Tomáš Cipra

Kalman filter with outliers and missing observations

The discrete Kalman filter which enables the treatment of incomplete data and outliers is described. The incomplete, or missing observations are included in such a way as to transform the Kalman filter to the case when observations have changing dimensions. In order to treat outliers, the Kalman filter is made robust using the M-estimation principle. Some special cases are considered including a convergence result for recursive parameter estimation in AR(1) process with innovation outliers and missing observations.

Financial and insurance formulas

Financial Formulas.- Simple Interest and Discount.- Compound Interest and Discount.- Continuous Interest and Discount.- Classical Analysis of Interest Rates.- Systems of Cash Flows.- Annuities.- Depreciation.- Financial Instruments.- Derivative Securities.- Utility Theory.- Rate of Return and Financial Risk.- Portfolio Analysis and CAPM Model.- Arbitrage Theory.- Financial Stochastic Analysis.- Insurance Formulas.- Insurance Classification.- Actuarial Demography.- Classical Life Insurance.- Modern Approaches to Life Insurance.- Pension Insurance.- Classical Non-Life Insurance.- Risk Theory in Insurance.- Health Insurance.- Reinsurance.- Formulas of Related Disciplines.- Mathematical Compendium.- Probability Theory.- Descriptive and Mathematical Statistics.- Econometrics.- Index Numbers.- Stochastic Processes.- Statistical Analysis of Time Series.

On practical implementation of robust kalman filtering

A parallel algorithm for Kalman filtering with contaminated observations is developed. This algorithm is suitable for the parallel computer implementation allowing to treat dynamic linear systems with large number of state variables in a robust recursive way. The implementation is based on the square root version of the Kalman filter. It represents a great improvement over serial implementations reducing drastically computational costs for each state update and avoiding numerical instability problems.

Kalman filter with a non-linear non-Gaussian observation relation

The dynamic linear model with a non-linear non-Gaussian observation relation is considered in this paper. Masreliezs theorem (see Masreliezs (1975)) of approximate non-Gaussian filtering with linear state and observation relations is extended to the case of a non-linear observation relation relation that can be approximated by a second-order Taylor expansion.ResumenEl modelo lineal dinámico con observación nolineal y no-Gausiano se estudia en este artículo. Se extiende el teorema de Masreliez (ver. Masreliez (1975)) como una aproximación de filtrado no-Gausiano con ecuación de estado lineal y ecuación de observaciones también lineal, al caso en que la ecuación de observaciones nolineal pueda aproximarse mediante la extesión de Taylor de segundo orden.

On conditional covariance modelling

A novel approach to conditional covariance modelling is introduced in the context of multivariate financial time series analysis. In particular, a class of multivariate generalized autoregressive conditional heteroscedasticity models is proposed. The suggested modelling technique is based on a specific dynamic orthogonal transformation derived by the LDL factorization of the conditional covariance matrix. An observed time series is transformed into a particular form that can be further treated by means of a discrete-time state space model under corresponding assumptions. The calibration can be performed by the associated Kalman recursive formulas, which are numerically effective. The introduced procedure has been investigated by extensive Monte Carlo experiments and empirical financial applications; it has been compared with other methods commonly used in this framework. The outlined methodology has demonstrated its capabilities, and it seems to be at least competitive in this field of research.

Simple correlated arma processes

The generalization of the simple correlated autoregressive processes introduced by RISAGEE ( 1980, 1981 ) is presented in the paper. The correlation structure of the model is investigated for the purpose of its identification. Various methods of estimation are discussed. Verification of the model is based on the portmanteau statistics whose behaviour is derived using the results for the simple correlated autoregressive processes and the prin¬ciple 0f Box and PIEBCE concerning the relation of estimated residual correlations in auto-regressive and ARMA models.

Self-weighted recursive estimation of GARCH models

ABSTRACT The generalized autoregressive conditional heteroscedasticity (GARCH) processes are frequently used to investigate and model financial returns. They are routinely estimated by computationally complex off-line estimation methods, for example, by the conditional maximum likelihood procedure. However, in many empirical applications (especially in the context of high-frequency financial data), it seems necessary to apply numerically more effective techniques to calibrate and monitor such models. The aims of this contribution are: (i) to review the previously introduced recursive estimation algorithms and to derive self-weighted alternatives applying general recursive identification instruments, and (ii) to examine these methods by means of simulations and an empirical application.

Robust recursive estimation in nonlinear time series

Aase (1983) has dealt with recursive estimation in nonlinear time series of autoregressive type including its asymptotic properties. This contribution modifies the results for the case of nonlinear time series with outliers using the principle of M-estimation from robust statistics. Strong consistency of the robust recursive estimates is preserved under corresponding assumptions. Several types of such estimates are compared by means of a numerical simulation.

Tests of periodicity with missing observations

The paper deals with testing of periodicity for time series with missing obser-vations. Two schemes of missing observations are considered: regularly missing obser-vation and observations missing randomly according to the BERNOULLI model

Autoregressive processes in optimization

Presentation de processus autoregressifs du premier ordre pour la programmation lineaire stochastique de structure dynamique