Mathematical Methods for Physics and Engineering - Matematica.NET

31.8 EXERCISES31.6 Prove that the sample mean is the best linear unbiased estimator of the populationmean µ as follows.(a) If the real numbers a 1 ,a 2 ,...,a n satisfy the constraint ∑ ni=1 a i = C, whereCis a given constant, show that ∑ ni=1 a2 i is minimised by a i = C/n for all i.(b) Consider the linear estimator ˆµ = ∑ ni=1 a ix i . Impose the conditions (i) that itis unbiased and (ii) that it is as efficient as possible.31.7 A population contains individuals of k types in equal proportions. A quantity Xhas mean µ i amongst individuals of type i and variance σ 2 , which has the samevalue for all types. In order to estimate the mean of X over the whole population,two schemes are considered; each involves a total sample size of nk. In the firstthe sample is drawn randomly from the whole population, whilst in the second(stratified sampling) n individuals are randomly selected from each of the k types.Show that in both cases the estimate has expectationµ = 1 kk∑µ i ,i=1but that the variance of the first scheme exceeds that of the second by an amount1k∑(µk 2 i − µ) 2 .n31.8 Carry through the following proofs of statements made in subsections 31.5.2 and31.5.3 about the ML estimators ˆτ and ˆλ.i=1(a) Find the expectation values of the ML estimators ˆτ and ˆλ given, respectively,in (31.71) and (31.75). Hence verify equations (31.76), which show that, eventhough an ML estimator is unbiased, it does not follow that functions of itare also unbiased.(b) Show that E[ˆτ 2 ]=(N+1)τ 2 /N and hence prove that ˆτ is a minimum-varianceestimator of τ.31.9 Each of a series of experiments consists of a large, but unknown, number n(≫ 1) of trials in each of which the probability of success p is the same, but alsounknown. In the ith experiment, i =1, 2,...,N, the total number of successes isx i (≫ 1). Determine the log-likelihood function.Using Stirling’s approximation to ln(n − x), show thatd ln(n − x)dnand hence evaluate ∂( n C x )/∂n.≈1+ln(n − x),2(n − x)By finding the (coupled) equations determining the ML estimators ˆp and ˆn,show that, to order n −1 , they must satisfy the simultaneous ‘arithmetic’ and‘geometric’ mean constraintsˆnˆp = 1 NN∑N∏ (x i and (1 − ˆp) N =i=1i=11 − x iˆn).1299