A0402
Title: Two-stage nonparametric estimation of multivariate densities
Authors: Ximing Wu - Texas AM University (United States)
Juan Lin - Department of Finance & WISE, Xiamen University (China) [presenting]
Abstract: In addition to slower convergence, nonparametric multivariate density estimation is plagued by the increasingly difficult specification of tuning parameters as dimension increases. We propose a two stage estimator of a $d$-variate density f that starts with a standard kernel density hat $f_0$. This pilot estimate can be refined via a multiplicative correction that estimates the density ratio $r= f/\hat{f_0}$, a challenging task due to its random denominator. Instead of direct estimation of $r$, we use a basis expansion to approximate $\log r$, which is achieved via minimizing the Kullback-Leibler discrepancy between $f$ and $\hat f_0$ subject to moment conditions associated with the basis functions. Thanks to the consistency of the pilot estimate, $\log r$ resides in a shrinking $d$-sphere around origin such that all coefficients of its basis approximation tend to zero as sample size increases. Thus in our penalized spline estimation of $\log r$, it suffices to use a single penalty parameter across all dimensions, effectively alleviating the curse of dimensionality. We derive the local and global convergence of this density estimator in terms of mean squared error and Kullback-Leibler discrepancy respectively. We use Monte Carlo simulations to demonstrate its good finite sample performance and present two real data examples.
