A finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface

We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove that the level 2 twist subgroup is normally generated in the mapping class group by a crosscap pushing map along a non-separating two-sided simple loop for genus $g\geq 5$ and $g=3$. As an application, we calculate the first homology group of the level 2 twist subgroup for genus $g\geq 5$ and $g=3$.Comment: 18 pages,16 figure

"Stochastic Volatility Model with Leverage and Asymmetrically Heavy-Tailed Error Using GH Skew Student's t-Distribution Models"

Bayesian analysis of a stochastic volatility model with a generalized hyperbolic (GH) skew Student's t-error distribution is described where we first consider an asymmetric heavy-tailed error and leverage effects. An efficient Markov chain Monte Carlo estimation method is described that exploits a normal variance-mean mixture representation of the error distribution with an inverse gamma distribution as the mixing distribution. The proposed method is illustrated using simulated data, daily S&P500 and TOPIX stock returns. The models for stock returns are compared based on the marginal likelihood in the empirical study. There is strong evidence in the stock returns high leverage and an asymmetric heavy-tailed distribution. Furthermore, a prior sensitivity analysis is conducted whether the results obtained are robust with respect to the choice of the priors.

Block Sampler and Posterior Mode Estimation for Asymmetric Stochastic Volatility Models (Published in "Computational Statistics and Data Analysis", 52-6, 2892-2910. February 2008. )

This article introduces a new efficient simulation smoother and disturbance smoother for asymmetric stochastic volatility models where there exists a correlation between today`s return and tomorrow`s volatility. The state vector is divided into several blocks where each block consists of many state variables. For each block, corresponding disturbances are sampled simultaneously from their conditional posterior distribution. The algorithm is based on the multivariate normal approximation of the conditional posterior density and exploits a conventional simulation smoother for a linear and Gaussian state space model. The performance of our method is illustrated using two examples (1) simple asymmetric stochastic volatility model and (2) asymmetric stochastic volatility model with state-dependent variances. The popular single move sampler which samples a state variable at a time is also conducted for comparison in the first example. It is shown that our proposed sampler produces considerable improvement in the mixing property of the Markov chain Monte Carlo chain.