Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.8 EXERCISES

31.6 Prove that the sample mean is the bestlinear unbiased estimatorof the population
meanμas follows.

(a) If the real numbersa 1 ,a 2 ,...,ansatisfy the constraint

∑n

i=1ai=C,whereC

is a given constant, show that

∑n

i=1a

2

iis minimised byai=C/nfor alli.

(b) Consider the linear estimatorμˆ=

∑n

i=1aixi. Impose the conditions (i) that it

isunbiasedand (ii) that it is asefficientas possible.

31.7 A population contains individuals ofktypes in equal proportions. A quantityX
has meanμiamongst individuals of typeiand varianceσ^2 , which has the same
value for all types. In order to estimate the mean ofXover the whole population,
two schemes are considered; each involves a total sample size ofnk. In the first
the sample is drawn randomly from the whole population, whilst in the second
(stratified sampling)nindividuals are randomly selected from each of thektypes.
Show that in both cases the estimate has expectation

μ=

1

k

∑k

i=1

μi,

but that the variance of the first scheme exceeds that of the second by an amount

1

k^2 n

∑k

i=1

(μi−μ)^2.

31.8 Carry through the following proofs of statements made in subsections 31.5.2 and
31.5.3 about the ML estimatorsˆτandλˆ.

(a) Find the expectation values of the ML estimatorsτˆandˆλgiven, respectively,

in (31.71) and (31.75). Hence verify equations (31.76), which show that, even

though an ML estimator is unbiased, it does not follow that functions of it

are also unbiased.

(b) Show thatE[ˆτ^2 ]=(N+1)τ^2 /Nand hence prove thatˆτis a minimum-variance

estimator ofτ.

31.9 Each of a series of experiments consists of a large, but unknown, numbern
(1) of trials in each of which the probability of successpis the same, but also
unknown. In theith experiment,i=1, 2 ,...,N, the total number of successes is
xi(1). Determine the log-likelihood function.
Using Stirling’s approximation to ln(n−x), show that

dln(n−x)

dn

≈

1

2(n−x)

+ln(n−x),

and hence evaluate∂(nCx)/∂n.

By finding the (coupled) equations determining the ML estimatorspˆandnˆ,

show that, to ordern−^1 , they must satisfy the simultaneous ‘arithmetic’ and

‘geometric’ mean constraints

ˆnˆp=

1

N

∑N

i=1

xi and (1−pˆ)N=

∏N

i=1

(

1 −

xi

nˆ

)

.