Nonparametric Density Estimation and Clustering with Application to Cosmology

We present a clustering method based on nonparametric density estimation. We use Kernel smoothing and orthogonal series estimators to estimate the density f and then we extract the connected components of the level set using a modified Cuevas et al (2000) algorithm. We extend an idea due to Stein (1981) and Beran and Dümbgen (1998) to construct confidence sets for the level set {f > δc} using the asymptotic distribution of loss function. Specifically, we show the stochastic convergence of the pivot process, Bn(λp) = √ n(Lp(λp) − Ŝp(λp)) where Lp(λp) and Sp(λp) are the loss function and the estimated risk function with the smoothing parameter λp. Inverting the pivot provides a confidence set for the coefficient of the orthogonal series estimator and furthermore one can construct a confidence set for functionals of f . We consider applications in astronomy and other fields. Acknowledgment This is joint work with Larry Wasserman, Chris Genovese and Bob Nichol. References[1] Beran, R. and Dümbgen. (1998). Modulation of Estimators and Confidence Sets. Ann.Statist.,26, 1826-1856.[2] Cuevas, A., Febrero, M. and Fraiman, R. (2000). Estimation the number of clusters.The Canadian Journal of Statistics, 28, 367-382.[3] Jang, W. and Wasserman, L. (2003). Confidence Sets for Densities and Clusters. Inpreparation.[4] Stein, C (1981). Estimation of the mean of a multivariate normal distribution. Ann.Statist.,9, 1135-1151.