Jana Špirková

Weak monotonicity of Lehmer and Gini means

We analyze directional monotonicity of several mixture functions in the direction ( 1 , 1 ź , 1 ) , called weak monotonicity. Our particular focus is on power weighting functions and the special cases of Lehmer and Gini means. We establish limits on the number of arguments of these means for which they are weakly monotone. These bounds significantly improve the earlier results and hence increase the range of applicability of Gini and Lehmer means. We also discuss the case of affine weighting functions and find the smallest constant which ensures weak monotonicity of such mixture functions.

Induced weighted operators based on dissimilarity functions

Based on the minimization considering dissimilarity function D ( x , y ) = f ( x ) - f ( y ) 2 induced ordered weighted averaging operators IOWA and induced ordered generalized mixture operators IOM g are discussed. In general, these operators need not be monotone non-decreasing, thus they need not be the aggregation operators. This paper introduces conditions for their monotonicity and weak monotonicity. Several examples are included.

Mixture Utility in General Insurance

Aggregation functions and utility functions belong to very interesting parts of modern decision making theory. We develop basic concept of the connection of aggregation functions theory and utility theory to determine gross annual premium in general insurance. We introduce specific values of the gross annual premium on the basis of aggregation of the person’s utility functions which were determined empirically based on a short personal interview. Moreover, by specific utility function we determine minimal gross annual premium acceptable for the insurer.

Calibration of Utility Function and Mixture Premium

Calibration of the different types of utility functions of money is discussed in this paper. This calibration is based on an expected utility maximization of different alternatives of investment strategies which are offered to persons in a short questionnaire. Investigated utility functions have different Arrow-Pratt absolute and relative risk aversion coefficients. Moreover, the paper proposes a basic concept of uncertainty modeling by the chosen utility functions for the determination of so-called maximum mixture premium in non-life insurance. This concept is based on aggregation of maximum premiums by so-called mixture function with the selected weighting function. A case study is included.

Calibration of Utility Function

Utility theory belongs to a very interesting part of a modern decision making theory. We develop the basic concept of utility theory on a determination of a personal utility function which was founded empirically by short personal interview. We suggest a calibration of theoretical utility function by expected utility maximization criterion. Moreover, on the basis of calibration of utility functions we determine a model of specific values of a maximal and minimal annual premium acceptable for an insured and for an insurer, too.

Mortality Rates Smoothing Using Mixture Function

The paper offers a description of the new method of a smoothing of the mortality rates using the so-called moving mixture functions. Mixture functions represent a special class of weighted averaging functions where weights are determined by continuous, input values dependent, weighting functions. If they are increasing, they form an important class of aggregation functions. Such mixture functions are more flexible than the standard weighted arithmetic mean, and their weighting functions allow one to penalize or reinforce inputs based on their magnitude. The advantages of this method are that the weights of the input values depend on ourselves and coefficients of weighting functions can be changed each year so that the mean square error is minimized. Moreover, the paper offers the impact of this method on the amount of whole life pension annuities.

How to Calibrate a Questionnaire for Risk Measurement

Utility functions content parameters related to risk aversion coefficients which represent natural extensions of utility function properties. They measure how much utility we gain (or lose) as we add (or subtract) from our wealth. We set up these parameters for a person based on her/his answers to a questionnaire constructed to identify individual risk behavior. Calibration of such a questionnaire, and subsequently of utility functions, is based on an expected utility maximization of different alternatives of investment strategies. In the paper, we present questionnaire calibration methodology which we illustrate using absolute and relative risk aversion coefficients of two selected utility functions which have common, as well as different properties.

Directional and Ordered Directional Monotonicity of Mixture Functions

In this contribution, we discuss the concepts of so-called fusion functions, pre-aggregation functions and their directional and ordered directional monotonicity in the context of mixture functions. Mixture functions represent a special class of weighted averaging functions whose weights are determined by continuous weighting functions which depend on the input values. They need not be monotone, in general. If they are monotone increasing, they also belong to the important class of aggregation functions. If the are directionally monotone, they belong to the class of pre-aggregation functions.