UCF Math Department's
Probability and Statistics Seminar
Spring 2008

Fridays in MAP 233 from 3:30 PM - 4:30 PM, unless otherwise noted.

Organized by Jason Swanson

Schedule and Abstracts

We consider the solution u(x,t) to a stochastic heat equation. For fixed x, the process F(t) = u(x,t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Ito sense. We show that for sufficiently differentiable functions g, a stochastic integral \int g(F) dF exists as a limit of discrete, midpoint style Riemann sums, where the limit is taken in distribution in the Skorohod space of cadlag functions. Moreover, we show that this integral satisfies a change of variables formulas with a correction term that is an ordinary Ito integral with respect to a Brownian motion that is independent of F.

Recently there has been considerable interest in variable selection via regularized methods mostly developed for single-index or semi-parametric models. We propose a penalized objective function using transformations of the response as the multivariate pseudo-response and utilizing an adaptive lasso as the operator. Under mild assumptions on the marginal distribution of the predictors, the proposed approach selects variables consistently without restrictive model assumptions. Simulations confirm these findings.

So far, likelihood-based interval estimate for quantiles has not been studied in literature for interval censored Case 2 data and partly interval-censored data, and in this context the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval-censored data. The 4th-order expansion of the weighted empirical log-likelihood ratio is derived, and the theoretical coverage accuracy equation for the proposed WELRCI is established, which generally guarantees at least the first-order accuracy. In particular for right censored data, we show that the coverage accuracy is at least $n^{-1/2}$, and our simulation studies show that in comparison with empirical likelihood-based methods, the smoothing used in WELRCI generally gives a shorter confidence interval with comparable coverage accuracy. For interval censored data, it is interesting to find that with an adjusted rate $n^{-1/3}$, the weighted empirical log-likelihood ratio has an asymptotic distribution completely different from that by the empirical likelihood approach, and the resulting WELRCI perform favorably in available comparison simulation studies.