Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.4 SOME BASIC ESTIMATORS (known) constants, it is immediately clear thatE[νˆr]=νr,andsoνˆris an unbiased estimator ofνr. It is ...

STATISTICS 31.4.6 Population covarianceCov[x, y]and correlationCorr[x, y] So far we have assumed that each of ourNindependent sa ...

31.4 SOME BASIC ESTIMATORS Since the number of terms in the double sum on the RHS isN(N−1), we have E[Vxy]=E[xiyi]− 1 N^2 (NE[xi ...

STATISTICS 31.4.7 A worked example To conclude our discussion of basic estimators, we reconsider the set of experi- mental data ...

31.5 MAXIMUM-LIKELIHOOD METHOD Substituting these values into (31.50), we obtain σˆx= ( N N− 1 ) 1 / 2 sx±(Vˆ[σˆx])^1 /^2 =12. 2 ...

STATISTICS 0 0 246810 12 1416 18 20 0.5 1 N=5 L(x;τ) τ^00246810 12 1416 18 20 0.5 1 N=10 L(x;τ) τ 0 0 246810 12 1416 18 20 0.5 1 ...

31.5 MAXIMUM-LIKELIHOOD METHOD a a a a L(x;a) L(x;a) L(x;a) L(x;a) aˆ aˆ ˆa aˆ (a) (b) (c) (d) Figure 31.6 Typical shapes of one ...

STATISTICS maximum that occurs at a stationary point (the likelihood function is then termed unimodal). In this case, the ML est ...

31.5 MAXIMUM-LIKELIHOOD METHOD
In an experiment,Nindependent measurementsxiof some quantity are made. Suppose that the random m ...

STATISTICS Differentiating with respect toμand setting the result equal to zero gives ∂lnL ∂μ = 1 TV−^1 (x−μ 1 )=0. Thus, the ML ...

31.5 MAXIMUM-LIKELIHOOD METHOD 31.5.2 Transformation invariance and bias of ML estimators An extremely useful property of ML est ...

STATISTICS 31.5.3 Efficiency of ML estimators We showed in subsection 31.3.2 that Fisher’s inequality puts a lower limit on the ...

31.5 MAXIMUM-LIKELIHOOD METHOD 31.5.4 Standard errors and confidence limits on ML estimators The ML method provides a procedure ...

STATISTICS 0 0 0. 1 0. 2 0. 3 0. 4 246810 12 14 P(τˆ|τ) τˆ Figure 31.7 The sampling distributionP(τˆ|τ) for the estimatorˆτfor t ...

31.5 MAXIMUM-LIKELIHOOD METHOD whereHdenotes our hypothesis of an assumed functional form. Now, using Bayes’ theorem(see subsect ...

STATISTICS Thus, a Bayesian statistician considers the ML estimatesaˆMLof the parameters to be the values that maximise the post ...

31.5 MAXIMUM-LIKELIHOOD METHOD 0 0 0. 1 0. 2 0. 3 0. 4 246810 12 14 L(x;τ) τ Figure 31.8 The likelihood functionL(x;τ) (normalis ...

STATISTICS By comparing this result with that given towards the end of subsection 31.5.4, we see that, as we might expect, the B ...

31.5 MAXIMUM-LIKELIHOOD METHOD By substitutingσ=1/u(so thatdσ=−du/u^2 ) and integrating by parts either (N−2)/ 2 or (N−3)/2 time ...

STATISTICS and the matrixV−^1 is given by ( V−^1 ) ij=− ∂^2 lnL ∂ai∂aj ∣ ∣ ∣ ∣ a=ˆa . Moreover, in the limit of largeN, this mat ...