Stat 5101 Lecture Notes - School of Statistics

7.3. SAMPLING DISTRIBUTIONS OF SAMPLE MOMENTS 189The k-th moment isA k,n = E n (X k )= 1 n∑Xi k .ni=1The central moments of this empirical distribution areM k,n = E n{[X − En (X)] k} = E n{(X − Xn ) k} = 1 nn∑(X i − X n ) kAs with any distribution, the first central moment is zero, and the second isV n =var n (X)=E n{(X−Xn ) 2} = 1 ni=1n∑(X i − X n ) 2 . (7.16)If there were any logic to statistics V n would be called the “sample variance,”but Lindgren, agreeing with most other textbooks, uses that term for somethingslightly differentS 2 n =nn − 1 V n = 1n − 1i=1n∑(X i − X n ) 2 . (7.17)The n − 1 rather than n in the definition makes all of the formulas involving S 2 nugly, and makes S 2 n not satisfy any of the usual rules involving variances. So bewarned, and be careful! For example, V n obeys the parallel axis theorem, hencei=1V n = E n (X 2 ) − E n (X) 2 = 1 nn∑Xi 2 − X 2 n.Clearly S 2 n cannot satisfy the same rule or it would be V n . The only way tofigure out the analogous rule for S 2 n is to remember the rule for V n (which makessense) and derive the one for S 2 n.Sn 2 =nn − 1 V n[= n 1n − 1 n= 1n − 1i=1i=1]n∑Xi 2 − X 2 ni=1n∑Xi 2 −nn − 1 X2 nNo matter how you try to write it, it involves both n and n − 1, and makes nosense.Since Sn 2 is so ugly, why does anyone use it? The answer, as with so manyother things, is circular. Almost everyone uses it because it’s the standard,and it’s the standard because almost everyone uses it. And “almost everyone”includes a lot of people, because Sn 2 is a topic in most introductory statisticscourses.