Statistics, Optimization & Information Computing

Convergence Analysis of a Stochastic Progressive Hedging Algorithm for Stochastic Programming

2020-08-07T14:03:47+08:00

Stochastic programming is an approach for solving optimization problems with uncertainty data whose probability distribution is assumed to be known, and progressive hedging algorithm (PHA) is a well-known decomposition method for solving the underlying model. However, the per iteration computation of PHA could be very costly since it solves a large number of subproblems corresponding to all the scenarios. In this paper, a stochastic variant of PHA is studied. At each iteration, only a small fraction of the scenarios are selected uniformly at random and the corresponding variable components are updated accordingly, while the variable components corresponding to those not selected scenarios are kept untouch. Therefore, the per iteration cost can be controlled freely to achieve very fast iterations. We show that, though the per iteration cost is reduced significantly, the proposed stochastic PHA converges in an ergodic sense at the same sublinear rate as the original PHA.

CQ-free optimality conditions and strong dual formulations for a special conic optimization problem

2020-08-06T23:32:29+08:00

In this paper, we consider a special class of conic optimization problems, consisting of set-semidefinite
(or K-semidefinite) programming problems, where the set K is a polyhedral convex cone.

For these problems, we introduce the concept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. This study provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions in the form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of a linear cost function, we reformulate the K-semidefinite problem in a regularized form and construct its dual. We show that the pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.

Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments

2020-08-06T23:32:30+08:00

We introduce a stochastic sequence $\zeta(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of the stochastic sequence $\zeta(k)$ based on its observations at points $ k<0$. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of the functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.

Sample Paths Properties of Stochastic Processes from Orlicz Spaces, with Applications to Partial Differential Equations

2020-08-06T23:32:30+08:00

In the present paper we obtain conditions for stochastic processes from Orlicz spaces to have almost sure bounded and continuous sample paths, the study is concerned with the processes defined on unbounded domains. Estimates for the distributions of suprema of the processes are also presented. Conditions are given in terms of entropy integrals and majorant characteristics of Orlicz spaces. Possible applications to solutions of partial differential equations are discussed. Examples of processes are given for which conditions of the main results are satisfied.

A Note on CCMV Portfolio Optimization Model with Short Selling and Risk-neutral Interest Rate

2020-08-06T23:32:31+08:00

In this paper, first we present some drawbacks of the cardinality constrained mean-variance (CCMV) portfolio optimization with short selling and risk-neutral interest rate when the lower and upper bounds of the assets contributions are -1/K and 1/K(K denotes the number of assets in portfolio). Then, we present an improved variant using absolute returns instead of the returns to include short selling in the model. Finally, some numerical results are provided using the data set of the S&P 500 index, Information Technology, and the MIBTEL index in terms of returns and Sharpe ratios to compare the proposed models with those in the literature.

Volatility Modelling of the BRICS Stock Markets

2020-08-06T23:32:31+08:00

Volatility modelling is a key factor in equity markets for risk and portfolio management. This paper focuses on the use of a univariate generalized autoregressive conditional heteroscedasticity (GARCH) models for modelling volatility of the BRICS (Brazil, Russia, India, China and South Africa) stock markets. The study was conducted under the assumptions of seven error distributions that include the normal, skewed-normal, Student’s t, skewed-Student’s t, generalized error distribution (GED), skewed-GED and the generalized hyperbolic (GHYP) distribution. It was observed that using an ARMA(1, 1)-GARCH(1, 1) model, volatilities of the Brazilian Bovespa and the Russian IMOEX markets can both be well characterized (or described) by a heavy-tailed Student’s t distribution, while the Indian NIFTY market’s volatility is best characterized by the generalized hyperbolic (GHYP) distribution. Also, the Chinese SHCOMP and South African JALSH markets’ volatilities are best described by the skew-GED and skew-Student’s t distribution, respectively. The study further observed that the persistence of volatility in the BRICS markets does not follow the same hierarchical pattern under the error distributions, except under the skew-Student’s t and GHYP distributions where the pattern is the same. Under these two assumptions, i.e. the skew-Student’s t and GHYP, in a descending hierarchical order of magnitudes, volatility with persistence is highest in the Chinese market, followed by the South African market, then the Russian, Indian and Brazilian markets, respectively. However, under each of the five non-Gaussian error distributions, the Chinese market is the most volatile, while the least volatile is the Brazilian market.

Overdisp: A Stata (and Mata) Package for Direct Detection of Overdispersion in Poisson and Negative Binomial Regression Models

2020-08-06T23:32:32+08:00

Stata has several procedures that can be used in analyzing count-data regression models and, more specifically, in studying the behavior of the dependent variable, conditional on explanatory variables. Identifying overdispersion in countdata models is one of the most important procedures that allow researchers to correctly choose estimations such as Poisson or negative binomial, given the distribution of the dependent variable. The main purpose of this paper is to present a new command for the identification of overdispersion in the data as an alternative to the procedure presented by Cameron and Trivedi [5], since it directly identifies overdispersion in the data, without the need to previously estimate a specific type of count-data model. When estimating Poisson or negative binomial regression models in which the dependent variable is quantitative, with discrete and non-negative values, the new Stata package overdisp helps researchers to directly propose more consistent and adequate models. As a second contribution, we also present a simulation to show the consistency of the overdispersion test using the overdisp command. Findings show that, if the test indicates equidispersion in the data, there are consistent evidence that the distribution of the dependent variable is, in fact, Poisson. If, on the other hand, the test indicates overdispersion in the data, researchers should investigate more deeply whether the dependent variable actually exhibits better adherence to the Poisson-Gamma distribution or not.

Tail distribution of the integrated Jacobi diffusion process

2020-08-06T23:32:32+08:00

In this paper, we study the distribution of the integrated Jacobi diffusion processes with Brownian noise and fractional Brownian noise. Based on techniques of Malliavin calculus, we develop a unified method to obtain explicit estimates for the tail distribution of these integrated diffusions.