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Matej Bel University, Faculty of Economics, Department of Applied Informatics, Banská Bystrica, Slovakia

The University of Economics, Faculty of Informatics and Statistics, Department of Statistics and Probability, Prague, Czech Republic

Wroclaw University of Economics, Department of Statistics and Economic Cybernetics, Wroclaw, Poland

**4 ^{th} International Scientific Conference**

APPLICATIONS OF MATHEMATICS AND STATISTICS IN ECONOMY

### September 13 – September 14, 2001

Zadov, Czech Republic

The Characteristic Features of Financial Time Series

Josef Arlt – Markéta Arltová; University of Economics, Prague

Time series of prices as well as time series based on prices or time series, which describe prices and their dynamism, are called financial time series. These time series have some typical properties. There are two basic assumptions: normality and linearity of log returns of the financial time series. The distributions of log returns are usually skewed and more peaked that the normal distribution. Linear models cannot express the characteristic features of these time series as they suppose only correlation dependence. The solution of the problems of non-normality and non-linearity in the financial time series can be in a very close connection.

Survey Sampling Methods

Jitka Bartošová; University of Economics, Prague

The aim of the survey sampling methods is the point – or interval – estimate of population characteristics. The presentation concerns with the influence of population probability distribution together with influence of sample size on precision of estimated characteristics.

Robust Estimators of Scale

Dagmar Blatná, University of Economics, Prague

Some of the estimators of scale are based on functions of the order statistics of a sample, others are derived from nonparametric rank tests for scale and some ones are derived from robust estimators of location parameter. Basic types of scale estimators will be presented.

Premium Principles and Reinsurance Contracts

Stanisław Heilpern; University of Economics, Wroclaw

This paper is devoted the premium principles. We study the premium principles strictly connected with the reinsurance contracts. We present the basic properties of insurance premium. We favor these properties witch characterized the reinsurance contracts, i.e.: the scale-invariance, subadditivity, commonotonic additivity and preservation of stop-loss order. In this paper we prefer the premium principle based on Choquet integral with respect to the monotone set functions. This premium principle satisfies the all properties mentioned above. We present the axiomatic approach and we show that it is the only continuous, commonotonic additive functional premium, which preserves the stop-loss order. We pay attention to the family of functional premiums induced by Choquet integrals with respect to the distorted probabilities. In the case of the concave distorted functions, these premium principles are risk adjusted. We prefer the proportional hazard transform principle, which is the best from the reinsurance point of view.

Digital Signature

Miroslav Hužvár; Matej Bel University, Banská Bystrica

Digital signature is one of the security tools that play the central role in the development of electronic commerce. This contribution deals with the theoretical principles of digital signatures, as well as with some aspects of their practical applications.

Some Remarks on Differential Geometry

Miloš Kaňka; University of Economics, Prague

In my contribution an example of using of Cartan’s method of moving frame in case of surfaces in 3-dimensional Euclidean space will be given.

Acceptance Sampling from Statistical Point of View

Jindřich Klůfa; University of Economics, Prague

Acceptance sampling plans for inspection by variables with given producer’s and consumer’s risk are considered in this paper. We report on an algorithm allowing the calculation of these plans. These plans were compared with the corresponding plans for inspection by attributes.

Prognosis – Software Package as a Modern Didactic Tool

Maria Kolenda; University of Economics, Wroclaw

The intention of the authors is a clear presentation of time series forecasting methodology and showing difficulties concerned with making a “good forecast”.

Once More to the Methodology of the Alternative Parametric Allocation

Josef Kozák – Richard Hindls – Stanislava Hronová; University of Economics, Prague

We have formulated the problem of parametric allocation of yearly aggregates of dependent variable into seasons in the cited paper [KHH]. In this connection we have mentioned one special approach called alternative parametric allocation. The purpose of the following remarks is to supply and put this approach in concrete terms.

The Law of Large Numbers

Ondřej Krakovič; Univerity of Economics, Prague

The author raises the problem of the law of large numbers and tries to explain psychological aspect of this problem using examples and to analyze it from the standpoint of the theory of probability, as well as from the standpoint of common sense.

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Approaches to Investigation of Seasonality in Tourism

Peter Laco; Matej Bel University, Banská Bystrica

Summary of approaches to measuring seasonality will be discussed. Approach of economists and approach of statisticians will be compared. Forms and measuring of seasonality as well as factors that cause seasonality will be characterized. Character of seasonality in various kinds of hotels in Slovakia will be also reflected.

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Ways of the Ruin Risk Reduction by the Optimum Retention Level of Reinsurance Contract Setting

Kinga Migdal; University of Economics, Wroclaw

One of the reasons for reinsurance is to improve the financial situation of a cadent by giving up part of its risk. The purpose of this paper is to analyze how to find the optimum retention level of reinsurance contract (optimum in the sense of minimizing the probability of ruin). Two basic models of reinsurance were considered: proportional quota-share reinsurance and non-proportional excess of loss reinsurance.

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Insurance Optimality Problems

Walenty Ostasiewicz – Stanislawa Ostasiewicz; University of Economics, Wroclaw

The paper presents an overview of the main problems concerning the optimal decisions in an insurance business. Particularly, there are discussed two famous theorems by Borch and Arrow about the optimal deductibility, the optimal problems of risk exchange and the other problems of expected utility maximization by insured as well as by insurer.

Czech Macroeconomic Time Series

Luboš Marek; University of Economics, Prague

I study the behavior of the Czech macroeconomic time series such as Gross domestic product, Inflation rate and Unemployment rate, Import, Export and so on in my paper. The problem with the number of observations and with the comparability analysis will be discussed. I describe the general class of suitable models for these series and I compare the different analysis methods.

Marginal Homogeneity Testing and Mantel-Haenszel Statistics

Iva Pecáková; University of Economics, Prague

When measurements of categorical response variables are obtained in time, under multiple conditions or from more observers, the resultant samples are dependent. The symmetry* *and the marginal homogeneity of joint distribution have special importance in the analysis of repeated measurements. Some tests of marginal homogeneity are presented in this paper.

On the Selection of Model’s Variables by Means of Genetical and Neural Methods

Mieczysław Rymarczyk; University of Economics, Wroclaw

The main problem explored in this paper deals with data smoothing and with identifying of a relationship between the set of input variables and the output variable. Assume that given are: a set of independent variables ** X**ÎR

^{n}, a dependent variable YÎR

^{1}, and a relationship between them is given by Y =

*f(*Depending on the form of function, the two cases are studied:

**X**, b).(a) linear dependency – optimization is carried out by means of a genetical algorithm for the choice of variables, given the function of fit expressing the goal: “*number of variables as small as possible and the function’s goodness of fit to data as large as possible*“.

(b) nonlinear dependency – procedure combines genetical and neuronal optimization. The structure of neuronal network, including the proper set of inputs, is genetically optimized, whereas the quality of fit is obtained in the process of neuronal optimizing the weights of net linkages.

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Methods for Variables Clustering and the Use of them in Statistical Packages

Hana Řezanková – Dušan Húsek; University of Economics, Prague

The paper is aimed on the comparison of different algorithms for variables clustering. Factor and cluster analysis, multidimensional scaling and the use of neural networks are considered. Attention to fuzzy clustering and the algorithms for binary variables are paid. The use of some methods mentioned above in statistical packages is shown.

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The Volatility of Financial Time Series

Jiří Trešl; University of Economics, Prague

The volatility of selected financial time series, namely of stock returns and exchange rates is presented and analyzed. As a rule, relatively quiet periods are accompanied by sudden bursts of instability. Statistical analysis reveals some kind of ant persistent behavior typical for turbulent fluid flow. Simple relaxation model is employed and discussed.

František Weyr and Statistics

Prokop Závodský; University of Economics, Prague

This year, we recall 50th annual of the death of F.Weyr, our leading legal theoretician in the period between World Wars. He dealt with statistics as well; he has been the chairman of State Office for Statistics during 1920-1929 and, further, the longstanding member of International Statistical Institute. This contribution is devoted to the contribution of F.Weyr to the development of statistical theory and practice in our country.

Socio-economic Profile of Income Poverty: Czech Republic and the EU

Martin Zelený; University of Economics, Prague

The paper compares socio-economic profile of income poverty in Czech Republic with those in the European Union countries. I employ the EU poverty measurement methodology developed by Eurostat used for construction of EU structural indicators of social cohesion and apply it on microdata from a large national representative survey of Czech households. Based on this exercise, I construct a socio-economic profile of income poverty in Czech Republic and compare it with societies in the EU countries. My focus is especially on identifying of vulnerable groups with high-income poverty risk both in Czech and EU societies and assessing the differences and similarities. Taking into account possible dependence on chosen poverty line definition, I go on with sensitivity analysis of our results by varying the poverty line parameters. I come to a simple poverty risk model as a way to evaluate sensitivity of results for a given poverty line and to give clearer picture of vulnerable groups affected by income poverty.

On Some Properties of Dynamic Mundell-Fleming Model

Emília Zimková-Rudolf Zimka; MatejBel University, Banská Bystrica

Contribution deals with a dynamic macroeconomic model of an open economy. The development of output and the rate of exchange is studied in the neighborhood of its equilibrium. The conditions for the equilibrium stability are found.