Why Cannot We Have a Strongly Consistent Family of Skew Normal (and Higher Order) Distributions

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 skewnormal 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. 1 Formulation of the Problem Need to select a distribution based on the first few moments. In many practical situations, we only have a partial information about the probability distribution. For example, often, all we know is the values of the first few moments. Most probabilistic and statistical techniques assume that we know the exact form of a probability distribution; see, e.g., [5]. In situations when we only have partial information about the probability distribution, there are many probability distribuThongchai Dumrongpokaphan Department of Mathematics, Faculty of Science, Chiang Mai University, Thailand, e-mail: [email protected] Vladik Kreinovich Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, Texas 79968, USA, e-mail: [email protected]