Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.5 MAXIMUM-LIKELIHOOD METHOD

31.5.4 Standard errors and confidence limits on ML estimators

The ML method provides a procedure for obtaining a particular set of estimators

ˆaMLfor the parametersaof the assumed populationP(x|a). As for any other set

of estimators, the associated standard errors, covariances and confidence intervals

can be found as described in subsections 31.3.3 and 31.3.4.

A company measures the duration (in minutes) of the 10 intervalsxi,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 distribution

P(x|τ)=(1/τ)exp(−x/τ), find the ML estimate of the meanτand quote an estimate of

the standard error on your result.

As shown in (31.71) the (unbiased) ML estimatorˆτin this case is simply the sample mean
̄x=3.77. Also, as shown in subsection 31.5.3,τˆis a minimum-variance estimator with
V[ˆτ]=τ^2 /N. Thus, the standard error inτˆis simply

σˆτ=

τ

√

N

. (31.78)

Since we do not know the true value ofτ, however, we must instead quote an estimateσˆˆτ
of the standard error, obtained by substituting our estimateτˆforτin (31.78). Thus, we
quote our final result as

τ=τˆ±

τˆ

√

N

=3. 77 ± 1. 19. (31.79)

For comparison, the true value used to create the sample wasτ=4.

For the particular problem considered in the above example, it is in fact possible

to derive the full sampling distribution of the ML estimatorτˆusing characteristic

functions, and it is given by

P(τˆ|τ)=

NN

(N−1)!

τˆN−^1

τN

exp

(

−

Nˆτ

τ

)

, (31.80)

whereNis the size of the sample. This function is plotted in figure 31.7 for the

caseτ= 4 andN= 10, which pertains to the above example. Knowledge of the

analytic form of the sampling distribution allows one to placeconfidence limits

on the estimateˆτobtained, as discussed in subsection 31.3.4.

Using the sample values in the above example, obtain the68%central confidence interval

on the value ofτ.

For the sample values given, our observed value of the ML estimator isˆτobs=3.77. Thus,
from (31.28) and (31.29), the 68% central confidence interval [τ−,τ+] on the value ofτis
found by solving the equations
∫τˆobs

−∞

P(ˆτ|τ+)dˆτ=0. 16 ,

∫∞

τˆobs

P(ˆτ|τ−)dˆτ=0. 16 ,