Georgi N. Boshnakov

A periodic Levinson-Durbin algorithm for entropy maximization

A recursive algorithm is presented for the computation of the first-order and second-order derivatives of the entropy of a periodic autoregressive process with respect to the autocovariances. It is an extension of the periodic Levinson-Durbin algorithm. The algorithm has been developed for use at one of the steps of an entropy maximization method developed by the authors. Numerical examples of entropy maximization by that method are given. An implementation of the algorithm is available as an R package.

Confidence characteristics of distributions

We introduce the confidence characteristic of a distribution as the distribution corresponding to the decomposition concentration function. We show that the decomposition concentration function has very good differentiability properties for any distribution, that the Lebesgue measures of the shortest confidence regions of a distribution and its confidence characteristic are the same, and that similar properties hold for the entropies and the level sets of their densities.

Periodically Correlated Solutions to a Class of Stochastic Difference Equations

The class of the periodically correlated processes sets up one of the possible frameworks for description and modeling of time series having pseudo-periodic behaviour. The mean and the autocovariance functions of the processes from this class are periodic. Many of the concepts of the stationary theory admit generalization to the periodic case. There is a duality between the multivariate stationary processes and the periodically correlated processes which makes the investigation of these two classes theoretically equivalent. A survey on these questions (mainly from an algorithmic point of view) and a lot of references may be found in Boshnakov [Bo].

Generation Of Time Series Models With Given Spectral Properties

We give a method for generation of periodically correlated and multivariate ARIMA models whose dynamic characteristics are partially or fully specified in terms of spectral poles and zeroes or their equivalents in the form of eigenvalues/eigenvectors of associated model matrices. Our method is based on the spectral decomposition of multi-companion matrices and their factorization into products of companion matrices. Generated models are needed in simulation but may also be used in estimation, e.g. to set sensible initial values of parameters for nonlinear optimization. Copyright 2009 The Authors. Journal compilation 2009 Blackwell Publishing Ltd

The asymptotic covariance matrix of the multivariate serial correlations

We show that the entries of the asymptotic covariance matrix of the serial covariances and serial correlations of a multivariate stationary process can be expressed in terms of the autocovariances corresponding to the tensor square of its spectral density. The tensor convolution introduced in the paper may be of some interest on its own.

Bartlett's formulae—Closed forms and recurrent equations

We show that the entries of the asymptotic covariance matrix of the sample autocovariances and autocorrelations of a stationary process can be expressed in terms of the square of its spectral density. This leads to closed form expressions and fast computational algorithms.

Heavy-tailed mixture GARCH volatility modeling and Value-at-Risk estimation

This paper presents a heavy-tailed mixture model for describing time-varying conditional distributions in time series of returns on prices. Student-t component distributions are taken to capture the heavy tails typically encountered in such financial data. We design a mixture MT(m)-GARCH(p,q) volatility model for returns, and develop an EM algorithm for maximum likelihood estimation of its parameters. This includes formulation of proper temporal derivatives for the volatility parameters. The experiments with a low order MT(2)-GARCH(1,1) show that it yields results with improved statistical characteristics and economic performance compared to linear and nonlinear heavy-tail GARCH, as well as normal mixture GARCH. We demonstrate that our model leads to reliable Value-at-Risk performance in short and long trading positions across different confidence levels.

Multi-companion matrices

In this paper, we introduce and study the class of multi-companion matrices. They generalize companion matrices in various ways and possess a number of interesting properties. We find explicit expressions for the generalized eigenvectors of multi-companion matrices such that each generalized eigenvector depends on the corresponding eigenvalue and a number of quantities which are functionally independent of the eigenvalues of the matrix and (up to a uniqueness constraint) of each other. Moreover, we obtain a parameterization of a multi-companion matrix through the eigenvalues and these additional quantities. The number of parameters in this parameterization is equal to the number of non-trivial elements of the multi-companion matrix. The results can be applied to statistical estimation, simulation and theoretical studies of periodically correlated and multivariate time series in both discrete- and continuous-time.

Sufficient statistics for ARMA models with some fixed parameters

The likelihood function of ARMA processes with some fixed parameters is considered. An expression for it and a sufficient statistic are obtained. It is shown how the proposed approach can be used to obtain closed form expressions for the likelihood functions of some ARMA models. Applications to parameter estimation, hypothesis testing, speech processing and related problems are discussed.

Maximum entropy models for general lag patterns

The maximum entropy problem for autocovariances given over a class of subsets of is solved. A more general problem when prediction coefficients and prediction error variances are given instead of covariances is considered and solved, as well. Two notions about maximum entropy in time series context are introduced and some misconceptions in the literature are discussed.