Estimating the Division Kernel of a Size-Structured Population

We consider a size-structured population describing the cell divisions. The cell population is described by an empirical measure and we observe the divisions in the continuous time interval [0, T ]. We address here the problem of estimating the division kernel h (or fragmentation kernel) in case of complete data. An adaptive estimator of h is constructed based on a kernel function K with a fully data-driven bandwidth selection method. We obtain an oracle inequality and an exponential convergence rate, for which optimality is considered

A local proof of the dimensional Pr\'ekopa's theorem

The aim of this paper is to find an expression for second derivative of the function $\phi(t)$ defined by \phi(t) = \lt(\int_V \vphi(t,x)^{-\beta} dx\rt)^{-\frac1{\be -n}},\qquad \beta\not= n, where $U\subset \R$ and $V\subset \R^n$ are open bounded subsets, and \vphi: U\times V\to \R_+ is a $C^2-$smooth function. As a consequence, this result gives us a direct proof of the dimensional Pr\'ekopa's theorem based on a local approach.Comment: 9 pages, to appear in J. Math. Anal. App