Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSHowever, since the sample values x i are assumed to be independent, we haveE[x r i x r j]=E[x r i ]E[x r j]=µ 2 r . (31.52)The number of terms in the sum on the RHS of (31.51) is N(N −1), and so we findV [m r ]= 1 N µ 2r − µ 2 r + N − 1Nµ2 r = µ 2r − µ 2 rN . (31.53)Since E[m r ]=µ r and V [m r ] → 0asN →∞,therth sample moment m r is alsoa consistent estimator of µ r .◮Find the covariance of the sample moments m r and m s for a sample of size N.We obtain the covariance of the sample moments m r and m s in a similar manner to thatused above to obtain the variance of m r . From the definition of covariance, we haveCov[m r ,m s ]=E[(m r − µ r )(m s − µ s )][( )( )]= 1 ∑ ∑N E x r 2 i − Nµ r x s j − Nµ sij= 1 N 2 E ⎡⎣ ∑ ix r+si+ ∑ i⎤∑∑ ∑x r i x s j − Nµ r x s j − Nµ s x r i + N 2 µ r µ s⎦Assuming the x i to be independent, we may again use result (31.52) to obtainj≠iCov[m r ,m s ]= 1 N [Nµ 2 r+s + N(N − 1)µ r µ s − N 2 µ r µ s − N 2 µ s µ r + N 2 µ r µ s ]= 1 N µ r+s + N − 1Nµ rµ s − µ r µ s= µ r+s − µ r µ s.NWe note that by setting r = s, we recover the expression (31.53) for V [m r ]. ◭ji31.4.5 Population central moments ν rWe may generalise the discussion of estimators for the second central moment ν 2(or equivalently σ 2 ) given in subsection 31.4.2 to the estimation of the rth centralmoment ν r . In particular, we saw in that subsection that our choice of estimatorfor ν 2 depended on whether the population mean µ 1 is known; the same is truefor the estimation of ν r .Let us first consider the case in which µ 1 is known. From (30.54), we may writeν r asν r = µ r − r C 1 µ r−1 µ 1 + ···+(−1) k r C k µ r−k µ k 1 + ···+(−1) r−1 ( r C r−1 − 1)µ r 1.If µ 1 is known, a suitable estimator is obviouslyˆν r = m r − r C 1 m r−1 µ 1 + ···+(−1) k r C k m r−k µ k 1 + ···+(−1) r−1 ( r C r−1 − 1)µ r 1,where m r is the rth sample moment. Since µ 1 and the binomial coefficients are1250