Feb 26

2:30 pm

## Some Functionals of Brownian Motion Connected with Estimation of Montone and Convex Functions

### Jon Wellner

Seminar

Department of Statistics, University of Washington - Department of Mathematics Probability Seminar

Estimation and testing problems for monotone functions in "Gaussian white noise" lead to several interesting functions of two-sided Brownian motion $W$ plus a parabola: the slope process of the greatest convex minorant is now well-understood, thanks to the work of Groeneboom (1983), (1989). In particular, the distribution of the slope process at $0$, say $Z_0$, has been computed analytically and numerically in Groeneboom (1985) and Groeneboom and Wellner (2001). In recent work with Moulinath Banerjee, we have found that the analogue of a chi-square distribution in regular problems is played by the distribution of $$ Z_1 \equiv \int \{ S (t)^2 - S^0 (t)^2 \} dt $$ where $S $ is the slope process of the greatest convex minorant of $W(t) +t^2$ and $S^0$ is the slope process of the one-sided greatest convex minorants constrained to be greater than or equal to zero to the right of zero, and constrained to be less than or equal to zero to the left of zero. An analytical description of the distribution of $Z_1$ is still unknown.

For estimation of a convex function in Gaussian white noise, the maximum likelihood estimator turns out to be the second derivative of a certain ``invelope'' of integrated (two-sided) Brownian motion plus $t^4$. The value $Z_2$ of this second derivative at zero describes the limiting distribution in statistical problems of interest. Almost nothing is known about the distribution of $Z_2$.

I will discuss some of the statistical background for these problems and some of the many open questions.