A quantile-copula approach to conditional density estimation

arXiv:0709.3192v3 [stat.ME] 12 Jun 2008 A quan tile- opula approa h to onditional densit y estimation. Olivier P . F augeras L.S.T.A, Université Paris 6 175, rue du Chevaler et, 75013 Paris, F r an e T el:+(33) 1 44 27 85 62 F ax:+(33) 1 44 27 33 42 Abstra t W e presen t a new non-parametri estimator of the onditional densit y of the k ernel t yp e. It is based on an e ien t transformation of the data b y quan tile transform. By use of the opula represen tation, it turns out to ha v e a remark able pro du t form. W e study its asymptoti prop erties and ompare its bias and v arian e to omp etitors based on nonparametri regression. A omparativ e n umeri al sim ulation is pro vided. Key wor ds: onditional densit y, k ernel estimation, opula, quan tile transform, nonparametri regression, 1991 MSC: 62G007, 62M20, 62M10 1 In tro du tion 1.1 Motivation Let ((Xi, Yi); i = 1, . . . , n) b e an indep enden t iden ti ally distributed sample from real-v alued random v ariables (X, Y ) sitting on a giv en probabilit y spa e. F or predi ting the resp onse Y of the input v ariable X at a giv en lo ation x, it is of great in terest of estimating not only the onditional mean or r e gr ession fun tion E(Y |X = x), but the full

onditional density f(y|x). Indeed, estimat- ing the onditional densit y is m u h more informativ e, sin e it allo ws not only to re al ulate the onditional exp e ted v alue E(Y |X) and onditional v arian e Email addr ess: olivier.faugeras gmail. om (Olivier P . F augeras ). Preprin t submitted to Elsevier No v em b er 1, 2018 from the densit y , but also to pro vide the general shap e of the onditional den- sit y . This is esp e ially imp ortan t for m ulti-mo dal or sk ew ed densities, whi h often arise from nonlinear or non-Gaussian phenomenas, where the exp e ted v alue migh t b e no where near a mo de, i.e. the most lik ely v alue to app ear. Moreo v er, for situations in whi h onden e in terv als are preferred to p oin t estimates, the estimated onditional densit y is an ob je t of ob vious in terest. 1.2 Estimation by kernel smo othing A natural approa h to estimate the onditional densit y f(y|x) of Y giv en X = x w ould b e to exploit the iden tit y f(y|x) = fXY (x, y) fX(x) (1) where fXY and fX denote the join t densit y of (X, Y ) and X , resp e tiv ely . By in tro du ing P arzen-Rosen blatt k ernel estimators of these densities, namely ˆfn,XY (x, y) : = 1 n n X i=1 K′ h′(Xi −x)Kh(Yi −y) ˆfn,X(x) : = 1 n n X i=1 K′ h′(Xi −x) where Kh(.) = 1/hK(./h) and K′ h′(.) = 1/h′K′(./h′) are (res aled) k ernels with their asso iated sequen e of bandwidth h = hn and h′ = h′ n going to zero as n →∞, one an onstru t the quotien t ˆf R n (y|x) := ˆfn,XY (x, y) ˆfn,X(x) and obtain an estimator of the onditional densit y . Su h an estimator w as rst studied b y Rosen blatt [26℄, and more re en tly b y Hyndman et al. [17 ℄, who sligh tly impro v ed on Rosen blatt’s k ernel based estimator. 1.3 Estimation by r e gr ession te hniques As p oin ted out b y n umerous authors, see e.g. F an and Y ao [7℄

hapter 6, this approa h is equiv alen t to the one arising from onsidering this onditional densit y estimation problem in a regression framew ork. Indeed, let F(y|x) b e the um ulativ e onditional distribution fun tion of Y giv en X = x. It stems from the fa t that E  1|Y −y|≤h|X = x  = F(y + h|x) −F(y −h|x) ≈2h.f(y|x) 2 as h →0 , that, if one repla e the exp e tation in the ab o v e expression b y its empiri al oun terpart, one an apply the usual lo al a v eraging metho ds and p erform a regression estimation on the syn theti data ((1/2h)1|Yi−y|≤h ; i = 1, . . . , n). By a Bo

hner t yp e theorem, one an ev en repla e the transformed data b y its smo othed v ersion Y ′ i := Kh(Yi −y) := 1 hK Yi −y h  . In parti ular, the p opular Nadara y a-W atson regression estimator ˆf NW n (y|x) := Pn i=1 Y ′ i K′ h′(Xi −x) Pn i=1 K′ h′(Xi −x) redu es itself to the same estimator of the onditional densit y of the double k ernel t yp e as b efore ˆf NW n (y|x) := Pn i=1 Kh(Yi −y)K′ h′(Xi −x) Pn i=1 K′ h′(Xi −x) = ˆf R n (y|x). T aking adv an tage of this regression form ulation, F an, Y ao and T ong [8℄ pro- p osed a onditional densit y estimator whi h generalizes the k ernel one b y use of the lo al p olynomial te hniques. In parti ular, it allo ws to ta kle with the bias issues of the k ernel smo othing. Ho w ev er, and unlik e the former, it is no longer guaran teed to ha v e p ositiv e v alue nor to in tegrate to 1 with resp e t to y . With these issues in mind, Hyndman and Y ao [18℄ built on lo al p oly- nomial te hniques and suggested t w o impro v ed metho ds, the rst one based on lo ally tting a log-linear mo del and the se ond one on onstrained lo al p olynomial mo deling. An o v erview an b e found in F an and Y ao [7℄ ( hapter 6 and 10). V ery re en tly , Gy ör and K ohler [15℄ studied a partitioning t yp e estimate and studied its prop erties in to