| In this work, certain probability approximation schemes are applied to two different contexts: one under stochastic volatility models in financial econometrics and the other about the hierarchical clustering of directional data on the unit (hyper)sphere. In both cases, approximations play an important role in improving the computational efficiency.; In the first part, we study stochastic volatility models. As an indispensable part of Bayesian inference using MCMC, we need to compute the option prices for each iteration at each time. To facilitate the computation, an approximation scheme is proposed for numerical computation of the option prices based on a central limit theorem, and some error bounds for the approximations are obtained.; The second part of the work originates from studying microarray data. After pre-processing the microarray data, each gene is represented by a unit vector. To study their patterns, we adopt hierarchical clustering and introduce the idea of linking by the size of a spherical cap. In this way, each cluster is represented by a spherical cap. By studying the distribution of direction data on the unit (hyper)sphere, we can assess the significance of observing a big cluster using Poisson approximations. |
