New Multivariate Product Density Estimators

✍ Scribed by Luc Devroye; Adam Krzyżak

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 173 KB
• Volume: 82
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.2001.2021

✦ Synopsis

Let X be an R d -valued random variable with unknown density f. Let X 1 , ..., X n be i.i.d. random variables drawn from f. The objective is to estimate f(x), where x=(x 1 , ..., x d ). We study the pointwise convergence of two new density estimates, the Hilbert product kernel estimate

where X i =(X i1 , ..., X id ), and the Hilbert k-nearest neighbor estimate

where

), and X (k) is the kth nearest neighbor of x when points are ordered by increasing values of the product < d j=1 |x j -X (k) j |, and k=o(log n), k Q .. The auxiliary results needed permit us to formulate universal consistency results (pointwise and in L 1 ) for product kernel estimates with different window widths for each coordinate, and for rectangular partitioning and tree estimates. In particular, we show that locally adapted smoothing factors for product kernel estimates may make the kernel estimate inconsistent even under standard conditions on the bandwidths.