Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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)
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