Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.4 SOME BASIC ESTIMATORS

(known) constants, it is immediately clear thatE[νˆr]=νr,andsoνˆris an unbiased

estimator ofνr. It is also possible to obtain an expression forV[ˆνr], though the

calculation is somewhat lengthy.

In the case where the population meanμ 1 isnotknown, the situation is more

complicated. We saw in subsection 31.4.2 that the second sample momentn 2 (or

s^2 )isnotan unbiased estimator ofν 2 (orσ^2 ). Similarly, therthcentralmoment of

a sample,nr, is not an unbiased estimator of therth population central moment

νr. However, in all cases the bias becomes negligible in the limit of largeN.

As we also found in the same subsection, there are complications in calculating

the expectation and variance ofn 2 ; these complications increase considerably for

generalr. Nevertheless, we have derived already in this chapterexactexpressions

for the expectation value of the first few sample central moments, which are valid

for samples of any sizeN. From (31.40), (31.43) and (31.46), we find

E[n 1 ]=0,

E[n 2 ]=

N− 1

N

ν 2 , (31.54)

E[n^22 ]=

N− 1

N^3

[(N−1)ν 4 +(N^2 − 2 N+3)ν 22 ].

By similar arguments it can be shown that

E[n 3 ]=

(N−1)(N−2)

N^2

ν 3 , (31.55)

E[n 4 ]=

N− 1

N^3

[(N^2 − 3 N+3)ν 4 +3(2N−3)ν^22 ]. (31.56)

From (31.54) and (31.55), we see that unbiased estimators ofν 2 andν 3 are

ˆν 2 =

N

N− 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 thatσ̂^2 =Ns^2 /(N−1) is an

unbiased estimator ofσ^2.

Unfortunately, the pattern that appears to be emerging in (31.57) and (31.58)

isnotcontinued for higherr, as is seen immediately from (31.56). Nevertheless,

in the limit of largeN, the bias becomes negligible, and often one simply takes

ˆνr=nr. For largeN, it may be shown that

E[nr]≈νr

V[nr]≈

1

N

(ν 2 r−νr^2 +r^2 ν 2 ν^2 r− 1 − 2 rνr− 1 νr+1)

Cov[nr,ns]≈

1

N

(νr+s−νrνs+rsν 2 νr− 1 νs− 1 −rνr− 1 νs+1−sνs− 1 νr+1)