Mathematical Methods for Physics and Engineering - Matematica.NET

31.4 SOME BASIC ESTIMATORSSince the number of terms in the double sum on the RHS is N(N − 1), we haveE[V xy ]=E[x i y i ] − 1 N 2 (NE[x iy i ]+N(N − 1)E[x i y j ])= E[x i y i ] − 1 N (NE[x iy 2 i ]+N(N − 1)E[x i ]E[y j ])= E[x i y i ] − 1 ( ) N − 1E[xi y i ]+(N − 1)µ x µ y = Cov[x, y],NNwhere we have used the fact that, since the samples are independent, E[x i y j ]=E[x i ]E[y j ]. ◭It is possible to obtain expressions for the variances of the estimators (31.59)and (31.60) but these quantities depend upon higher moments of the populationP (x, y) and are extremely lengthy to calculate.Whether the means µ x and µ y are known or unknown, an estimator of thepopulation correlation Corr[x, y] is given byy]Ĉorr[x, y] =Ĉov[x, , (31.61)ˆσ x ˆσ ywhere Ĉov[x, y], ˆσ x and ˆσ y are the appropriate estimators of the population covarianceand standard deviations. Although this estimator is only asymptoticallyunbiased, i.e. for large N, it is widely used because of its simplicity. Once againthe variance of the estimator depends on the higher moments of P (x, y) andisdifficult to calculate.In the case in which the means µ x and µ y are unknown, a suitable (but biased)estimator isV xyĈorr[x, y] =N =NN − 1 s x s y N − 1 r xy, (31.62)where s x and s y are the sample standard deviations of the x i and y i respectivelyand r xy is the sample correlation. In the special case when the parent populationP (x, y) is Gaussian, it may be shown that, if ρ = Corr[x, y],E[r xy ]=ρ − ρ(1 − ρ2 )+O(N −2 ),2N(31.63)V [r xy ]= 1 N (1 − ρ2 ) 2 +O(N −2 ), (31.64)from which the expectation value and variance of the estimator Ĉorr[x, y] maybe found immediately.We note finally that our discussion may be extended, without significant alteration,to the general case in which each data item consists of n numbersx i ,y i ,...,z i .1253