A generalized reflection method of boundary correction in kernel density estimation

Abstract

Abstract:The kernel method of estimation of curves is now popular and widely used in statistical applications. Kernel estimators suffer from boundary effects, however, when the support of the function to be estimated has finite endpoints. Several solutions to this problem have already been proposed. Here the authors develop a new method of boundary correction for kernel density estimation. Their technique is a kind of generalized reflection involving transformed data. It generates a class of boundary corrected estimators having desirable properties such as local smoothness and nonnegativity. Simulations show that the proposed method performs quite well when compared with the existing methods for almost all shapes of densities. The authors present the theory behind this new methodology, and they determine the bias and variance of their estimators.Une methode de reflexion generalisee pour la correction des effets de frontiere en estimation par noyau Rhunk' : L'estimation de courbes par la mkthode du noyau est dorknavant populaire et largement utiliske en statistique appliqube. Les estimateurs B noyau sont toutefois sujets A des effets de bord, lorsque le support de la fonction h estimer est fiN. Quelques solutions B ce probltme ont dkjA 6tk propostes. Les auteurs dkveloppent ici une nouvelle mtthode de correction des estimations A noyau de densitk. Leur technique fait intervenir une sorte de rbflexion gbntralis& de donnQs transformbes. Elle gbntre une classe d'estimateurs comgts pour les bords posskdant de bonnes propriktks, dont une rkgularitk locale et la non-nkgativitk. Des simulations montrent que la mkthode proposbe se comporte bien par rapport aux techniques existantes, et ce pour presque toutes les formes de densitks. Les auteurs prksentent la thborie sous-jacente A cette nouvelle mtthodologie et ils dtterminent le biais et la variance de leurs estimateurs.

The efficiency of top agents: An analysis through service strategy in tennis

“…This standard approach, however, leads to a downward bias near the boundary of the support, in our case near one. To avoid the boundary effect we use a local linear fitting method, described in Karunamuni and Alberts (2005). Repeating this for all R replications we find the median and the 2.5 and 97.5 percentiles in Figure 4.…”

Miscellanea Kernel-Type Density Estimation on the Unit Interval

“…The usual version of f for density estimation on the whole real line R takes K(x, X; h) = h −1 K R (h −1 (x − X)) where K R is a symmetric unimodal probability density function with support R or some finite interval such as [−1, 1]. This is, however, not available for estimation on the unit interval without correction for boundary effects, although many boundary correction schemes exist by now (e.g., Müller, 1991, Jones, 1993, Cheng et al, 1997, Zhang et al, 1999, Karunamuni & Alberts, 2005. See Silverman (1986), Wand & Jones (1995) and Simonoff (1996) for general background on kernel density estimation.…”

Are Economic Agents Successful Optimizers? An Analysis through Service Strategy in Tennis

“…This standard approach, however, leads to a downward bias near the boundary of the support, in our case near one. To avoid the boundary effect we use a local linear fitting method, described in Karunamuni and Alberts (2005). Repeating this for all R replications we find the median and the 2.5 and 97.5 percentiles in Figure 2.…”