Generalized sech-Probability Density: Robust Properties, Eva luation, and Application Probability Density Function

Abstract

For an adequate probabilistic description of the objects of statistical research, it is convenient to use a generalized probability density function that provides keeping the same mathematical model when the density parameters that affect the function shape change. We present the definition of a generalized sechk-density of probability distribution and the main characteristics: distribution function, characteristic function, variance, kurtosis, differential entropy, Fisher information regarding the offset parameter. It is shown that generalized sechk-density is a convenient probabilistic model that can be used in applied statistics in the range of excess values from zero to three. Sechk-density is a smooth function minimizing Fisher’s information on the density class. The robust properties of the generalized sechk-density are studied as applied to the algorithm of estimating mixing with known and unknown density parameters. A subclass of reduced robust estimation procedures - naturally lowered estimates - is introduced into the consideration. The quantitative characteristics of the sensitivity to a large error and sensitivity to a change in the asymptotic dispersion are presented. The considered examples of possible use of sechk-density allow us to speak about the expediency of its application in theoretical and applied statistics.