Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

maximum that occurs at a stationary point (the likelihood function is then termed

unimodal). In this case, the ML estimators of the parametersai,i=1, 2 ,...,M,

may be foundwithoutevaluating the full likelihood functionL(x;a). Instead, one

simply solves theMsimultaneous equations

∂L

∂ai

∣

∣

∣

∣

a=ˆa

=0 fori=1, 2 ,...,M. (31.68)

Since lnzis a monotonically increasing function ofz(and therefore has the

same stationary points), it is often more convenient, in fact, to maximise the

log-likelihood function,lnL(x;a), with respect to theai. Thus, one may, as an

alternative, solve the equations

∂lnL

∂ai

∣

∣

∣

∣

a=aˆ

=0 fori=1, 2 ,...,M. (31.69)

Clearly, (31.68) and (31.69) will lead to the same ML estimatesaˆof the parameters.

In either case, it is, of course, prudent to check that the pointa=aˆis a local

maximum.

Find the ML estimate of the parameterτin the previous example, in terms of the measured

valuesxi,i=1, 2 ,...,N.

From (31.67), the log-likelihood function in this case is given by

lnL(x;τ)=

∑N

i=1

ln

(

1

τ

e−xi/τ

)

=−

∑N

i=1

(

lnτ+

xi

τ

)

. (31.70)

Differentiating with respect to the parameterτand setting the result equal to zero, we find

∂lnL

∂τ

=−

∑N

i=1

(

1

τ

−

xi

τ^2

)

=0.

Thus the ML estimate of the parameterτis given by

τˆ=

1

N

∑N

i=1

xi, (31.71)

which is simply the sample mean of theNmeasured intervals.

In the previous example we assumed that the sample valuesxiwere drawn

independently from thesameparent distribution. The ML method is more flexible

than this restriction might seem to imply, and it can equally well be applied to

the common case in which the samplesxiare independent but each is drawn

from adifferentdistribution.