Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions

Although there is an extensive literature on the upper bound for cumulative standard normal distribution, there are relatively not sharp for all values of the interested argument x. The aim of this paper is to establish a sharp upper bound for standard normal distribution function, in the sense that its maximum absolute difference from phi(x) is less than for all values of x. The established bound improves the well-known Polya upper bound and it can be used as an approximation for Phi(x) itself with a very satisfactory accuracy. Numerical comparisons between the proposed upper bound and some other existing upper bounds have been achieved, which show that the proposed bound is tighter than alternative bounds found in the literature.Comment: 11 pages, 3 figure

Finite-dimensional nonparametric priors: theory and applications

The investigation of flexible classes of discrete prior has been an active research line in Bayesian statistics. Several contributions were devoted to the study of nonparametric priors, including the Dirichlet process, the Pitman–Yor process and normalized random measures with independent increments (NRMI). In contrast, only few finite-dimensional discrete priors are known, and even less come with sufficient theoretical guarantees. In this thesis we aim at filling this gap by presenting several novel general classes of parametric priors closely connected to well-known infinite-dimensional processes, which are recovered as limiting case. A priori and posteriori properties are extensively studied. For instance, we determine explicit expressions for the induced random partition, the associated urn schemes and the posterior distributions. Furthermore, we exploit finite-dimensional approximations to facilitate posterior computations in complex models beyond the exchangeability framework. Our theoretical and computational findings are employed in a variety of real statistical problems, covering toxicological, sociological, and marketing applications

Robust portfolio selection involving options under a “ marginal+joint ” ellipsoidal uncertainty set

AbstractIn typical robust portfolio selection problems, one mainly finds portfolios with the worst-case return under a given uncertainty set, in which asset returns can be realized. A too large uncertainty set will lead to a too conservative robust portfolio. However, if the given uncertainty set is not large enough, the realized returns of resulting portfolios will be outside of the uncertainty set when an extreme event such as market crash or a large shock of asset returns occurs. The goal of this paper is to propose robust portfolio selection models under so-called “ marginal+joint” ellipsoidal uncertainty set and to test the performance of the proposed models. A robust portfolio selection model under a “marginal + joint” ellipsoidal uncertainty set is proposed at first. The model has the advantages of models under the separable uncertainty set and the joint ellipsoidal uncertainty set, and relaxes the requirements on the uncertainty set. Then, one more robust portfolio selection model with option protection is presented by combining options into the proposed robust portfolio selection model. Convex programming approximations with second-order cone and linear matrix inequalities constraints to both models are derived. The proposed robust portfolio selection model with options can hedge risks and generates robust portfolios with well wealth growth rate when an extreme event occurs. Tests on real data of the Chinese stock market and simulated options confirm the property of both the models. Test results show that (1) under the “ marginal+joint” uncertainty set, the wealth growth rate and diversification of robust portfolios generated from the first proposed robust portfolio model (without options) are better and greater than those generated from Goldfarb and Iyengar’s model, and (2) the robust portfolio selection model with options outperforms the robust portfolio selection model without options when some extreme event occurs