Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

However, since the sample valuesxiare assumed to be independent, we have

E[xrixrj]=E[xri]E[xrj]=μ^2 r. (31.52)

The number of terms in the sum on the RHS of (31.51) isN(N−1), and so we find

V[mr]=

1

N

μ 2 r−μ^2 r+

N− 1

N

μ^2 r=

μ 2 r−μ^2 r

N

. (31.53)

SinceE[mr]=μrandV[mr]→0asN→∞,therth sample momentmris also

a consistent estimator ofμr.

Find the covariance of the sample momentsmrandmsfor a sample of sizeN.

We obtain the covariance of the sample momentsmrandmsin a similar manner to that
used above to obtain the variance ofmr. From the definition of covariance, we have

Cov[mr,ms]=E[(mr−μr)(ms−μs)]

=

1

N^2

E

[(

∑

i

xri−Nμr

)(

∑

j

xsj−Nμs

)]

=

1

N^2

E





∑

i

xri+s+

∑

i

∑

j=i

xrixsj−Nμr

∑

j

xsj−Nμs

∑

i

xri+N^2 μrμs





Assuming thexito be independent, we may again use result (31.52) to obtain

Cov[mr,ms]=

1

N^2

[Nμr+s+N(N−1)μrμs−N^2 μrμs−N^2 μsμr+N^2 μrμs]

=

1

N

μr+s+

N− 1

N

μrμs−μrμs

=

μr+s−μrμs

N

.

We note that by settingr=s, we recover the expression (31.53) forV[mr].

31.4.5 Population central momentsνr

We may generalise the discussion of estimators for the second central momentν 2

(or equivalentlyσ^2 ) given in subsection 31.4.2 to the estimation of therth central

momentνr. In particular, we saw in that subsection that our choice of estimator

forν 2 depended on whether the population meanμ 1 is known; the same is true

for the estimation ofνr.

Let us first consider the case in whichμ 1 is known. From (30.54), we may write

νras

νr=μr−rC 1 μr− 1 μ 1 +···+(−1)krCkμr−kμk 1 +···+(−1)r−^1 (rCr− 1 −1)μr 1.

Ifμ 1 is known, a suitable estimator is obviously

ˆνr=mr−rC 1 mr− 1 μ 1 +···+(−1)krCkmr−kμk 1 +···+(−1)r−^1 (rCr− 1 −1)μr 1 ,

wheremris therth sample moment. Sinceμ 1 and the binomial coefficients are