Sep 25

3:30 pm

## Research Day - Part I

### Jon Wellner & Russell Steele

Seminar

University of Washington - Department of Statistics

**Jon Wellner:**

*"Maximum Likelihood Estimators and LR Statistics for some Non-Regular Problems: Monotone and Convex Functions"*

For regular statistical problems it is well-known that maximum likelihood estimators have asymptotically normal distributions, and likelihood ratio statistics are asymptotically chi-square. For non-regular problems these classical limit distributions fail to hold. In this talk I will briefly discuss recent progress concerning the asymptotic distribution theory of maximum likelihood estimators and likelihood ratio statistics for a class of non-regular problems connected with estimation of a monotone or convex function. For monotone functions, the limiting distributions can be described in terms of the slope (process) of the greatest convex minorant of two-sided Brownian plus a parabola. For convex functions, the limiting distributions are described in terms of a certain ``invelope'' of two-sided integrated Brownian motion +t4. I will also mention a few of the many open problems connected with this area of research.

**Russell Steele:**

*"Bayes Factors for Finite Mixture Models from the EM Algorithm via Importance Sampling."*

We present a general method for calculating the Bayes factors for finite mixture models via importance sampling of the mixture component labels. The importance sampling function uses conditional group probabilities obtained via the EM algorithm and the complete data-likelihood to sample from vital regions of the parameter space. The integration method requires less computational time than Markov Chain Monte Carlo and is far easier to implement, involving only sampling from multinomial distributions and from the prior.

Refreshments to follow in the Statistics lounge, 3rd floor, Padelford.