Mathematical Methods for Physics and Engineering - Matematica.NET

31.5 MAXIMUM-LIKELIHOOD METHOD31.5.4 Standard errors and confidence limits on ML estimatorsThe ML method provides a procedure for obtaining a particular set of estimatorsâ ML for the parameters a of the assumed population P (x|a). As for any other setof estimators, the associated standard errors, covariances and confidence intervalscan be found as described in subsections 31.3.3 and 31.3.4.◮A company measures the duration (in minutes) of the 10 intervals x i , i =1, 2,...,10,between successive telephone calls made to its switchboard to be as follows:0.43 0.24 3.03 1.93 1.16 8.65 5.33 6.06 5.62 5.22.Supposing that the sample values are drawn independently from the probability distributionP (x|τ) =(1/τ)exp(−x/τ), find the ML estimate of the mean τ and quote an estimate ofthe standard error on your result.As shown in (31.71) the (unbiased) ML estimator ˆτ in this case is simply the sample mean¯x =3.77. Also, as shown in subsection 31.5.3, ˆτ is a minimum-variance estimator withV [ˆτ] =τ 2 /N. Thus, the standard error in ˆτ is simply√ τ . (31.78)Nσˆτ =Since we do not know the true value of τ, however, we must instead quote an estimate ˆσˆτof the standard error, obtained by substituting our estimate ˆτ for τ in (31.78). Thus, wequote our final result asτ = ˆτ ±ˆτ √N=3.77 ± 1.19. (31.79)For comparison, the true value used to create the sample was τ =4.◭For the particular problem considered in the above example, it is in fact possibleto derive the full sampling distribution of the ML estimator ˆτ using characteristicfunctions, and it is given byP (ˆτ|τ) =NN ˆτ N−1 ((N − 1)! τ N exp − Nˆτ ), (31.80)τwhere N is the size of the sample. This function is plotted in figure 31.7 for thecase τ = 4 and N = 10, which pertains to the above example. Knowledge of theanalytic form of the sampling distribution allows one to place confidence limitson the estimate ˆτ obtained, as discussed in subsection 31.3.4.◮Using the sample values in the above example, obtain the 68% central confidence intervalon the value of τ.For the sample values given, our observed value of the ML estimator is ˆτ obs =3.77. Thus,from (31.28) and (31.29), the 68% central confidence interval [τ − ,τ + ] on the value of τ isfound by solving the equations∫ ˆτobs−∞∫ ∞P (ˆτ|τ + ) dˆτ =0.16,ˆτ obsP (ˆτ|τ − ) dˆτ =0.16,1263