A scalar inference function suggested by Fabián ( 2001 Fabián , Z. ( 2001 ). Induced cores and their use in robust parametric estimation . Commun. Statist. Theor. Meth. 30 ( 3 ): 537 – 556 . [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ) is used for an introduction of measures of the central tendency and variability of continuous distributions. A number of examples shows that new measures are plausible namely for such distributions, for which the mean and/or the variance do not exist. The estimates of both measures are new characteristics of random samples taken from the distribution.

A continuous random variable can be uniquely represented by the score function of distribution, i.e. a scalar-valued function describing the influence of a data item on the typical value of the distribution. There are at most two relevant scalar-valued scores based on two basic transformations: the “natural” one, useful for estimating parameters of parametric distributions by the generalized moment method; and the “universal” one, which is a basis for the score function of distribution. The score mean, the zero of the score function of distribution, represents the typical value of the distribution in the sense of the value which is most likely to be sampled. Both scores are often identical to each other. The derivative of the score function of distribution is the weight function of this distribution. The main result is that sums of score random variables obey the central limit theorem and can be used for directly estimating the score mean without use of any current estimator.

The t-mean is a new characteristic of the central tendency of continuous distributions. In this article, we introduce a t-difference in the sample space, which is used for a construction of confidence intervals for the t-mean.

A new approach to theory of continuous probability distributions, based on scalar-valued score functions, offers the score mean as a finite typical value of distributions including the heavy-tailed ones. In the article, we define the score variance as a finite measure of variability of distributions with respect to the typical value and discuss its properties and methods of its estimation. By means of the square root of the score variance we introduce a generalized Rao distance in the sample space.

1. Classical mathematical statistics does not work well with heavy-tailed distributions. Robust statistics solves a different problem: how to reduce effects of possible contamination of the data wi...