Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Size: px

Start display at page:

Download "Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at"

Arnold Harris
5 years ago
Views:

Four (Pathological) Examples i Asymptotic Statistics Authors(s): Roger W. Koeker ad Gilbert W. Bassett Source: The America Statisticia, Vol. 38, No. 3 (Aug., 1984), pp.

1 Four (Pathological) Examples i Asymptotic Statistics Authors(s): Roger W. Koeker ad Gilbert W. Bassett Source: The America Statisticia, Vol. 38, No. 3 (Aug., 1984), pp Published by: Taylor & Fracis, Ltd. o behalf of the America Statistical Associatio Stable URL: Accessed: :03 UTC Your use of the JSTOR archive idicates your acceptace of the Terms & Coditios of Use, available at JSTOR is a ot-for-profit service that helps scholars, researchers, ad studets discover, use, ad build upo a wide rage of cotet i a trusted digital archive. We use iformatio techology ad tools to icrease productivity ad facilitate ew forms of scholarship. For more iformatio about JSTOR, please cotact support@jstor.org. America Statistical Associatio, Taylor & Fracis, Ltd. are collaboratig with JSTOR to digitize, preserve ad exted access to The America Statisticia

3 E >0, But for strog covergece it is ecessary that for ay (see Feller 1968, Vol. 1, p. 201, lemma 2). Note that the evets i brackets are mutually idepedet by the iid hypothesis o the u's ad the desig assumptio. The (1.6) becomes, for E= 1 ad some o, Now suppose F is symmetric ad that i the tails, which diverges. Thus for F satisfyig (1.7) ad xi = 2', we have weak cosistecy, but ot strog. Sice if either side coverges, I u I F{du} = [1 - F(u) + F(-u)]du, (1.9) existece of a first absolute momet for u is sufficiet to assure strog covergece. Note, however, that (1.7) is worse tha mere lack of a first momet. The Cauchy, for example, has tails like 1 - llu. Here the 11 estimator, which is geerally immue to the tail behavior of u, is early udoe because of the explosive desig. Fixig the tail behavior (1.7) ad assumig a strictly positive desity for u at the media, the results of Bassett ad Koeker (1978) imply weak covergece of 3 for the less explosive class of desigs satisfyig for some 0 < D < x. More explosive desigs tha xi = 2' will also serve to restore strog cosistecy for a fixed error distributio. Take F(u) with tails give by (1.7). By makig the desig more explosive, we ca obviously make the left side of (1.8) coverge. This illustrates the rather delicate iteractio of desig ad error coditios i 11 cosistecy argumets. Examples of weak but ot strog covergece may also be costructed for the least squares estimator. These examples are also somewhat pathological, sice if O< Eu2 < oc, the r3 = Exiyi1/x2 is weakly cosistet if ad oly if it is strogly cosistet, ad the latter obtais if ad oly if Jx2 o (see Lai, Robbis, ad Wei 1979). Relaxig the fiite variace coditio creates some scope for excitemet. Take the sample mea: If tails are sufficietly fat so that E I u = oc, the strog cosistecy fails by the Kolmogorov strog law, but weak cosistecy may still be salvaged. Let F'(u)= 1/(u2logu) i the tails. The O = y - 0 i probability (see Chug 1974, p. 111). As i the 11 case, cosistecy of the least-squares estimator may be salvaged by makig the desig sufficietly explosive. For example, Che, Lai, ad Wei (1981) showed that if F is Cauchy i (1.1) ad for some > 3, the,b -~ 0 almost surely. 210? The America Statisticia, August 1984, Vol. 38, No. 3 =1 2 E (1 - F(x,)) ::- 2 '1 (1.8) = 1 =o EPr[ I u,x, I> El < x (1.6) F(u)= 1 - l/log2(u). (1.7) (log )l(>x?) = 0 (1) i=1 1 _-2 D Agai cosider the model with ui iid F ad F symmetric about 0, but ow let xi= N/k for i = k(k - 1)/2 for some iteger k = 0 otherwise. Thus, (1/)E7=I x7 2-1, ad if F has media 0 ad cotiuous, positive desity f(o) at the media, the by Bassett ad Koeker (1978), Thus a fortiori, 3 is cosistet for,. Now cosider, however, the limitig behavior of the objective fuctio V(b). I particular, we wish to cosider the limitig behavior of -'[V(b) - V(0)] or, equivaletly, settig 8 = b - Typically, i oliear estimatio problems, we would require that Ev,(8) have a uique miimum at 8 = 0. Such a coditio is required, for example, i Oberhofer's (1982) paper o 11 cosistecy; see Gallat, Burguete, ad Souza (1983) for a detailed discussio of this geeral approach. But i the preset example, lettig = k(k - 1)/2 for some iteger k, Ev,,(8) (8 1 ) Xi Thus the mea of the objective fuctio coverges to zero i ay eighborhood of 0, ad thus the typical idetifiability coditio fails. Nevertheless, \/?- 1 coverges i distributio! This example differs from similar examples ivolvig the least-squares estimator (e.g., see Wu 1981, where least-squares cosistecy is salvaged without Cesaro idetifiability, but covergece i distributio occurs at rates slower tha the covetioal 1/V' ). We tur to a example i which the desig i (1.1) is simply xi-1, 50,B is simply the sample media from a radom sample from F. Whe F has a cotiuous ad strictly positive desity, say f(0), at the media, it is well kow that 3. PATHOLOGICAL CONVERGENCE CONSISTENCY WITHOUT CESARO IDENTIFIABILITY VG-r - N(0, (4f(0))'). (2.3) V(8) =(11) ui - xi 8- uij =(2181 lk(k - 1))[1 + V2_+...+ Vk] c(218 1k Vk )Ik (k - 1) \'?(,B-1) -~N(O,(4f2(0))-1). (3.1) <(11)z Xi 81 IN DISTRIBUTION Yi=X13i+ Ui, (2.1) i= -

4 What happes whe f(o) = O? Suppose, for a= (log 2) -1/, so F has a uique media of zero, ad ot oly is f(o) = 0, but all higher order derivatives of F are also zero at zero. so 3 < z if ad oly if the gradiet The sample media a miimizes where sg(u) = 1 if u -0 ad = -1 otherwise. Note that sice F is absolutely cotiuous, y = z with probability 0. Let {A} be a sequece such that X ->x as -c. The where ad Now (g - Eg)/(varg)"12 is a stadardized biomial radom variable with probability of success p - 1/2 as z/ 0, ad therefore it coverges i distributio to a stadard Gaussia law. Pr[\ 3 < z] = Pr[g(z/X) > 0] We would like to choose {X} so that - Eg/(varg)"12 coverges to somethig that is bouded away from 0. Sice var g- for ay A xc, [1-2F(z/X)]=j [-2eIz2 * e - (3.8) must stay away from 0. Choosig X = (log -/? )1/2 yields where Z is a stadard Gaussia radom variable. The desity fuctio of the limitig distributio is where + deotes the stadard Gaussia desity. The desity is depicted i Figure 1. Thus ot oly is the rate of covergece cosiderably slower tha the typical rate i/iv?, the limitig distributio is also bizarre. F(x) =1 a < x g(z) - Esg(yi - z) >? (34) varg = 4[F(zXA)(1 - F(z/1))1 (3.7) Pr[X, p < z]- Pr[Z > 2e -z 2] (3.9) f (x) = 4)(2e -x-2) 14e -x-2x -31 (3.10) = + e -x-2 0<X a =2 X=0 =0 x <-, (3.2) V(b) =ElYi-b 1, (3.3) = Pr[(g(Z /X)- Eg)/lvarg Eg = [1-2F(z /X)] (3.6) >-Eg/v'arg], (3.5) = -x-2 -a?x<o 1=1 C) The America Statisticia, August 1984, Vol. 38, No Figure 1. The Limitig Desity of the Normalized Sample Media by Radom Samplig From the Distributio Fuctio (3.2). 4. CONSISTENCY WITH HARMONIC DESIGNS Whe,x2 < c, covergece of x typically fails, as does the cosistecy of the least-squares estimator. The preset example shows, however, that the cosistecy of, the 11 estimator, may be salvaged if F has sufficiet So mass at the media. Cosider the harmoic desig xi = 1/i. I this case xi2 -rr2/6 (Kopp 1956, p. 173), so -l 'x? 0. As i the previous sectio, with 8 > 0, where ow where Now which coverges to zero if ad oly if the deomiator diverges. Suppose i a eighborhood of 0, F is symmetric with the for some o > 0, the deomiator becomes which diverges. Thus by the Chebyshev iequality, h(5) coverges i probability to zero, ad hece the left side of (4.3) coverges to oe. A idetical argumet implies that Pr[-(,B -,B) < 5-> 1 as well. varh() =(Eg(b) )2 - [j(1i )(1-2F(bli ))] Thus we have a example i which the usual desig coditio Ex - x is violated, but the weak cosistecy of,b is salvaged by makig the desity of u sufficietly large ear the media. The least-squares estimator is icosistet uder these coditios, of course, uless F is actually degeerate at 0. Pr[p3-3 < 8] = Pr[g,(8) > 0], (4.1) Pr[P - < 8] = Pr[h() > -11, (4.3) h()= (g()- Eg(M))lEg(b) (4.4) F(u) = 1/2 + 1/log2(1/u), u >0; (4.6) [Received September Revised March ] g()= ->Esg(ui - xi 8)xi. (4.2) (Eg(ti2 = [Z2(i 1og2(i/ ))-12, (4.7) x

5 B3ASSETT, G., ad KOENKER, R. (1978), "Asymptotic Theory of Least Absolute Error Regressio," Joural of the America Statistical Associatio, 73, CHEN, G.J., LAI, T.L., ad WEI, C.Z. (1981), "Covergece Systems ad Strog Cosistecy of Least Squares Estimates i Regressio Models," Joural of Multivariate Aalysis, 11, CHUNG, K.L. (1974), A Course i Probability Theory, New York: Academic Press. FELLER, W. (1968), A Itroductio to Probability Theory ad Its Applicatios (3rd ed.), New York: Joh Wiley. 212? The America Statisticia, August 1984, Vol. 38, No. 3 REFERENCES GALLANT, A.R., BURGUETE, J.F., ad SOUZA, G. (1983), "O the Uificatio of the Asymptotic Theory of Noliear Ecoometric Models" (with discussio), Ecoometric Reviews, 1, KNOPP, K. (1956), Ifiite Sequeces ad Series, New York: Dover. LAI, T.L., ROBBINS, H., ad WEI, C.Z. (1979), "Strog Cosistecy of Least Squares Estimates i Multiple Regressio II," Joural of Multivariate Aalysis, 9, OBERHOFER, W. (1982), "The Cosistecy of Noliear Regressio Miimizig the l, Norm," Aals of Statistics, 10, WU, CHIEN-FU (1981), "Asymptotic Theory of Noliear Least- Squares Estimatio," Aals of Statistics, 9,

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

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 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