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