The Levy’s continuity theorem is a very important tool in the statistical machinery. For example, it will give us two simple proofs to two classical statistical problems: The Law of Large Numbers and the Central Limit Theorem.

In the framework of the Journées de Statisque 2012 in Bruxelles, I presented the paper “Efficient estimation of conditional covariance matrices” made under Jean-Michel Loubes and Clement Marteau direction. You could check the program and the slides of the presentation. Today I will present you some ideas about the problem studied and the solution found …

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 $latex \displaystyle \mathrm{MISE}\approx\frac{\left(\mu_{2}(K)\right)^{4}}{4}\left[\sum_{j=1}^{d}h_{j}^{4}\int f_{jj}^{2}(x)dx+\sum_{j\neq k}h_{j}^{2}h_{k}^{2}\int f_{jj}f_{kk}dx\right]+\frac{\left(\int K^{2}(x)dx\right)^{d}}{nh_{1}\cdots h_{d}} &fg=000000$

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, $latex \displaystyle \mathrm{MISE}\triangleq\mathop{\mathbb E}_{p}\int\left(\hat{p} _{n}(x)-p(x)\right)^{2}dx &fg=000000$ …