Content deleted Content added
m add cat |
Added Silverman's rule of thumb |
||
Line 80:
Thus <math>\hat{\bold{H}}_{\operatorname{SCV}} = \operatorname{argmin}_{\bold{H} \in F} \, \operatorname{SCV} (\bold{H})</math> is the SCV selector.<ref>{{Cite journal| doi=10.1007/BF01205233 | author1=Hall, P. | author2=Marron, J. | author3=Park, B. | title=Smoothed cross-validation | journal=Probability Theory and Related Fields | year=1992 | volume=92 | pages=1–20}}</ref><ref name="DH2005">{{Cite journal| doi=10.1111/j.1467-9469.2005.00445.x | author1=Duong, T. | author2=Hazelton, M.L. | title=Cross validation bandwidth matrices for multivariate kernel density estimation | journal=Scandinavian Journal of Statistics | year=2005 | volume=32 | pages=485–506}}</ref>
These references also contain algorithms on optimal estimation of the pilot bandwidth matrix <strong>G</strong> and establish that <math>\hat{\bold{H}}_{\operatorname{SCV}}</math> converges in probability to '''H'''<sub>AMISE</sub>.
=== Rule of thumb ===
Silverman's rule suggests using <math>\sqrt{\mathbf{H}_{ii}} = \left(\frac{4}{d+2}\right)^{\frac{1}{d+4}} n^{\frac{-1}{d+4}} \sigma_i</math> where <math>\sigma_i</math> is the standard deviation of the ith variable and <math>\mathbf{H}_{ij} = 0, i\neq j</math>.
==Asymptotic analysis==
| |||