Kernel regression with Weibull-type tails supporting information

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1 Kerel regressio with Weibull-type tails supportig iformatio Tertius de Wet Yuri Goegebeur Armelle Guillou Michael Osma May 8, 205 I order to prove Lemma, we eed the followig prelimiary result that we state, for otatioal coveiece, i the uivariate framework. Lemma S. Let U deote the tail quatile fuctio of a distributio fuctio F, i.e. Uy = if{x : F x /y}, y >. If for some positive fuctio a U u U lim = l u, u > 0, p a the, with V U0,, U lim p a E [ V ] l U l U > p =. V p Departmet of Statistics ad Actuarial Sciece, Uiversity of Stellebosch, Private Bag X, Matielad 7602, South Africa tdewet@su.ac.za. Departmet of Mathematics ad Computer Sciece, Uiversity of Souther Demark, Campusvej 55, 5230 Odese M, Demark yuri.goegebeur@imada.sdu.dk. Istitut Recherche Mathématique Avacée, UMR 750, Uiversité de Strasbourg et CNRS, 7 rue Reé Descartes, Strasbourg cedex, Frace armelle.guillou@math.uistra.fr. Departmet of Mathematics ad Computer Sciece, Uiversity of Souther Demark, Campusvej 55, 5230 Odese M, Demark mosma@imada.sdu.dk.

2 Proof of Lemma S We have [ E l U V = p = It is well-kow that implies U lim p a p [ l U l U [ l U z p p p u p l U V > p ] l U p ] dz z 2. p ] du [ ] u l U l U = l u, u > 0, p p see for istace de Haa ad Ferreira 2006, p 0. Now let ãy := ay/uy, ad cosider a fuctio ã 0 ã. Thus U E a [ V ] l U l U > p V p ã 0 = ã ã 0 = ã ã 0 = ã + l U z l z dz z 2 + l U dz ã z 2 0 l U z l U l z dz ã z 2 0 l U z l U l z dz ã z 2. 0 As for the itegral i the right-had side, we use Propositio B.2.7 of de Haa ad Ferreira 2006 p 382, so that for all ε, δ > 0 ad for p sufficietly large l U z l U l z ã 0 2 dz z 2 ε = ε δ, z δ 2 dz

3 provided δ <. Collectig the terms establishes the result of Lemma S. A ote about Assumptio F I this sectio we elaborate o the smoothess coditio F. We will focus o the case t = 0 ad show that the assumptio lim sup z F ω ; x h z F ω ; x = 0, 2 ca be satisfied by imposig some more structure o F ad some additioal coditios o h ad ω. I particular, assume that l.; x is ormalized, i.e. y l εv; x y; x = cx exp dv, v where cx > 0 ad εv; x 0 as v, ad impose the followig Hölder cotiuity coditios θx θy M θ x y η θ, cx cy M c x y ηc, sup εv; x εv; y M ε x y ηε, v where. deotes the Euclidea orm o R q, ad M θ, M c, M ε, η θ, η c, η ε are positive costats. Usig the well kow iequality expx exp x x we obtai F ω ; x h z F ω ; x exp ω /θx l ω ; x ω/θx hz /θx ω /θx l ω ; x ω/θx hz /θx l ω ; x h z l ω ; x l ω ; x h z l ω ; x. Furthermore, ω/θx hz /θx l ω ; x h z l ω ; x ω /θx hz /θx l ω ; x h z l ω ; x + l ω ; x h z l ω ; x. 3 We ow examie the two terms i the right had side of 3 separately. The Hölder cotiuity of θx ad the iequality expx exp x x lead for large to ω /θx hz /θx Mθ h η θ l ω Mθ h η θ l ω exp θxθx θxθx 3

4 with θx := if z θx z. Thus, by assumig that h η θ l ω 0 as, we get for large the iequality where C is a positive costat. ω /θx hz /θx C h η θ l ω, 4 Cocerig l we get the followig decompositio l ω ; x h z l ω ; x cx h z ω cx exp εv; x h z εv; x dv v ω + exp εv; x h z εv; x dv v. 5 Agai by the Hölder cotiuity coditios we obtai cx h z cx M ch ηc cx, 6 ad, for large ω exp εv; x h z εv; x dv v C 2h ηε l ω, 7 for C 2 a positive costat. Combiig 3, 4, 5, 6, 7, we get thus for large that ω /θx l ω ; x l ω ; x h z ω/θx hz /θx l ω ; x with C 3, C 4 > 0. C 3 h ηc ω /θx Thus 2 will be satisfied if oe assumes additioally that ad as. h ηc ω /θx l ω ; x 0 h η θ η ε ω /θx l ω ; x l ω 0, l ω ; x + C 4 h η θ η ε ω /θx l ω ; x l ω, 4

5 Proof of Lemma 2 From the rule of repeated expectatios we obtai Straightforward calculatios lead to m K, t; x = E[K h x Xmω, t; X] = K h x umω, t; ugudu R q = Kzmω, t; x h zgx h zdz. m K, t; x mω, t; xgx mω, t; x Kz gx h z gx dz +gx Kz mω, t; x h z mω, t; x dz + Kz mω, t; x h z mω, t; x gx h z gx dz =: T + T 2 + T 3. Usig G ad K we obtai T mω, t; xc g h Kzd0, zdz = mω, t; xgxoh. Cocerig T 2, usig F ad K T 2 = gxmω, t; x Kz mω, t; x h z mω, t; x = gxmω, t; xoφω, h ; x. dz Similar argumets lead to T 3 = mω, t; xgxoh Φω, h ; x. Now, collectig all the terms establishes Lemma 2. Proof of Lemma 4 5

6 Let r := h q F ω ; xgx. By straightforward calculatios we ca write ˆ F ω ; x r F ω ; x = gx ĝ x r + gx ĝ x r gx ĝ x r T 0 =: T 4 + T 5 T 6. x, K ET 0 x, K F ω ; xgx ET 0 x, K F ω ; xgx F ω ; xgx ĝ x gx gx From Lemma 3 we obtai that [ T 0 x, K r F ω ; xgx E ] T 0 x, K F ω ; xgx D N0, K 2 2, ad hece, also usig Parze s results o kerel desity estimatio Parze, 962, summarized i Lemma 3 i Daouia et al. 203, T 4 D N0, K 2 2. Cocerig T 5, we use Lemma 3 i Daouia et al. 203 ad Lemmas ad 2 from the preset paper to obtai T 5 = O P r h +O P r Φω, h ; x. Fially, T 6 = O P r h +O P F ω ; x by Lemma 3 i Daouia et al Combiig these results establishes our Lemma 4. Proof of Theorem 2 First cosider θ 2 x; t, K, K 2 := l F ω ; xt t+ x, K t + T t x, K 2 6

7 ad write θ2 r x; t, K, K 2 θx F ω ; xgx = l F ω ; x t T t x, K 2 {r l F ω ; x t+ [ θxr l F ω ; x t [ +r [ =: E l F ω ; x t+ T t+ T t+ x, K t + F ω ; xgx E T t x, K 2 F ω ; xgx E F ω ; xgx l F ω ; x t T t x, K 2 {T 7 + T 8 + T 9 }. ] T t+ x, K t + F ω ; xgx ] T t x, K 2 F ω ; xgx x, K θxt + E l F ]} ω ; x t T t x, K 2 t + F ω ; xgx From Lemma 3, we have D T 7 + T 8 N 0, θ2t+2 xγ2t + [ 22t + K t + K 2 2 ] 2 22t + K K 2. t + Cocerig T 9, usig Lemmas ad 2, we have T 9 = r {θ t xγt + b l F ω ; x; x + o + O l F ω ; x ε +Oh + OΦω, h ; x}, ad hece uder our assumptios, we have T 9 λ gx θ t xγt +. Accordig to Lemma 3, l F ω ; x t T t x, K 2 / F ω ; xgx = θ t xγt + + o P. Collectig the above results gives r θ 2 x; t, K, K 2 θx D N λ gx, θ 2 xγ2t + [ 22t + K t + Γ t + K 2 2 ] 2 22t + K K 2. t + To complete the proof we cosider r ˆθ 2 x; t, K, K 2 θx = r θ 2 x; t, K, K 2 θx ˆ F ω ; x T +r t+ x, K l F ω ; x t + T t x, K 2. 7

8 Agai usig Lemmas, 2, ad 3 we obtai that r ˆθ 2 x; t, K, K 2 θx = r θ 2 x; t, K, K 2 θx + O P l F ω ; x. Refereces [] Daouia, A., Gardes, L., Girard, S., 203. O kerel smoothig for extremal quatile regressio. Beroulli, 9, [2] de Haa, L., Ferreira, A., Extreme Value Theory: A Itroductio. Spriger. [3] Parze, E., 962. O estimatio of a probability desity fuctio ad mode. Aals of Mathematical Statistics, 33,