Estimating third central moment C3 for privacy case under interval and fuzzy uncertainty

Some probability distributions (e.g., Gaussian) are symmetric, some (e.g., lognormal) are non-symmetric (skewed). How can we gauge the skeweness? For symmetric distributions, def 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 · Σ_i=1ⁿ(x_i - E)³, where E =^def 1/n · Σ_i=1ⁿx_i. In many practical situations, we do not know the exact values of x%. 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.