Bandwidth selectors for kernel estimation of density partial derivatives and mode estimates of multivariate data

• Main Authors: Chih-YuanHsu, 許志遠
• Other Authors: Tiee-Jian Wu
• Format: Others
• Language: en_US
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/59743997547569053256

博士 === 國立成功大學 === 統計學系碩博士班 === 101 === Based on a random sample of size n from an unknown multivariate density f, two research topics are investigated: (i) bandwidth selection in multivariate kernel density partial derivatives, and (ii) mode estimation of a multivariate density. On the topic (i), a bandwidth selector is proposed, which extends the ones of Wu (1997) and Wu and Tsai (2004). The bandwidth selector is asymptotically normal with the optimal root n relative convergence rate and achieves the (conjectured) “lower bound” on the covariance matrix. On the topic (ii), two mode estimates are proposed. The first one is a multivariate extension to the one of Bickel (2003), which is an application of the joint Box-Cox transform. Also, we show that the estimate is quite efficient when the sample size n is relatively small. However, the first one may not be consistent due to the restriction of the transform method. To solve the non-consistent problem, the second estimate is proposed, which is based on a weighted average of a parametric density estimate and a nonparametric density estimate. It is shown that the estimate not only keeps the strengths of the first one at small n but also overcomes the non-consistent drawback at large n.