Publication Date



Technical Report: UTEP-CS-16-42


In many practical situations, the only information that we have about the probability distribution is its first few moments. Since many statistical techniques requires us to select a single distribution, it is therefore desirable to select, out of all possible distributions with these moments, a single "most representative" one. When we know the first two moments, a natural idea is to select a normal distribution. This selection is strongly consistent in the sense that if a random variable is a sum of several independent ones, then selecting normal distribution for all of the terms in the sum leads to a similar normal distribution for the sum. In situations when we know three moments, there is also a widely used selection -- of the so-called skew-normal distribution. However, this selection is not strongly consistent in the above sense. In this paper, we show that this absence of strong consistency is not a fault of a specific selection but a general feature of the problem: for third and higher order moments, no strongly consistent selection is possible.