Tag Archives: Nonparametric statistics

Multivariate kernel density estimation

Briefly, we shall see the definition of a kernel density estimator in the multivariate case. Suppose that the data is d-dimensional so that $latex {X_{i}=(X_{i1},\ldots,X_{id})}&fg=000000$. We will use the product kernel $latex \displaystyle \hat{f}_{h}(x)=\frac{1}{nh_{1}\cdots h_{d}}\left\{ \prod_{j=1}^{d}K\left(\frac{x_{j}-X_{ij}}{h_{j}}\right)\right\} . &fg=000000$ The risk is given by $latex \displaystyle \mathrm{MISE}\approx\frac{\left(\mu_{2}(K)\right)^{4}}{4}\left[\sum_{j=1}^{d}h_{j}^{4}\int f_{jj}^{2}(x)dx+\sum_{j\neq k}h_{j}^{2}h_{k}^{2}\int f_{jj}f_{kk}dx\right]+\frac{\left(\int K^{2}(x)dx\right)^{d}}{nh_{1}\cdots h_{d}} &fg=000000$

Choosing the smoothing parameter

Two popular methods to find the bandwidth $latex {h}&fg=000000$ for the nonparametric density estimator are the plug-in method and the method cross-validation. The first one we will focus in the “quick and dirty” plug-in method introduced by Silverman (1986). In cross-validation we will minimize a modified version of the quadratic risk of $latex {\hat{f}_{h}}&fg=000000$. The …

A global measure of risk for kernel estimators in Nikolski classes

Sergey Mikhailovich Nikolsky (Russian: Серге́й Миха́йлович Нико́льский; 30 April 1905 – 9 November 2012) was a Russian mathematician. He was born in Talitsa, which was at that time located in the Governorate of Perm, Russia. He had been an Academician since November 28, 1972. He also had won many scientific prizes. At the age of 92 he was still actively giving lectures in Moscow Institute of Physics and Technology. In 2005, he was only giving talks at scientific conferences, but was still working in MIPT, at the age of 100.

Photos of Sergey Nikolskii from The Russian Academy of Sciences The MSE  gives an error of the estimator $latex {\hat{p}_{n}}&fg=000000$ at an arbitrary point $latex {x_{0}}&fg=000000$, but it is worth to study a global risk for $latex {\hat{p} _{n}}&fg=000000$. The mean integrated squared error (MISE) is an important global measure, $latex \displaystyle \mathrm{MISE}\triangleq\mathop{\mathbb E}_{p}\int\left(\hat{p} _{n}(x)-p(x)\right)^{2}dx &fg=000000$ …