Liang Peng, Qiwei Yao Estimating conditional means with heavy tails

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1 Liang Peng, Qiwei Yao Etimating conditional mean with heavy tail Article (Acceted verion) (Refereed) Original citation: Peng, Liang and Yao, Qiwei (217) Etimating conditional mean with heavy tail. Statitic & Probability Letter, ISS DOI: 1.116/j.l Reue of thi item i ermitted through licening under the Creative Common: 217 Elevier B.V. CC BY-C-D 4. Thi verion available at: htt://erint.le.ac.u/7382/ Available in LSE Reearch Online: Aril 217 LSE ha develoed LSE Reearch Online o that uer may acce reearch outut of the School. Coyright and Moral Right for the aer on thi ite are retained by the individual author and/or other coyright owner. You may freely ditribute the URL (htt://erint.le.ac.u) of the LSE Reearch Online webite.

2 Acceted Manucrit Etimating conditional mean with heavy tail Liang Peng, Qiwei Yao PII: S (17) DOI: htt://dx.doi.org/1.116/j.l Reference: STAPRO 791 To aear in: Statitic and Probability Letter Received date: 29 January 217 Revied date: 21 March 217 Acceted date: 21 March 217 Pleae cite thi article a: Peng, L., Yao, Q., Etimating conditional mean with heavy tail. Statitic and Probability Letter (217), htt://dx.doi.org/1.116/j.l Thi i a PDF file of an unedited manucrit that ha been acceted for ublication. A a ervice to our cutomer we are roviding thi early verion of the manucrit. The manucrit will undergo coyediting, tyeetting, and review of the reulting roof before it i ublihed in it final form. Pleae note that during the roduction roce error may be dicovered which could affect the content, and all legal diclaimer that aly to the journal ertain.

3 *Manucrit Clic here to view lined Reference Etimating Conditional Mean with Heavy Tail 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 method uch a local olynomial etimator for the conditional mean admit non-normal limiting ditribution (Hall, Peng and Yao 22). Thi comlicate the related inference a the conventional tet and confidence interval baed on aymtotic normality are no longer alicable, and the tandard boottra method often fail. By utilizing the middle art of data nonarametrically and the tail art arametrically baed on extreme value theory, thi aer rooe a new etimation method for conditional mean, reulting in aymtotically normal etimator even when the conditional ditribution ha infinite variance. Conequently the tandard boottra method could be emloyed to contruct, for examle, confidence interval regardle of the tail heavine. The ame idea can be alied to etimating the difference between a conditional mean and a conditional median, which i a ueful meaure in data exloratory analyi. Key word: Aymtotic normality; conditional mean; extreme value theory; heavy tail. 1 Introduction Mean and median are two imortant location arameter in data exloratory analyi and the difference between them i indicative for the ewne of the underlying ditribution. 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 admit a table law limit. Therefore, in order to contruct a valid confidence interval for the mean, one ha to now if the variance of the underlying ditribution i finite or not. When the ditribution i heavy 1

4 tailed with infinite variance, the tandard boottra method doe not wor, and a ubamle boottra method hould be emloyed to contruct a valid confidence interval for the mean; ee Hall and Jing (1998). A different and unified aroach ha been rooed by Peng (21) which contruct a mean etimator by uing the middle art of data nonarametrically and the tail art arametrically baed on extreme value theory. The reulted etimator for the mean i alway aymtotically normal regardle of the tail heavine of the underlying ditribution. Hence one could imly emloy the tandard boottra method or emirical lielihood method to contruct the confidence interval for the mean even when the underlying ditribution ha an infinite variance; ee Peng (24). Thi idea ha been taen further for etimating exected hortfall in ri management by ecir and Meraghni (29). Thi aer aim to further extend thi idea for etimating a conditional mean and the difference between a conditional mean and a conditional median, which are ueful quantitie in data exloratory analyi. 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 1 F (t x) = y α(x), y > 1 F (t x) lim t 1 F (t x)+f ( t x) = (x) [, 1], (1) where m(x) i an unnown mooth function and α(x) > 1. Lie mean and median, the conditional mean E(Y i X i = 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 P (ɛ i >t) = y β, y > P (ɛ lim i >t) t P ( ɛ i >t) = [, 1], (3) for ome β > 1. Under model (2) and condition (3), model (1) hold with α(x) β. Furthermore, the limiting ditribution of a local moothing etimator for m(x) i normal or non-normal, reectively, when β > 2 or β < 2. Thi mae interval etimation nontrivial. 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. Conequently the tandard boottra method can be emloyed to contruct a confidence interval for the conditional median even when β i le than 2. 2

5 In thi aer, we ee a new etimator for conditional mean E(Y i X i = x), and the difference between thi conditional mean and the conditional median of Y i given X i = x under the general etting (1). The new etimator i alway aymtotically normal rovided α(x) > 1. Therefore the tandard boottra method can be emloyed to contruct confidence interval for the conditional mean in a traightforward manner. We organize the aer a follow. Section 2 reent the new method and the 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 admit a normal limiting ditribution regardle of Var(Y i X i = x) being finite or not. Suoe that 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 + 1 / F (y x) dy := m 1 (x) + m 2 (x) + m 3 (x), 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 for ome contant c 1 and c 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 and ˆα 2 = 1 i=1 log + (Ȳ, i+1) log + (Ȳ, +1)} 1 i=1 3 (4)

6 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). Moreover one may emloy a different in ˆα 1 and ˆα 2. 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 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. 4

7 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). (6) Theorem 1. Suoe (5) and Condition A1) A3) hold. α = α(x ), β = β(x ), and further aume that a n, h 2 (log ) 2 = o(1), 2β α = o( +2β ), σ(/ ) h2 = o(1), Put = nh, (7) where Then a n, σ 2 () = σ(/) ˆm(x ) m(x )} = 2α (α 1) 2 1α (α 1) 2 / / ( B ( ) (u v uv) dh (u)dh (v). B ( )) d 2 α 1 ( B (1 ) B (1 )) d 1 α 1 B () dh () σ(/) + o (1) d (, 1 + (2 α )(2α 2 2α +1) 2(α 1) α α 1 }I(α < 2)), where m(x ) = E(Y i X i = x ), B n( ) B (1 ) and 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). 5

8 Remar 1. If α(x ) > 2, then a σ 2 (/) E(Y 2 i X i = x ) (E(Y i X i = x )) 2 <. 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). Remar 3. When α > 2, the term m 1 (x) and m 3 (x) in (4) become a maller order than the term m 2 (x). Therefore the aymtotic limit of the new etimator i indeendent of the tail index α. 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. Theorem 2. Under condition of Theorem 1 and that the conditional denity function g(y x) = df (y x) dy i oitive and continuou at y = F ( 1 2 x ) and x = x, we have, a n, σ(/) ˆθ(x ) θ(x )} (α 1) 2 ( B ( ) 1α (α 1) 2 / B / () dh () = 2α d (, σ 2 θ ), σ(/) B ( )) d 2 α 1 ( B (1 ) B (1 )) d 1 α 1 B (1/2) σ(/)g(f ( 1 2 x ) x ) + o (1) B n( ) 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 ) 6

9 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 ˆm(.1j) m(.1j)} 2} 1/2. (8) j= 9 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. 7

10 (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. 4 Proof Proof of Theorem 1. Write ˆm 1 (x ) / H (v) dv = H (U, )( ˆα 1 + α α 1 ( H (U, ) H (/) ) ˆα 1 1 α α 1 ) + ( H (/) α α 1 / H (v) dv ) = H ˆα (U, ) 1 α 1 } (ˆα 1 1)(α 1) i=1 log H (U,i ) H (U, ) log(u,i/u, ) 1/α + H ˆα (U, ) 1 α 1 } (ˆα 1 1)(α 1) i=1 log(u,i/u, ) 1/α 1/α } + H (/) α H (U, ) α 1 H (/) ( U,) 1/α + H (/) α α 1 ( U,) 1/α 1 } + H (/) α α 1 } / H (v) dv := I 1 + I 2 + I 3 + I 4 + I 5, ˆm 3 (x ) 1 / H (v) dv = H ˆα (U, +1 ) 2 α 1 } (ˆα 2 1)(α 1) i=1 log H (U, i+1 ) H (U, +1 ) log( 1 U, i+1 1 U, +1 ) 1/α + H ˆα (U, +1 ) 2 α 1 } (ˆα 2 1)(α 1) i=1 log( 1 U, i+1 1 U, +1 ) 1/α 1/α + H (1 /) α H (U, +1 ) α 1 ( H (1 /) (1 U, +1) ) } 1/α + H (1 /) α ( α 1 (1 U, +1) ) } 1/α 1 + H (1 /) α α 1 } 1 1 / H (v) dv := III 1 + III 2 + III 3 + III 4 + III 5 8

11 and 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 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. f 1 (x ), P (Ȳ,1 n δ, Ȳ, n δ ) 1 (13) 9

12 Write σ 2 () = 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 H(v)1 H(u)} dudv v +2 H (1 ) H(u)1 H(v)} dudv = 2 H () vh(v) dv H () H(u) du}2 +2 H (1 ) v1 H(v)} dv H (1 ) (1 H(u)) du} 2 = IV 1 () + IV 2 () + IV 3 () + IV 4 (). 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 α 2 = 1 2 (α 1) 2 B ( ) σ(/) I 4 = 2 1 α 1 σ(/) III α 2 = 1 (α 1) 2 σ(/) III 4 = 1 1 α 1 σ(/) II 1 σ(/) II 2 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

13 which imlie that σ(/) II 3 σ(/) II 4 σ(/) II 5 = = 1 B (1 ) + o (1), = 2 / / σ(/) ˆm(x ) H (v) dv} = 2α (α 1) 2 1α (α 1) 2 / / ( B ( ) B ( ) + o (1), B () dh (v) + o (1), σ(/) B ( )) d 2 α 1 ( B (1 ) B (1 )) d 1 α 1 B () dh () σ(/) + o (1) d (, 1 + (2 α )(2α 2 2α +1) 2(α 1) α α 1 }I(α < 2)) by noting that and = = E / B ( ) / / / / σ(/) (1 ) dh () σ(/) B () dh () σ(/) } B n( ) B (1 ) H (/) (1 H(u)) du + H (1 /) (1 H(u)) du} 2 α 2(c 2/α 1 (x )+c 2/α 2 (x )) }1/2 c 1/α = = E / B (1 ) / / / / σ(/) It follow from A3) that = σ(/) dh () 2 (x )I(α < 2) B () dh () σ(/) } H (/) H(u) du + H (1 /) H(u) du} 2 α 2(c 2/α 1 (x )+c 2/α 2 (x )) }1/2 c 1/α H (v) dv F (v x ) dv H(v) dv + = = O(h 2 ). 1 (x )I(α < 2). (15) (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) Hence the theorem follow from (15), (16) and (7). (16) 11

14 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 ) + o (1). Acnowledgment We than a reviewer for helful comment. Peng reearch wa artly uorted by the Simon Foundation. Yao reearch wa artially uorted by an UK EPSRC grant. Reference [1] Cörgő, M., Cörgő, S., Horváth, L. and Maon, D.M. (1986). Weighted emirical and quantile rocee. Annal of Probability 14, [2] Diercx, G., Goegebeur, Y. and Guillou, A. (214). Local robut and aymtotically unbiaed etimation of conditional Pareto-tye tail. Tet 23, [3] De Haan, L. and Ferreira, A. (26). Extreme Value Theory: An Introduction. Sringer. [4] Hall, P. (1992). The Boottra and Edgeworth Exanion. Sringer. [5] Hall, P. and Jing, B. (1998). Comarion of boottra and aymtotic aroximation to the ditribution of a heavy-tailed mean. Statitica Sinica 8, [6] Hall, P., Peng, L. and Yao, Q. (22). Prediction and nonarametric etimation for time erie with heavy tail. Journal of Time Serie Analyi 23, [7] Hill, A.M. (1975). A imle general aroach to inference about the tail of a ditribution. Annal of Statitic 3, [8] ecir, A. and Meraghni, D. (29). Emirical etimation of the roortional hazard remium for heavy-tailed claim amount. Inurance: Mathematic and Economic 45, [9] Peng, L. (21). Etimating the mean of a heavy tailed ditribution. Statitic & Probability Letter 52,

15 [1] Peng, L. (24). Emirical lielihood confidence interval for a mean with a heavy tailed ditribution. Annal of Statitic 32, [11] Qi, Y. (28). Boottra and emirical lielihood method in extreme. Extreme 11,