Estimating Conditional Mean and Difference Between Conditional Mean and Conditional Median

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1 Etimating Conditional Mean and Difference Between Conditional Mean and Conditional Median Liang Peng Deartment of Ri Management and Inurance Georgia State Univerity and Qiwei Yao Deartment of Statitic, London School of Economic Abtract When a conditional ditribution ha an infinite variance, commonly emloyed ernel moothing etimator uch a local olynomial etimator for the conditional mean have a nonnormal limit, which comlicate interval etimation ince one ha to emloy different method for the cae of finite variance and infinite variance. By etimating the middle art nonarametrically and the tail art arametrically baed on extreme value theory, thi aer rooe a new etimator for the conditional mean, which reult in a normal limit regardle of whether the conditional ditribution ha a finite variance or an infinite variance. Hence a naive boottra method could be emloyed to contruct a unified interval regardle of tail heavine. Similar reult hold for etimating the difference between conditional mean and conditional median, which i a ueful quantity in exloring data. Key word and hrae: limit Conditional mean, heavy tail, normal 1 Introduction Mean and median are two imortant location arameter in data exloratory analyi and the difference between mean and median give a good indication 1

2 of data ewne. When the underlying ditribution ha a finite variance, the amle mean ha a normal limit. However, when the underlying ditribution ha heavy tail with a finite mean, but an infinite variance, the amle mean ha a table law limit. Therefore, in order to give an interval for a mean, one ha to now whether the underlying ditribution ha a finite variance or an infinite variance, which can be done when ome additional aumtion on the underlying ditribution are imoed uch a heavy tail. Under thi etting, when the tail index i etimated to be le than two, a ubamle boottra method could be emloyed to contruct an interval for the mean; ee Hall and Jing (1998). Intead of uing different method to contruct a confidence interval for the mean of a heavy tailed ditribution, Peng (21) rooed to etimate the middle art nonarametrically and tail art arametrically baed on extreme value theory, which reult in an etimator alway with a normal limit. Hence one could imly emloy a boottra method or develo an emirical lielihood method to contruct an interval without earating the cae of finite variance and infinite variance; ee Peng (24). Thi idea ha been alied to etimating exected hortfall in ri management by ecir and Meraghni (29). Thi aer aim to further extend thi idea in Peng (21) to etimating a conditional mean and the difference between conditional mean and conditional median, which are ueful quantitie in exloring data. Suoe that (X i, Y i ) T } i a equence of indeendent and identically ditributed random vector and the conditional ditribution function F (y x) = P (Y i y X i = x) atifie 1 F (ty x) limt = y α(x), y > 1 F (t x) 1 F (t x) lim t = (x) [, 1], (1) 1 F (t x)+f ( t x) where α(x) > 1. Lie mean and median, the conditional mean E(Y X = x) i of imortance in many alication, which include the random deign regreion model a a ecial cae: Y i = m(x i ) + ɛ i, (2) where ɛ i are indeendent and identically ditributed random variable with zero mean and atify P (ɛ i >ty) limt lim t P (ɛ i = y β, y > >t) P (ɛ i >t) = [, 1], (3) P ( ɛ i >t) 2

3 for ome β > 1. Under model (2) and condition (3), model (1) hold with α(x) β, and it i well-nown that a local moothing etimator for m(x) ha a normal limit and a nonnormal limit for β > 2 and β < 2, reectively, which mae interval etimation nontrivial at all. However, when ɛ i ha a median zero, i.e., m(x) i a conditional median, Hall, Peng and Yao (22) howed that the leat abolute deviation etimator ha a normal limit for any β > 1, and o a boottra method can be emloyed to contruct an interval without nowing whether β i larger than 2 or le than 2. In thi aer, we ee new etimator for the conditional mean, and the difference between conditional mean and conditional median under condition (1), which hould alway have a normal limit for any α(x) > 1. Therefore a boottra method can be emloyed to contruct confidence interval traightforwardly. We organize thi aer a follow. Section 2 reent the new method and aymtotic reult. A imulation tudy i given in Section 3. All roof are ut in Section 4. 2 Main Reult Firt we rooe a new etimator for the conditional mean E(Y i X i = x), which give a normal limit regardle of whether the conditional ditribution ha a finite variance or an infinite variance. Suoe our obervation (X i, Y i ) T } n i=1 are indeendent and identically ditributed random vector with ditribution function F (x, y) and the conditional ditribution F (y x) of Y i given X i = x atifie (1). For a given h = h(n) >, define = n i=1 I( X i x h), let ( X j, Ȳj)} j=1 denote thoe data air (X i, Y i )} n i=1 uch that X i x h, and let Ȳ,1 Ȳ, denote the order tatitic of Ȳ1,, Ȳ. Obviouly, when h and hn, we have /(nh) f 1 (x), where f 1 denote the denity of X i. Therefore we write = [nh] and ay intead of. Similar to Peng (21), we write E(Y i X i = x) = y df (y x) = F (y x) dy = / F (y x) dy + / F (y x) dy + F (y x) dy / 1 / := m 1 (x) + m 2 (x) + m 3 (x), (4) 3

4 where F (y x) denote the generalized invere of the conditional ditribution F (y x), and = ( ) and / a. Baed on (4) we rooe to etimate the firt and third term by a arametric aroximation for F (y x) via extreme value theory and to etimate the econd term nonarametrically. More ecifically, when F (y x) c 1 y 1/α 1 and F (1 y x) c 2 y 1/α 2 a y, the tail indice α 1 and α 2 can be etimated by the well-nown Hill etimator (Hill (1975)) ˆα 1 = 1 log + ( Ȳ,i) log + ( Ȳ,)} 1 i=1 and ˆα 2 = 1 log + (Ȳ, i+1) log + (Ȳ, +1)} 1 i=1 with log + x = log(x 1). In our imulation, we et ˆα 1 = when all Ȳ,i > 1 for 1 i, and ˆα 2 = when all Ȳ,i < 1 for + 1 i. ote that a and m 1 (x) F (/ x) m 3 (x) F (1 / x) y 1/α(x) dx = y 1/α(x) dx = α(x) α(x) 1 α(x) α(x) 1. Therefore the three term in (4) can be etimated earately by ˆm 1 (x) = Ȳ, ˆα 1 ˆα 1 1, ˆm 2(x) = 1 i=+1 Ȳ,i, ˆm 3 (x) = Ȳ, +1 ˆα 2 ˆα 2 1, which lead to our new etimator for the conditional mean m(x) = E(Y i X i = x) a ˆm(x) = ˆm 1 (x) + ˆm 2 (x) + ˆm 3 (x). ote that one could alo ue other tail index etimator intead of the Hill etimator uch that the one in Diercx, Goegebeur and Guillou (214). Lie the tudy of extreme value tatitic, in order to derive the aymtotic limit for ˆm 1 (x) and ˆm 3 (x), one need to ecify an aroximate rate in (1), which i generally called a econd order condition in extreme value theory; ee De Haan and Ferreira (26). Here we imly aume that there 4

5 exit oitive moothing function d(x), c 1 (x), c 2 (x), α(x) > 1, β(x) uch that for y large enough 1 F (y x) c 1 (x)y α(x) + F ( y x) c 2 (x)y α(x) d(x)y α(x) β(x) (5) uniformly in x x h. ote that β(x) i lightly maller than the ocalled econd order arameter in extreme value theory, which can be een from the inequality for a econd order regular variation in De Haan and Ferreira (26). Furthermore we aume the following regularity condition: A1) the marginal denity f 1 of X i i oitive and continuou at x ; A2) function c 1 (x), c 2 (x) and α(x) have a continuou econd order derivative at x, and function d(x) and β(x) have a continuou firt order derivative at x ; A3) the conditional mean function m(x) = F (y x) dy + (1 F (y x)) dy ha a continuou econd order derivative at x. To how that the new etimator alway ha a normal limit, we rely on the following aroximation. Let H(y) denote the ditribution function Ȳi with x = x, i.e., the conditional ditribution of Y i given X i x h. Put U i = H(Ȳi) for i = 1,,, and o U 1,, U are i.i.d. random variable with uniform ditribution on (, 1). Let U,1 U, denote the order tatitic of U 1,, U. Define G (v) = 1 i=1 1(U i v), α (v) = G (v) v}, Q () = U,1, Q () = U,i if i 1 < i, and β () = Q () }. Then it follow from Cörgő, Cörgő, Horváth and Maon (1986) that there exit a equence of Brownian bridge B (u)} uch that for any ν [, 1/4) and λ > uu,1 u, u ν α (u) B (u) u 1/2 ν (1 u) 1/2 ν = O (1) u λ/ 1 λ/ ν β ()+B () 1/2 ν (1 ) 1/2 ν = O (1). Theorem 1. Suoe (5) and Condition A1) A3) hold. Put = [nh], α = α(x ), β = β(x ), and further aume that a n (6), / = o(1), h 2 (log ) 2 = o(1), 2β α = o( +2β ), σ(/ ) h2 = o(1), (7) 5

6 where Then a n, σ 2 () = σ(/) ˆm(x ) m(x )} = 2α () 2 1α () 2 / / ( B (1 ) B () dh () ( B ( ) + o σ(/) (1) d (, 1 + (2 α )(2α 2 2α +1) where m(x ) = E(Y i X i = x ), and (u v uv) dh (u)dh (v). B ( )) d 2 B (1 )) d 1 2() α }I(α < 2)), B n( ) 2 α 1 = 2(c 2/α 1 (x ) + c 2/α 2 (x )) }1/2 c 1/α 1 (x )I(α < 2) 2 α 2 = 2(c 2/α 1 (x ) + c 2/α 2 (x )) }1/2 c 1/α 2 (x )I(α < 2). Remar 1. If α(x ) > 2, then a σ 2 (/) E(Y 2 i X i = x ) (E(Y i X i = x )) 2 <. B (1 ) In thi cae, we require nhh 2, which give the ame rate of convergence a the local moothing etimator of a conditional mean without aymtotic bia. It alo follow from the roof of the above theorem that the above H(y) can be relaced by F (y x ). Remar 2. It follow from the above theorem that a naive boottra method can be emloyed to contruct a confidence interval for the conditional mean regardle of tail heavine. We refer to Hall (1992) for an overview on boottra method. A review aer on alying boottra method to extreme value tatitic i Qi (28). ext we conider etimating the difference between conditional mean and conditional median, i.e., θ(x) = E(Y i X i = x) F (1/2 x). Baed on the above etimator for m(x), the rooed etimator for θ i ˆθ(x) = ˆm(x) Ȳ,[/2], and it aymtotic limit i given in the theorem below. 6

7 Theorem 2. Under condition of Theorem 1 and that the conditional denity df (y x) function g(y x) = i oitive and continuou at y = F ( 1 x dy 2 ) and x = x, we have, a n, σ(/) ˆθ(x ) θ(x )} = 2α ( B ( () 2 ) 1α () 2 / B / () dh () d (, σ 2 θ ), σ(/) ( B (1 ) B ( )) d 2 B n( ) B (1 )) d 1 B (1/2) σ(/)g(f ( 1 2 x ) x ) + o (1) B (1 ) where σ 2 θ equal to the variance in Theorem 1 when α 2, and i when α > g 2 (F (1/2 x ) x ) (u v uv) df (u x )df (v x ) /2 u df (u x )+ 1/2 (1 u) df (u x ) g(f (1/2 x ) x ) (u v uv) df (u x )df (v x ) 3 Simulation We conduct a mall cale imulation to illutrate the rooed method. To thi end, we let X i in (2) be indeendent U( 1, 1) random variable, and conider m(x) = x + 4 ex( 4x 2 ). Furthermore in (2) we let ɛ i be indeendent caled t-ditribution with d degree of freedom for d = 1.5 and 3. Then α(x) = d in (1). We re-cale ɛ i uch that it tandard deviation i.5. We et amle ize n = 1 or 3, and chooe = 5, 1, 2, 3, 4 and 5. We ue bandwidth h =.2 when n = 1, and h =.1 when n = 3. Thi effectively et the amle ize 2 and 3, reectively, in the local etimation for m(x) for each given x. We etimate m( ) on a regular grid of the 19 oint between -.9 and.9, and calculate the root mean quare error: 1 rmse = 19 9 } 1/2. ˆm(.1j) m(.1j)} 2 (8) j= 9 7

8 For each etting, we relicate the exercie 5 time. To comare the erformance with conventional nonarametric regreion, we alo calculate three nearet neighbor etimate, namely etimate m(x) by the mean of Y i correonding to thoe X i within, reectively, h-, h/2- and h/4-ditance from x. Table 1 reort the mean and the tandard deviation of rmse for different etting over 5 relication. A we exected, the etimation error decreae when amle ize n increae from 1 to 3, and the error alo decreae when the tail index, reflected by the degree of freedom (df), increae. With t 1.5 -ditributed error, = 3 give a mallet tandard deviation, and both = 2 and = 3 erform well. But with t 3 -ditributed error, = 5 lead to the mot accurate etimate, which i in line with the theorem that tail art do not lay a role aymtotically in cae of finite variance and o a maller i referred. For the model with t 1.5 -ditributed error, the nearet neighbor etimator i no longer aymtotically normal. Indeed our newly rooed etimator with either = 2 or = 3 erform better than the nearet neighbor etimator. However for the model with t 3 -ditributed error, the nearet neighbor etimator i aymtotically normal and i indeed erform better than the new method. 8

9 9 (n, h, df) ew Etimator Etimator = 5 = 1 = 2 = 3 = 4 = 5 h h/2 h/4 (1,.2, 1.5) Mean STD (3,.1, 1.5) Mean STD (1,.2, 3) Mean STD (3,.1, 3) Mean STD Table 1: Mean and tandard deviation (STD) of rmse defined in (8) for the rooed new etimator and the nearet neighbor () etimator in imulation with 5 relication.

10 4 Proof Proof of Theorem 1. Write ˆm 1 (x ) / H (v) dv = H (U, )( ˆα 1 α ( ) ˆα 1 1 H (U, ) H (/) ) + α + ( H (/) α / α H (v) dv ) 1 = H ˆα (U, ) 1 α } 1 (ˆα 1 1)() i=1 log H (U,i ) log(u H (U, ),i/u, ) 1/α + H ˆα (U, ) 1 α } 1 (ˆα 1 1)() i=1 log(u,i/u, ) 1/α 1/α } + H (/) α H (U, ) ( U H (/),) 1/α + H (/) α ( U,) 1/α 1 } + H (/) α } / α H (v) dv 1 := I 1 + I 2 + I 3 + I 4 + I 5, ˆm 3 (x ) 1 / H (v) dv = H ˆα (U, +1 ) 2 α + H (U, +1 ) + H (1 /) α + H (1 /) α log H (U, i+1 ) log( 1 U, i+1 H (U, +1 ) ˆα 2 α } 1 i=1 log( 1 U, i+1 1 U, +1 ) 1/α 1/α H (U, +1 ) ( (1 U H (1 /), +1) ) } 1/α ( (1 U, +1) ) } 1/α 1 1 (ˆα 2 1)() i=1 (ˆα 2 1)() + ( H (1 /) α 1 / H (v) dv ) := III 1 + III 2 + III 3 + III 4 + III 5 and 1 U, +1 ) 1/α ˆm 2 (x ) / H (v) dv / = / U, H (v) dg (v) + U, 1 / H (v) dg (v) +H (1 /)G (1 /) 1 + /} H (/)G (/) /} / G / (v) v} dh (v) := II 1 + II 2 + II 3 + II 4 + II 5. Uing Condition A1) A2), (5) and the fact that y δ 3h 1 Mh log y uniformly in y [n δ 1, n δ 2 ] for any given < δ 1 < δ 2 < 1 and δ 3 >, where 1 }

11 M > only deend on δ 1, δ 2, δ 3, ince h log n, we have = 1 H(y) c 1 (x )y α x +h x h 1 F (y z)}f 1(z) dz P ( X 1 x h) c 1 (x )y α(x ) x +h x h 1 F (y z) c 1(z)y α(z) }f 1 (z) dz P ( X 1 x h) x +h x h c 1(z)y α(z) c 1 (x )y α }f 1 (z) dz P ( X 1 x h) + M 1 y α y β + h 2 (log y) 2 } (9) uniformly in y [n δ 1, n δ 2 ] for any given < δ 1 < δ 2 < 1, where M 1 > i indeendent of y. Similarly H( y) c 2 (x )y α M 2 y α h 2 (log y) 2 + y β } (1) uniformly in y [n δ 1, n δ 2 ] for any given < δ 1 < δ 2 < 1, where M 2 > i indeendent of y. Therefore and H (1 t) c 1/α 1 (x )t 1/α M 3 t 1/α h 2 (log t) 2 + t β /α } (11) H (t) + c 1/α 2 (x )t 1/α M 4 t 1/α h 2 (log t) 2 + t β /α } (12) uniformly in t [n δ 1, n δ 2 ] for any given < δ 2 < δ 1 < 1, where M 3 > and M 4 > are indeendent of t. ote that nh for δ (, 1) large enough. Write σ 2 () = f 1 (x ), P (Ȳ,1 n δ, Ȳ, n δ ) 1 (13) H () H () H (1 ) H(u) H(v) H(u)H(v)} dudv + H (1 ) H(u) H(v) H(u)H(v)} dudv = 2 H(v)1 H(u)} dudv H () v v +2 H (1 ) H(u)1 H(v)} dudv = 2 vh(v) dv H(u) H () H du}2 () +2 H (1 ) v1 H(v)} dv H (1 ) (1 H(u)) du} 2 = IV 1 () + IV 2 () + IV 3 () + IV 4 (). 11

12 Then it follow from (11) (13) that IV 1 (/) (/) 1 2/α IV 3 (/) (/) 1 2/α 2c2/α 2 (x ) 2 α, 2c2/α 1 (x ) 2 α, IV 2 (/) (/) 1 2/α IV 4 (/) (/) 1 2/α,, when α < 2, and σ 2 (/) σ 2 < if α > 2 if α = 2, where σ 2 = (u v uv) df (u x )df (v x ). Therefore, (/) 1 2/α σ 2 (/) 2 α 2(c 2/α 1 (x ) + c 2/α 2 (x )) I(α < 2). (14) ow uing (6), (9) (14) and (7), we can how that σ(/) I 1 + I 3 + I 5 + III 1 + III 3 + III 5 } = o (1), σ(/) I 2 = 2 α () 2 σ(/) I 4 = 2 1 σ(/) III 2 = 1 α () 2 σ(/) III 4 = 1 1 σ(/) II 1 σ(/) II 2 σ(/) II 3 σ(/) II 4 σ(/) II 5 = B ( ) B ( )} d + o (1), B ( ) + o (1), B (1 ) B (1 )} d + o (1), B (1 ) + o (1), = 2 B ( ) + o (1), = 1 B (1 ) + o (1), = 1 B (1 ) + o (1), = 2 / / B ( ) + o (1), B () dh (v) + o (1), σ(/) 12

13 which imlie that ˆm(x σ(/) ) H (v) dv} = 2α () 2 1α () 2 / / ( B (1 ) B () dh () ( B ( ) + o σ(/) (1) d (, 1 + (2 α )(2α 2 2α +1) B ( )) d 2 B (1 )) d 1 2() α by noting that E B ( 1 / ) / and = = / / / σ(/) (1 ) dh () σ(/) }I(α < 2)) B () dh () σ(/) } B n( ) B (1 ) (1 H(u)) du + H (1 /) (1 H(u)) du} H (/) 2 α 2 (x )I(α < 2) 2(c 2/α 1 (x )+c 2/α 2 (x )) }1/2 c 1/α = = E B (1 1 / ) / / / / σ(/) It follow from A3) that H (v) dv σ(/) dh () B () dh () σ(/) } H(u) du + H (1 /) H(u) du} H (/) 2 α 2(c 2/α 1 (x )+c 2/α 2 (x )) }1/2 c 1/α 1 (x )I(α < 2). (15) F (v x ) dv = H(v) dv + (1 H(v)) dv F (v x ) dv (1 F (v x )) dv x +h x h f 1(z) F (y z) dy+ (1 F (y z)) dy F (y x ) dy (1 F (y x )) dy}dz P ( X 1 x h) = = O(h 2 ). Hence the theorem follow from (15), (16) and (7). (16) Proof of Theorem 2. The theorem eaily follow from the exanion in the roof of Theorem 1 and the fact that Ȳ,[/2] H ( 1 2 )} = (G (1/2) 1/2) g(f ( 1 2 x ) x ) 13 + o (1).

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