A variable bandwidth selector in multivariate kernel density estimation

Based on a random sample of size n from an unknown d-dimensional density f, the problem of selecting the variable (or adaptive) bandwidth in kernel estimation of f   is investigated. The common strategy is to express the variable bandwidth at each observation as the product of a local bandwidth factor and a global smoothing parameter. For selecting the local bandwidth factor a method based on cluster analysis is proposed. This method is direct and intuitively appealing. For selecting the global smoothing parameter a method that is an adaptation of the frequency domain approach of selecting the fixed bandwidth in Wu and Tsai [2004. Root nn bandwidths selectors in multivariate kernel density estimation. Probab. Theory Related Fields 129, 537–558] is used. For d=1d=1 and 2, extensive simulation studies have been done to compare the performance of our selector with the selectors of Abramson [1982. On bandwidth variation in kernel estimates—a square root law. Ann. Statist. 10, 1217–1223] and Sain and Scott [1996. On locally adaptive density estimation. J. Amer. Statist. Assoc. 91, 1525–1534] and Sain [2002. Multivariate locally adaptive density estimation. Comput. Statist. Data Anal. 39, 165–186], and the excellent performance of our selector at practical sample sizes is clearly demonstrated.