Titel_SS06

By equating the k sample moments to the k equations for the moments for the random
variable X a set of k equations with the k unknown distribution parameters are obtained,
the solution of which gives the point estimates of the distribution parameters.

The Method of Maximum Likelihood

This method may be somewhat more difficult to use than the method of moments but has a
number of very attractive properties, which makes this method especially applicable in
engineering risk and reliability analysis.

The principle of the method is that the parameters of the distribution function are fitted such
that the probability (likelihood) of the observed random sample is maximised.

Let the random variable of interest X have a probability density function fX( ; )x where
( , ,.., ) 12
T
 :: :k are the distribution parameters to be estimated.

If the random sample that will be used to estimate the distribution parameters
( , ,.., ) 12
T
 :: :k is collected in the vector ˆˆˆ ˆ( ,12,,.., )
T
x x xxn the likelihood L(xˆ)of the
observed random sample is defined as:

1

()ˆ (

n

Xi

i

Lf



x. xˆ ) (2.90)

The maximum likelihood point estimates of the parameters ( , ,.., ):: : 12 k T may now be
obtained by solving the following optimisation problem:

min(: L(xˆ)) (2.91)

Instead of the likelihood function it is advantageous to consider the log-likelihood l(xˆ) i.e.:

1

( ˆ) log( ( ))

n

Xi

i

lf



x xˆ  (2.92)

One of the most attractive properties of the maximum likelihood method is that when the
number of samples i.e. n is sufficiently large the distribution of the parameter estimates
converges towards a Normal distribution with mean values equal to the point estimates, i.e.:

( , ,.., 12 )
T
 :: :n (2.93)

The covariance matrix C;;for the point estimates may readily be obtained by:

 ^1

CH (2.94)

where is the Fischer information matrix with components determined by the second order
partial derivatives of the log-likelihood function taken in the maximum, i.e.:

H

*

(^2) ()ˆ
ij
ij
l
H
:: 





 

x

(2.95)

Example 2.4 – Parameter estimation

Consider again the experimental results of the concrete cube compressive strengths given in