Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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

N=20

L(x;τ)

τ

0

0

246810 12 1416 18 20

0.5

1

N=50

L(x;τ)

τ

Figure 31.5 Examples of the likelihood function (31.67) for samples of dif-

ferent sizeN. In each case, the true value of the parameter isτ=4andthe

sample valuesxiare indicated by the short vertical lines. For the purposes

of illustration, in each case the likelihood function is normalised so that its

maximum value is unity.

likelihood is given by

L(x;τ)=P(xi|τ)P(x 2 |τ)···P(xN|τ)

=

1

τ

exp

(

−

x 1

τ

) 1

τ

exp

(

−

x 2

τ

)

···

1

τ

exp

(

−

xN

τ

)

=

1

τN

exp

[

−

1

τ

(x 1 +x 2 +···+xN)

]

. (31.67)

which is to be considered as a function ofτ, given that the sample valuesxiare fixed.
The likelihood function (31.67) depends on just a single parameterτ.Plotsof

the likelihood function, considered as a function ofτ, are shown in figure 31.5 for

samples of different sizeN. The true value of the parameterτused to generate the

sample values was 4. In each case, the sample valuesxiare indicated by the short

vertical lines. For the purposes of illustration, the likelihood function in each

case has been scaled so that its maximum value is unity (this is, in fact, common

practice). We see that when the sample size is small, the likelihood function is

very broad. AsNincreases, however, the function becomes narrower (its width is

inversely proportional to

√

N) and tends to a Gaussian-like shape, with its peak

centred on 4, the true value ofτ. We discuss these properties of the likelihood

function in more detail in subsection 31.5.6.