Estimating Third Central Moment C3 for Privacy Case under Interval and Fuzzy Uncertainty

Authors

Ali Jalal-Kamali, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

3-2013

Comments

Technical Report UTEP-CS-13-13a

To appear in Proceedings of the joint World Congress of the International Fuzzy Systems Association and Annual Conference of the North American Fuzzy Information Processing Society IFSA/NAFIPS'2013, Edmonton, Canada, June 24-28, 2013.

Abstract

Some probability distributions (e.g., Gaussian) are symmetric, some (e.g., lognormal) are non-symmetric ({\em skewed}). How can we gauge the skeweness? For symmetric distributions, the third central moment C₃ = E[(x - E(x))³] is equal to 0; thus, this moment is used to characterize skewness. This moment is usually estimated, based on the observed (sample) values x₁, ..., x_n, as C₃ = (1/n) * ((x₁ - E)³ + ... + (x_n - E)³), where E = (1/n) * (x₁ + ... + x_n). In many practical situations, we do not know the exact values of x_i. For example, to preserve privacy, the exact values are often replaced by intervals containing these values (so that we only know whether the age is under 10, between 10 and 20, etc). Different values from these intervals lead, in general, to different values of C₃; it is desirable to find the range of all such possible values. In this paper, we propose a feasible algorithm for computing this range.