Estimators when the Correlation Coefficient. is Negative

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Estimators when the Correlation Coefficient. is Negative"

Φώτιος Ιωάννου
7 χρόνια πριν
Προβολές:

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

Σχετικά έγγραφα

CS 1675 Introduction to Machine Learning Lecture 7. Density estimation. Milos Hauskrecht 5329 Sennott Square

CS 1675 Introduction to Machine Learning Lecture 7. Density estimation. Milos Hauskrecht 5329 Sennott Square CS 675 Itroducto to Mache Learg Lecture 7 esty estmato Mlos Hausrecht mlos@cs.tt.edu 539 Seott Square ata: esty estmato {.. } a vector of attrbute values Objectve: estmate the model of the uderlyg robablty