Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.4 SOME BASIC ESTIMATORS

wheres^4 is given by

s^4 =

[∑

ix

2

i

N

−

(∑

ixi

N

) 2 ]^2

=

(

∑

ix

2

i)

2

N^2

− 2

(

∑

ix

2

i)(

∑

ixi)

2

N^3

+

(

∑

ixi)

4

N^4

. (31.45)

We will consider in turn each of the three terms on the RHS. In the first term, the sum
(

∑

ix

(^2) i) (^2) can be written as
(
∑
i
x^2 i

) 2

=

∑

i

x^4 i+

∑

ji,j=i

x^2 ix^2 j,

where the first sum containsNterms and the second containsN(N−1) terms. Since the
sample elementsxiandxjare assumed independent, we haveE[x^2 ix^2 j]=E[x^2 i]E[x^2 j]=μ^22 ,
and so

E





(

∑

i

x^2 i

) 2 

=Nμ 4 +N(N−1)μ^22.

Turning to the second term on the RHS of (31.45),

(

∑

i

x^2 i

)(

∑

i

xi

) 2

=

∑

i

x^4 i+

∑

i,j

j=i

x^3 ixj+

∑

i,j

j=i

x^2 ix^2 j+

∑

i,j,k

k=j=i

x^2 ixjxk.

Since the mean of the population has been assumed to equal zero, the expectation values
of the second and fourth sums on the RHS vanish. The first and third sums containN
andN(N−1) terms respectively, and so

E





(

∑

i

x^2 i

)(

∑

i

xi

) 2 

=Nμ 4 +N(N−1)μ^22.

Finally, we consider the third term on the RHS of (31.45), and write

(

∑

i

xi

) 4

=

∑

i

x^4 i+

∑

i,j

j=i

x^3 ixj+

∑

i,j

j=i

x^2 ix^2 j+

∑

i,j,k

k=j=i

x^2 ixjxk+

∑

i,j,k,l

l=k=j=i

xixjxkxl.

The expectation values of the second, fourth and fifth sums are zero, and the first and third
sums containNand 3N(N−1) terms respectively (for the third sum, there areN(N−1)/ 2
ways of choosingiandj, and the multinomial coefficient ofx^2 ix^2 jis 4!/(2!2!) = 6). Thus

E





(

∑

i

xi

) 4 

=Nμ 4 +3N(N−1)μ^22.

Collecting together terms, we therefore obtain

E[s^4 ]=

(N−1)^2

N^3

μ 4 +

(N−1)(N^2 − 2 N+3)

N^3

μ^22 , (31.46)

which, together with the result (31.43), may be substituted into (31.44) to obtain finally

V[s^2 ]=

(N−1)^2

N^3

μ 4 −

(N−1)(N−3)

N^3

μ^22

=

N− 1

N^3

[(N−1)ν 4 −(N−3)ν^22 ], (31.47)