Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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 ± 6. 7 , (31.65)

σˆy=

(

N

N− 1

) 1 / 2

sy±(Vˆ[σˆy])^1 /^2 =11. 2 ± 3. 6. (31.66)

Finally, we estimate the population correlation Corr[x, y], which we shall denote byρ.
From (31.62), we have

ρˆ=

N

N− 1

rxy=0. 60.

Under theassumptionthat the sample was drawn from a two-dimensional Gaussian
populationP(x, y), the variance of our estimator is given by (31.64). Since we do not know
the true value ofρ, we must use our estimateρˆ. Thus, we find that the standard error ∆ρ
in our estimate is given approximately by

∆ρ≈

10

9

(

1

10

)

[1−(0.60)^2 ]^2 =0. 05 .

31.5 Maximum-likelihood method

The population from which the samplex 1 ,x 2 ,...,xNis drawn is, in general,

unknown. In the previous section, we assumed that the sample values were inde-

pendent and drawn from a one-dimensional populationP(x), and we considered

basic estimators of the moments and central moments ofP(x). We didnot,how-

ever, assume a particular functional form forP(x). We now discuss the process

ofdata modelling, in which a specific form is assumed for the population.

In the most general case, it will not be known whether the sample values are

independent, and so let us consider the full joint populationP(x), wherexis the

point in theN-dimensional data space with coordinatesx 1 ,x 2 ,...,xN.Wethen

adopt thehypothesisHthat the probability distribution of the sample values has

some particular functional formL(x;a), dependent on the values of some set of

parametersai,i=1, 2 ,...,m. Thus, we have

P(x|a,H)=L(x;a),

where we make explicit the conditioning on both the assumed functional form and

on the parameter values.L(x;a) is called thelikelihood function. Hypotheses of this

type form the basis ofdata modellingandparameter estimation. One proposes a

particular model for the underlying population and then attempts to estimate from

the sample valuesx 1 ,x 2 ,...,xNthe values of the parametersadefining this model.

A company measures the duration (in minutes) of theNintervalsxi,i=1, 2 ,...,N

between successive telephone calls received by its switchboard. Suppose that the sample

valuesxiare drawn independently from the distributionP(x|τ)=(1/τ)exp(−x/τ),whereτ

is the mean interval between calls. Calculate the likelihood functionL(x;τ).

Since the sample values are independent and drawn from the stated distribution, the