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### Asymmetric recursive methods for time series

Applications of Mathematics

The problem of asymmetry appears in various aspects of time series modelling. Typical examples are asymmetric time series, asymmetric error distributions and asymmetric loss functions in estimating and predicting. The paper deals with asymmetric modifications of some recursive time series methods including Kalman filtering, exponential smoothing and recursive treatment of Box-Jenkins models.

### Dynamic credibility with outliers and missing observations

Applications of Mathematics

In actuarial practice the credibility models must face the problem of outliers and missing observations. If using the $M$-estimation principle from robust statistics in combination with Kalman filtering one obtains the solution of this problem that is acceptable in the numerical framework of the practical actuarial credibility. The credibility models are classified as static and dynamic in this paper and the shrinkage is used for the final ratemaking.

### Exponential smoothing for irregular data

Applications of Mathematics

Various types of exponential smoothing for data observed at irregular time intervals are surveyed. Double exponential smoothing and some modifications of Holt’s method for this type of data are suggested. A real data example compares double exponential smoothing and Wright’s modification of Holt’s method for data observed at irregular time intervals.

### Improvement of prediction for a larger number of steps in discrete stationary processes

Aplikace matematiky

Let $\left\{{W}_{t}\right\}=\left\{{\left({X}_{{t}^{\text{'}}}^{\text{'}},{Y}_{t}^{\text{'}}\right)}^{\text{'}}\right\}$ be vector ARMA $\left(m,n\right)$ processes. Denote by ${\stackrel{^}{X}}_{t}\left(a\right)$ the predictor of ${X}_{t}$ based on ${X}_{t-a},{X}_{t-a-1},...$ and by ${\stackrel{^}{X}}_{t}\left(a,b\right)$ the predictor of ${X}_{t}$ based on ${X}_{t-a},{X}_{t-a-1},...,{Y}_{t-b},{Y}_{t-b-1},...$. The accuracy of the predictors is measured by ${\Delta }_{X}\left(a\right)=\text{E}\left[{X}_{t}-{\stackrel{^}{X}}_{t}\left(a\right)\right]{\left[{X}_{t}-{\stackrel{^}{X}}_{t}\left(a\right)\right]}^{\text{'}}$ and ${\Delta }_{X}\left(a,b\right)=\text{E}\left[{X}_{t}-{\stackrel{^}{X}}_{t}\left(a,b\right)\right]{\left[{X}_{t}-{\stackrel{^}{X}}_{t}\left(a,b\right)\right]}^{\text{'}}$. A general sufficient condition for the equality ${\Delta }_{X}\left(a\right)={\Delta }_{X}\left(a,a\right)\right]$ is given in the paper and it is shown that the equality ${\Delta }_{X}\left(1\right)={\Delta }_{X}\left(1,1\right)\right]$ implies ${\Delta }_{X}\left(a\right)={\Delta }_{X}\left(a,a\right)\right]$ for all natural numbers $a$.

### Udělení národní ceny ČR prof. J. Andělovi

Aplikace matematiky

### Some problems of exponential smoothing

Aplikace matematiky

The paper deals with some practical problems connected with the classical exponential smoothing in time series. The fundamental theorem of the exponential smoothing is extended to the case with missing observations and an interpolation procedure in the framework of the exponential smoothing is described. A simple method of the exponential smoothing for multivariate time series is suggested.

### Periodic moving average process

Aplikace matematiky

Periodic moving average processes are representatives of the class of periodic models suitable for the description of some seasonal time series and for the construction of multivariate moving average models. The attention having been lately concentrated mainly on periodic autoregressions, some methods of statistical analysis of the periodic moving average processes are suggested in the paper. These methods include the estimation procedure (based on Durbin's construction of the parameter estimators...

### Investigation of periodicity for dependent observations

Aplikace matematiky

It is proved that Hannan's procedure for statistical test of periodicity in the case of time series with dependent observations can be combined with Siegel's improvement of the classical Fischer's test of periodicity. Simulations performed in the paper show that this combination can increase the power of Hannan's test when at least two periodicities are present in the time series with dependent observations.

### Improvement of Fisher's test of periodicity

Aplikace matematiky

Fisher's test of periodicity in time series and Siegel's version of this test for compound periodicities are investigated in the paper. An improvement increasing the power of the test is suggested and demonstrated by means of numerical simulations.

### Class of unimodal distributions and its transformations

Časopis pro pěstování matematiky

### Note on approximate non-Gaussian filtering with nonlinear observation relation

Commentationes Mathematicae Universitatis Carolinae

### Note on inappropriate trend and seasonal elimination

Commentationes Mathematicae Universitatis Carolinae

Kybernetika

Kybernetika

Kybernetika

### 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. Masreliez's theorem (see Masreliez's (1975)) of approximate non-Gaussian filtering with linear state and observation relations is extended to the case of a non-linear observation relation that can be approximated by a second-order Taylor expansion.

### Study on Kalman filter in time series analysis

Commentationes Mathematicae Universitatis Carolinae

### ARMA models with nonstationary white noise

Commentationes Mathematicae Universitatis Carolinae

### Exponential smoothing for irregular time series

Kybernetika

The paper deals with extensions of exponential smoothing type methods for univariate time series with irregular observations. An alternative method to Wright’s modification of simple exponential smoothing based on the corresponding ARIMA process is suggested. Exponential smoothing of order m for irregular data is derived. A similar method using a DLS **discounted least squares** estimation of polynomial trend of order m is derived as well. Maximum likelihood parameters estimation for forecasting...

### Exponential smoothing for time series with outliers

Kybernetika

Recursive time series methods are very popular due to their numerical simplicity. Their theoretical background is usually based on Kalman filtering in state space models (mostly in dynamic linear systems). However, in time series practice one must face frequently to outlying values (outliers), which require applying special methods of robust statistics. In the paper a simple robustification of Kalman filter is suggested using a simple truncation of the recursive residuals. Then this concept is applied...

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