帮顶

The Complications of the Fourth Central Moment Yadolah DODGE and Valentin ROUSSON This article illustrates the additional complications in the mathematical formulas involving the fourth central moment in comparison with those involving the first moment, the second central moment, and the third central moment.
KEY WORDS: Central moments; Cumulants; Descrip- tive statistics; Moments; Unbiased estimation.
1. INTRODUCTION
      In an introductory statistics course, moments and central moments of a random variable are interpreted descriptively. The first moment (the mean) and the second central moment (the variance) are introduced to describe, respectively, the location and the dispersion of a random variable. Then the third and the fourth central moments are said to be in turn the skewness and the kurtosis of the random variable. A nat- ural and legitimate question that students ask at this point is "What about the higher order central moments? What do they describe? Why do we not go beyond the fourth one?" The fact is that the fifth, the sixth, and other higher order central moments have not yet received any descriptive in- terpretation in the statistical literature and one can wonder why. The absence of interpretation for high order central mo- ments is perhaps simply due to the mathematical complica- tions involved. In this article we present some mathemat- ical formulas dealing with moments and central moments. Looking at these formulas we come to the conclusion that mathematical complications arise not with the fifth, but al- ready with the fourth central moment. In comparison, for- mulas involving the first moment, the second central mo- ment, and the third central moment are beautifully simple and certainly give joy to teachers and graduates of statistics.
2. MOMENTS AND CUMULANTS Let X be a real valued random variable with distribution F. The first moment ,u of X is defined by +C) At = t (x) xdF(x). -00 Similarly the moments of higher order p4 (r > 2) are de- fined by r+C utr =A ur(X) = xrdF(x) +00 and the central moments Atr (r > 1) are defined by +00 -tr = Atr(X) - j (x c - ft)dF(x). -00 The first central moment ft, is always equal to zero and as a result is not an intere


3. SUM OF INDEPENDENT RANDOM VARIABLES