Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.4 SOME BASIC ESTIMATORS

the form

σˆ=

(

N

N− 1

) 1 / 2

s,

wheresis the sample standard deviation. The expectation value of this estimator is given
by

E[σˆ]=

(

N

N− 1

) 1 / 2

E[(s^2 )^1 /^2 ]≈

(

N

N− 1

) 1 / 2

(E[s^2 ])^1 /^2 =σ.

An approximate expression for the variance ofσˆmay be found using (31.47) and is given
by

V[σˆ]=

N

N− 1

V[(s^2 )^1 /^2 ]≈

N

N− 1

[

d

d(s^2 )

(s^2 )^1 /^2

] 2

s^2 =E[s^2 ]

V[s^2 ]

≈

N

N− 1

[

1

4 s^2

]

s^2 =E[s^2 ]

V[s^2 ].

Using the expressions (31.43) and (31.47) forE[s^2 ]andV[s^2 ] respectively, we obtain

V[σˆ]≈

1

4 Nν 2

(

ν 4 −

N− 3

N− 1

ν 22

)

.

31.4.4 Population momentsμr

We may straightforwardly generalise our discussion of estimation of the popula-

tion meanμ(=μ 1 ) in subsection 31.4.1 to the estimation of therth population

momentμr. An obvious choice of estimator is therth sample momentmr.The

expectation value ofmris given by

E[mr]=

1

N

∑N

i=1

E[xri]=

Nμr

N

=μr,

and so it is an unbiased estimator ofμr.

The variance ofmrmay be found in a similar manner, although the calculation

is a little more complicated. We find that

V[mr]=E[(mr−μr)^2 ]

=

1

N^2

E





(

∑

i

xri−Nμr

) 2 



=

1

N^2

E





∑

i

x^2 ir+

∑

i

∑

j=i

xrixrj− 2 Nμr

∑

i

xri+N^2 μ^2 r





=

1

N

μ 2 r−μ^2 r+

1

N^2

∑

i

∑

j=i

E[xrixrj]. (31.51)