) αμ MSE ( ˆ μ) = Var (X)( ) Cov (X, Y) + α ( αμ αμ

6 50 S A Al-Hadhram αμ αμ MSE ( ˆ μ) = Var (X)( ) Cov (X, Y) + γ ( αμ αμ MSE ( ˆ μ) MSE ( ˆ μ) Settg = 0 ad = 0 we get that ether γ = 0 or α = 0 α γ provded that μ 0 Therefore, the geeral estmator, ˆμ, has mmum varace f we wrte t as ˆ μ = + ( μ ) whch s the regresso estmator Refereces [] SLLohr, Samplg: Desg ad Aalss, Dubur Press, 999 [] MN Murth, Product method of estmato, Sakha, A6(963), [3] LNUpadhaa, HP Sgh, Use of trasformed aular varable estmatg the fte populato mea, Bometrcal Joural 4(999), Receved: November, 009

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