Bull. Korean Math. Soc. 36 (1999), No. 3, pp. 451{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Seung Hoe Choi and Hae Kyung

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1 Bull. Korea Math. Soc. 36 (999), No. 3, pp. 45{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Abstract. This paper provides suciet coditios which esure the strog cosistecy of regressio quatiles estimators of oliear regressio models. The mai result is supported by the applicatio of a asymptotic property of the least absolute deviatio estimators as a special case of the proposed estimators. Some example is give to illustrate the applicatio of the mai result.. Itroductio I this paper we cosider the followig oliear regressio model (:) y t = f(x t o )+ t t = 2 where f is akow respose fuctio, x t which belog to bouded subspace of R q is a observed iput vector, the error term t are idepedet ad idetically distributed (i.i.d.) radom variables with ite variace. The parameter vector o which is iterior poit iis ukow ao be estimated. The most commo techique useo estimate the true parameters i model (.) is the method of least squares developed by Jerich (969) ad Wu (98). However, o the occasios of the error terms cotai some outliers or depart from ormal distributio the least squares method is poor estimators due to the extreme sesitivity of the least squares estimators to some outlier. To overcome this defect, the search for the robust procedures alterative the least squares method has geerated cosiderable iterest i statistical iferece. Received Jue 5, Mathematics Subject Classicatio: 62J02. Key words ad phrases: oliear regressio model, strog cosistecy, regressio quatiles estimator.

2 The Least Absolute Deviatio (LAD) estimators based o sample media is deed by ay vector miimizig the sum of absolute deviatios D = jy t ; f t j where f t = f(x t ). Oberhofer (982) gave suciet coditios for the weak cosistecy of the LAD estimators i oliear regressio models. Wag (995) derivehe asymptotic ormality of the oliear LAD estimators ad Kim ad Choi (995) ivestigatehe asymptotic properties of the oliear LAD estimators ad explaiehat the relative eciecy of the LAD estimators to the least square estimators is the same as the relative eciecy of the sample media to the sample mea. Meawhile, i case of a distributio fuctio of errors is positively skewed (or egatively skewed) other quatiles tha media (50th quatile) may reveal the iformatio about the ukow parameter o i model (.). Regressio quatiles which provide a atural geeralizatio of the otio of sample quatile to the geeral regressio model were proposed by Koeker ad Basset (982). Quatile-based estimators have log bee kow i the statistical literature as `L-estimators' for their relative eciecy for heavy-tailed error distributios. The -th regressio quatiles estimators (0 < < ) of the true parameter o based o (y t x t ), deoted by ^, is a parameter which miimizes the objective fuctio (:2) S ( ) = ' (y t ; f t ) where the \check" fuctio if 0 ' = ( ; ) if <0: Sice the check fuctio ' (x) rotates the absolute fuctio jxj by some 2 agle i the clockwise directio ( < ), the least absolute deviatio 2 estimators is a obviously importat special case of the regressio quatiles estimators. I some recet papers, aalysis of liear models usig quatiles estimatio has bee published by may authors : Basset ad Koeker (982, 986) ad Portoy (99). Basset ad Koeker (986) establishehe strog cosistecy of regressio quatiles statistics i 452

3 The strog cosistecy of oliear regressio quatiles estimators liear models with i.i.d. errors. Portoy (99) discussed asymptotic behavior of regressio quatiles uder more geeral heteroscedasticity ad depedece assumptios i liear models. The mai object of this paper is to provide simple ad suciet coditios for the strog cosistecy of the regressio quatiles estimators ^ i oliear regressio model (.). 2. Strog Cosistecy We start this sectio by itroducig some coditios which esure the strog cosistecy of the regressio quatile estimator i the oliear regressio model (.). Let P be a probability measure o R q ad H deote the distributio fuctio of iput vector x t. Let rf t = i t )] (p). Assumptio A The parameter space is compact subspace of R p. Assumptio B B : The respose fuctio f(x t ) ahe partial derivatives rf t are cotiuous o. B 2 : The distributio fuctio G(x) of the errors is cotiuously dieretiable withp desity g(x) which is strictly positive atg ; =0. B 3 : V ( o ) = rf t( o )r T f t ( o ) coverges to a positive deite matrix V ( o )as!. B 4 : P fx 2 :f(x o ) 6= f(x )g > 0 for each 6= o. Modifyig (.2), we have aother objective fuctio of the oliear regressio quatiles estimators (2:) Q ( ) =S ( ) ; S ( o ): Sice S ( o ) is idepedet of, the regressio quatiles estimators ^ deed i (.2) is equivalet to the miimizer of (2.). Before we proceeo cosider the mai result, we preset the followig lemma eeded i the proof of the mai theorem. Lemma 2.. Suppose that model (:) satises Assumptios A ad B. The for ay, Q ( ) ; EfQ ( )g = o p 453

4 where o p deotes covergece i probability. Proof. Dee the radom variable t asfollowig if yt f t = t 0 otherwise: The we ca rewrite Q ( ) = = = [' (r t ) ; ' (" t )] [( ; )(r t I frt 0g ; " t I f"t 0g) +(r t I frt >0g ; " t I f"t >0g)] [( ; t )r t +( t ( 0 ) ; )" t ] where r t = y t ; f t. Let X t = ( ; t )r t +( t ( 0 ) ; )" t. Accordig to Holder's iequality, we get jx t j( +2)j t j + j +jkrf( )kk ; o k wherek:k deote Euclidea orm ad = o +(; ) 0 : O the other had, Chebyshev's iequality gives P fjq ( ) ; EfQ ( )gj >g The proof follows from Assumptio A ad B. max VarX t t : 2 The followig theorem is the mai result of this sectio, which provides suciet coditios for the strog cosistecy of regressio quatiles estimators. Theorem 2.. For the model (:), suppose that Assumptios A ad B are fullled. The the regressio quatiles estimators ^ deed i (:2) is strogly cosistet for 0. Proof. For ay >0, it is suciet to show that (2:2) lim if fq ( )g > 0 a:e:! k; o 454

5 The strog cosistecy of oliear regressio quatiles estimators From the lemma 2. we have Q ( ) = E X t + o p where E deotes the expected value of the error term t. Note that E X t = = ( ; I fdt g)( ; ) + (I fdt ( o )g ; )dg R 0 ; dg ; dt ; dg+ G( ) ; where =f t ; f t ( o ). First weprovethat o is a local miimizer of Q( ) = lim By simple calculatio, we get Q( ) = lim! r 2 Q( ) = lim! h 0 rq( ) = lim!! P E X t. i dg+ G( ) ; : [rf t (G( ) ; )]: [rf t (G( ) ; )+g( )rf t r T f t ]: Furthermore, rq( o ) = 0 ad r 2 Q( o ) is positive deite matrix. Hece Q( ) attais a local miimum at o. Next we show that this local miimizer o is ideehe global miimizer. Let N ( o ) = f : k ; o k < g: Sice R = N c ( o ) \ is compact, there exists such that We cosider Q( ) = lim! = lim! Q( ) = if Q( : ): 2R h 0 0 i dg+ G( ) ; ( ; ) dg: 455

6 If < 0, the ; is positive i ( 0). Thus there exist ad 2 such that < < 2 < 0. From Assumptio B 2, sice g is strictly positive o [ 2 ], there exists a > 0 such that g > o [ 2 ]. Thus we obtai (2.3) lim! 0 ( ; )g d > 2 ( ; )g d 2 > ( ; ) d: Likewise if > 0, we have similar result. Thus, we have Q( ) > Q( o ) or because the last term is positive. Fially, from Assumptio B 4 ad above fact we obtai (2:4) if E x Q ( ) k; o if k; o! R X dgdh(x) where! = fx 2 jf(x ) 6= f(x o )g: I virtue of (2.3) ad (2.4), we get for sucietly large if E x Q ( ) 2 k; o where 2 is a positive realumber. The proof is completed. To see that the assumptios of the mai theorem are suciet tocover a class of oliear regressio fuctios, we ow cosider the followig example. Example. Let p be a dieretial fuctio from R m to R + : Cosider the model y t = f(x t o )+ t where o 2 =[0 a ] [0 a 2 ] a a 2 < ad f(x ) = p(x) 2. May authors cosider the oliear model with p(x) = e x ad p(x) = x for m =. Assume that f t g are i.i.d. radom variables with the cotiuous p.d.f g(x) Pad distributio fuctio G(x) for which G(0) =. The V = rf tr T f t coverges to V where V = p(x) 2 2 dg(x) p(x) 2 2 l p(x) dg(x) p(x) 2 2 l p(x) dg(x) ( p(x) 2 l p(x))2 dg(x) : 456

7 The strog cosistecy of oliear regressio quatiles estimators For a o-zero vector =( 2 ) V T = ( + l p(x) 2 ) 2 p(x) 2 2 dg(x) > 0 where ( 2 ) 2 0. It is ot haro verify that Assumptios A ad B are satised. Thus we ca guaratee the strog cosistecy of the regressio quatile estimators. Ackowledgemets. We thak to the referee for suggestios ad remarks i improvig this paper. Refereces [] Basset, G. ad Koeker, R., A empirical quatile fuctio for liear models with i.i.d. errors, Joural of the America Statistical Associatio 77 (982), [2], Strog cosistecy of regressio quatiles ad related empirical process, Ecoometric Theory 2 (986), [3] Jerich, R. I., Asymptotic properties of oliear least squares estimators, The Aal of Mathematical Statistics 40 (969), [4] Kim, H. K. ad Choi, S. H., Asymptotic properties of oliear Least Absolute Deviatio estimators, Joural of Korea Statistics Society 24 (995), [5] Oberhofer, W., The cosistecy of oliear regressio miimizig the L -orm, The Aals of Statistics 0 (982), [6] Portoy, S., Asymptotic behavior of regressio quatiles i o-statioary, depedet cases, Joural of Multivariate Aalysis 50 (99), [7] Wag, J., Asymptotic ormality of L -estimators i oliear regressio, Joural of Multivariate Aalysis 54 (995), [8] Wu, C. F., Asymptotic theory of oliear least squares estimatio, The Aals of Statistics 9 (98), Seug Hoe Choi, Departmet of Geeral Studies, Hakuk Aviatio Uiversity, Koyag4, Korea shchoi@mail.hagkog.ac.kr Hae Kyug Kim, Departmet of Mathematics, Yosei Uiversity, Seoul 20, Korea abc27@yosei.ac.kr 457