Mathematical Methods for Physics and Engineering - Matematica.NET

31.4 SOME BASIC ESTIMATORS(known) constants, it is immediately clear that E[ˆν r ]=ν r ,andsoˆν r is an unbiasedestimator of ν r . It is also possible to obtain an expression for V [ˆν r ], though thecalculation is somewhat lengthy.In the case where the population mean µ 1 is not known, the situation is morecomplicated. We saw in subsection 31.4.2 that the second sample moment n 2 (ors 2 )isnot an unbiased estimator of ν 2 (or σ 2 ). Similarly, the rth central moment ofa sample, n r , is not an unbiased estimator of the rth population central momentν r . However, in all cases the bias becomes negligible in the limit of large N.As we also found in the same subsection, there are complications in calculatingthe expectation and variance of n 2 ; these complications increase considerably forgeneral r. Nevertheless, we have derived already in this chapter exact expressionsfor the expectation value of the first few sample central moments, which are validfor samples of any size N. From (31.40), (31.43) and (31.46), we findE[n 1 ]=0,E[n 2 ]= N − 1N ν 2, (31.54)E[n 2 2]= N − 1N 3 [(N − 1)ν 4 +(N 2 − 2N +3)ν2].2By similar arguments it can be shown that(N − 1)(N − 2)E[n 3 ]=N 2 ν 3 , (31.55)E[n 4 ]= N − 1N 3 [(N 2 − 3N +3)ν 4 +3(2N − 3)ν2]. 2 (31.56)From (31.54) and (31.55), we see that unbiased estimators of ν 2 and ν 3 areˆν 2 =NN − 1 n 2, (31.57)ˆν 3 =N 2(N − 1)(N − 2) n 3, (31.58)where (31.57) simply re-establishes our earlier result that ̂σ 2 = Ns 2 /(N − 1) is anunbiased estimator of σ 2 .Unfortunately, the pattern that appears to be emerging in (31.57) and (31.58)is not continued for higher r, as is seen immediately from (31.56). Nevertheless,in the limit of large N, the bias becomes negligible, and often one simply takesˆν r = n r . For large N, it may be shown thatE[n r ] ≈ ν rV [n r ] ≈ 1 N (ν 2r − ν 2 r + r 2 ν 2 ν 2 r−1 − 2rν r−1 ν r+1 )Cov[n r ,n s ] ≈ 1 N (ν r+s − ν r ν s + rsν 2 ν r−1 ν s−1 − rν r−1 ν s+1 − sν s−1 ν r+1 )1251