5 Augut 0. Vol. o. 005-0 JATIT & LLS. All right reerved. ISS: 99-865 www.jatit.org E-ISS: 87-395 A EW OPTIMAL QUADRATIC PREDICTOR OF A RESIDUAL LIEAR MODEL I A FIITE POPULATIO MOHAMMED EL HAJ TIRARI, ABDELLATIF EL AFIA, RDOUA FAIZI Inea, Rabat-Intitut, BP 67, Rabat, Maroc Cret-Enai, Rue Blaie Pacal, BP 3703, 357 Bruz cedex, France Enia, Rabat - Intitut, BP 73, Rabat, Maroc E-mail: tirari@inea.ac.ma, elafia@enia.ma, faizi@enia.ma ABSTRACT In thi paper, we deal the problem related to the prediction of a reidual quadratic form of a uperpopulation model in the cae of a finite population. We propoe a new optimal quadratic predictor of thi quadratic form for a linear Gauian model Y=Xβ+e. The main reult of thi paper i ued to derive an optimal quadratic predictor of the reidual um of quare of a linear Gauian model. Keyword: Survey Sampling, Finite Population, Super-Population Model, Quadratic Prediction. ITRODUCTIO Linear model play a key role in improving the preciion of parameter etimator of a finite population ince they allow to take into account the auxiliary information available on population unit. Indeed, uing the predictive approach in the ampling theory ha enabled to conuct the Bet Linear Unbiaed Predictor (BLUP of the total of an interet variable (Royall, 97; Cael et al., 993; Valliant et al., 000. However, the optimality of thi predictor cannot be guaranteed unle the linear model adopted i valid. Therefore, analyi of the quality of the linear model play a role in the conuction of the predictor BLUP. Thi analyi i carried out through the reidue of the linear model, epecially the reidual um of quare. Thi i why it i very important to predict the reidual um of quare of a linear model or any other quadratic form of thee reidue. Under the predictive approach by auming the e= Ν 0,σ V, Liu linear model Y=Xβ+e ( Rong (007 were mainly intereted in predicting a poitive quadratic form Y QY Q i a ymmeic nonnegative definite maix atifying QX = 0. Thu, conidering the cla of invariant predictor (ee definition below, Liu Rong propoe an optimal quadratic predictor following the minimization criterion of the Mean Squared Error under the model. In thi paper, we focu on cae in which the function to predict i a quadratic reidue, of a multiple linear model, e Qe Q i a ymmeic maix. We propoe a new optimal predictor of thi quadratic form whoe expreion i much impler than that of the predictor of Liu Rong. Therefore, Section i devoted to notation definition ued in thi work. In Section 3, we put forward an optimal quadratic predictor of e Qe following the minimization of the Mean Squared Error under the model. Finally, in Section, the main reult of thi paper are applied to the problem of predicting the reidual um of quare of a linear Gauian model.. OTATIOS AD DEFIITIOS Given a finite population U = {,..., k,..., } compoed of unit, focu will laid on a variable of interet y = ( y,..., y. For thi reaon, a ample of n unit i elected from the population U the value of y are oberved for thee unit. We note the et of unit U unelected by r. We have value of p auxiliary variable X,..., X p which can be repreented by maix X= ( x,..., x,..., x k x k i the vector of value of p variable X,..., X p for a unit k U. To implify the notation in thi paper, for any maix A, we note A mn, if A i a real maix; 70
Journal of Theoretical Applied Information Technology 5 Augut 0. Vol. o. 005-0 JATIT & LLS. All right reerved. ISS: 99-865 www.jatit.org E-ISS: 87-395 y A n if A nn, it i ymmeic; A 3. THE OPTIMAL QUADRATIC n PREDICTOR y if A n i nonnegative definite. ote alo A, A, P = ( y A A AA A Given e Qe Q a reidual quadratic rk ( A repectively the anpoe, the generalized form of the multiple linear model (.. To predict thi quadratic form, we can ue the quadratic invere, the orthogonal projector, the rank of y predictor eae ˆ ˆ A n A, C ( A the ubpace panned by the column of eˆ ˆ A ( A the ace of A. = β Y X ˆβ i given by (.. In general, the maix A i choen in uch a way that the Under the approach baed on a model, we aume that the value y,..., predictor eae ˆ ˆ atifie an optimality criterion. A y of the interet choice criterion of the maix A may minimize variable y are the achievement of a rom vector the Mean Squared Error (MSE under the model Y = ( Y,..., Y whoe joint probability (.. Thi criterion wa ued by Liu Rong (007 to determine an invariant optimal quadratic diibution ξ i given by a uper-population predictor of a quadratic form Y QY maix model. In thi work, we conider the following Q atifie QX = 0. ote that a quadratic linear model: predictor YAY i aid to be invariant if maix Y=Xβ+e e = Ν0(,σV (. A atifie AX = 0. Thu, to predict a E ξ ( Y = β X Cov ξ ( Y = σ V. The quadratic form Y QY, Liu Rong (007 demonated that the optimal quadratic predictor in covariance maix V i aumed to be known the cla of invariant unbiaed quadratic β, σ are unknown parameter. To etimate * predictor i given by YAY β, we can ue the bet linear unbiaed etimator ˆβ * * A = λ X + X V QV X given by ˆ β= ( XV X XV Y (. ( QV QV * X V for a given ample of n unit elected from rk ( T rk ( X U, vector Y maice X V are defined through the following decompoition of Y, X + + ( + + X = T TX XTX XT. V : Y X e The predictor propoed by Liu Rong can be Y =, X=, e= Yr Xr er ued to predict a quadratic form of reidue e Qe of the multiple linear model (. a the invariance condition implie that YAY V Vr V = eae ˆ ˆ but not V = = the bet to ue in the cae of predicting e Qe. Vr Vr V Indeed, by conidering the cla of quadratic Y = ( Y,..., Y n, X = ( x,..., x n, predictor eae ˆ ˆ (out impoing the invariance V n V = ( V, V r condition, we propoe a quadratic predictor of. e Qe whoe expreion i much impler than that It i alo noted by T the maix defined by of the predictor of Liu Rong while alo being optimal in the cla of quadratic predictor. ote T = V + X UX that the prediction Mean Squared Error (MSE of p U defined o that C( T = C( X, V. 7 eae ˆ ˆ cannot be calculated exactly but can be approximated by that of the quadratic form eae e = β Y X β i the coefficient vector of the multiple linear model (.. In fact, an approximation of the prediction mean quared error under the model (. i given by
5 Augut 0. Vol. o. 005-0 JATIT & LLS. All right reerved. ISS: 99-865 www.jatit.org E-ISS: 87-395 MSEξ ( eae ˆ ˆ MSEξ ( eae We note that a C( V = C( V, we have Eξ ( e Ae e Qe V = VV V σ ( AVAV ( AV QV = ( AVV V QVV V σ ( AV QV = ( AVGV +σ ( QVQV G = V + σ ( ( V QVV. AV QV Thi expreion of MSE ξ ( eae i obtained Moreover, by conidering A = V AV by uing the fact that for any variable G = V GV, the minimization problem (3.3 can Zμ= Ω Ν,, we have be rewritten a follow ( ( Z AZ Z BZ μ( AWBW AWBμ Covξ, = + for all ymmeic maice A B (ee for example, Schott, 005, p.8. In addition, we have ( e A e e Qe = ( A V ( QV E ξ Hence, a quadratic form eae ˆ ˆ i an optimal predictor of e Qe in the cla of quadratic predictor if the maix A minimize ( AVAV ( AVQV + ( A V ( QV In what follow, we will adopt the approach ued by Liu Rong (007 o a to get an optimal quadratic predictor of the quadratic form e Qe. However, to look for maix A minimizing ( ˆ ˆ MSE ξ eae in the cla of quadratic predictor, we will be reicted to maice A atifying the C A C V. Thi reiction i condition ( ( reaonable neceary a eae ˆ ˆ i equal to ep ˆ AP e ˆ a probability equal to, uing V V ˆ the fact that C ( e V i atified almot urely. Thu, the problem of finding the optimal quadratic predictor ea ˆ, opte ˆ of e Qe i reduced to finding maix A, opt, which i the olution to the following minimization problem: min { ( C( y n C A A V ( AVAV ( AVQV + ( ( A V QV (3.3 7 min ( C( y A n C A V { ( A G (3. + ( ( A V QV Indeed, we have thi equivalence between (3.3 (3. a ( A G = ( AVA V + ( GVGV ( A V GV GVGV i independent of A. The expreion of maix A minimizing (3. i given in the following lemma: LEMMA : The maix A which i the unique olution of (3. i given by ( λ A = P P + G P (3.5, opt V V V ( QV ( GP V ( PV + (3.6 Proof: the maix A, opt given by (3.5 atifie that C( A = C( V n, opt. Moreover, for any maix A atifying C( A C( V, we have i equal to P B = B B = A A V, opt ( A ( ( G + A QV
5 Augut 0. Vol. o. 005-0 JATIT & LLS. All right reerved. ISS: 99-865 www.jatit.org E-ISS: 87-395 ( A, opt G + ( A, opt ( QV ( B Δ ( B + + + ( ( ( ( Δ = λ B + B, opt A QV = λ + + V V = ( B λ( + ( λ( + PV ( PV = 0 ( B ( B λ ( P ( GP ( QV A G + A QV i greater than ( A ( (, opt G + A, opt QV we have equality if only if B = 0, thi mean that A = A, opt, thu completing the proof. So, ( ( ( The reult of Lemma can be ued to determine the expreion of maix A, opt, minimizing (3.3. THEOREM : The optimal quadratic predictor, according to the minimization of the mean quared error under the model (., of e Qe y Q i given by ea ˆ, Proof: ince opt e ˆ, opt = λ + A V V V QV V (3.7 ( QV ( QVV V rk ( V = VAV minimizing (3.3 i given by A V A V + (3.8 A, the maix A,, opt =, opt = λv PV V + V PV V GV PV V = λv + V V QVV opt In addition, the expreion (3.8 of λ i deduced from (3.6. Thi complete the proof of Theorem. ote that the quadratic predictor propoed in Theorem ha an expreion much impler than that of Liu' Rong' predictor by being approximately optimal in the cla of quadratic predictor. Moreover, one can eaily demonate that MSE ξ ( ea ˆ, opteˆ MSEξ ( ea, opte ( rk ( V σ λ + + α α = ( ( QVQV V V QV V V QV. Hence, the ability to approximate the MSE of ea ˆ, opte ˆ enable to meaure it accuracy unlike that propoed by Liu Rong whoe complexity of it expreion make it impoible to calculate accuracy.. OPTIMAL QUADRATIC PREDICTOR OF THE RESIDUAL SUMS OF SQUARES In what follow, we will ue the optimal quadratic predictor given by (3.7 to provide a new predictor of the reidual um of quare (RSS of a regreion model given by ( ( e = Y - Xb Y - Xb which correpond to the quadratic form e Qe Q = I I i the identity maix. Therefore, Theorem allow u to deduce the following reult: COROLLARY : Under the model (., the optimal quadratic predictor of the reidual um of quare e i given by ea ˆ e ˆ e ˆ ˆ V I V V V V e, opt = λ0 + n + r r Proof: for reduced to ( V ( V ( VrV Vr λ0 = rk + ( V Q = I, the expreion of A, opt i, opt = λ0 + = λ0v + V ( V + VrVr V = λ0v + In + V VrVrV A V V VVV 0 ( V ( VV V rk ( V + ( V ( V ( VrV Vr = rk + ( V 73
5 Augut 0. Vol. o. 005-0 JATIT & LLS. All right reerved. ISS: 99-865 www.jatit.org E-ISS: 87-395 Thi complete the proof. We note that when V = diag ( v,..., v have ea ˆ e ˆ e ˆ ˆ V I e, opt = λ0 + n which get reduced to when V = I. 5. COCLUSIO 0 k U\ v n + k + ea ˆ eˆ = e ˆ n +, opt, we In thi paper, we propoed a new quadratic predictor of a reidual quadratic form of a linear Gauian model. Thi predictor wa found out to be optimal according to the minimization of mean quared error under the adopted linear model. The main interet of thi reult i ued to provide an optimal predictor for any quadratic function involving reidue of a linear model. Thi i the cae, for example, of the model' reidual um of quare, which i among the criteria ued to meaure the quality of the model. A uch, a new optimal quadratic predictor of the reidual um of quare of a linear Gauian model i propoed in thi pape REFRECES: [] C.-M Cael, C.-E. Särndal J. H. Wretman, Foundation of inference in urvey ampling, John Wiley Son, e edition, 993. [] X.-Q. Liu J.-Y. Rong, Quadratic prediction problem in finite population, Statitic Probability Letter, Vol. 77, 007, pp. 83-89. [3] J. R. Schott, Maix analyi for tatitic, Wiley, 005 (econd edition. [] R. M. Royall, Linear regreion model in finite population ampling theory ; In Foundation of Statitical Inference, Ed. Godambe Sprott, 97, pp. 59-79. Toronto: Holt, Rinchart Winton. [5] R. Valliant, A. H. Dorfman R. M.,Royall, Finite population ampling inference : a prediction approach, Wiley, 000. 7