Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Mixed Poisson Models for Claim Numbers 35
1.5 Statistical Inference for Discrete Distributions
1.5.1 Maximum Likelihood Estimators
Maximum likelihood is a method of estimation and inference for parametric models. The
maximum likelihood estimator is the value of the parameter (or parameter vector) that makes
the observed data most likely to have occurred given the data generating process assumed
to have produced the variable of interest.
The likelihood of a sample of observations is defined as the joint density of the data, with
the parameters taken as variable and the data as fixed (multiplied by any arbitrary constant
or function of the data but not of the parameters). Specifically, let N 1 N 2 N n be a set
of independent and identically distributed outcomes with probability mass function p·
where is a vector of parameters. The likelihood function is the probability of observing
the data N 1 = k 1 N n = k n , that is,
=
n∏
pk i
i=1
The key idea for estimation in likelihood problems is that the most reasonable estimate is
the value of the parameter vector that would make the observed data most likely to occur.
The implicit assumption is of course that the data at hand are reliable. More formally we
seek a value of that maximizes . The maximum likelihood estimator of is the
random variable ̂ for which the likelihood is maximum, that is
̂ ≥ for all
It is usually simpler mathematically to find the maximum of the logarithm of the likelihood
L = ln =
n∑
ln pk i
rather than the likelihood itself. The function L is usually referred to as the log-likelihood.
Because the logarithm is a monotonic transformation, the log-likelihood will be maximized
at the same parameter value that maximizes the likelihood (although the shape of the loglikelihood
is different from that of the likelihood).
When working with counting variables, it is often easier to use the observed frequencies
i=1
f k = #observations equal to k k = 0 1 2 (1.46)
In other words, f k is the number of times that the value k has been observed in the sample.
Denoting the largest observation as
the log-likelihood becomes
k max = max k i
i=1n
k∑
max
L = f k ln pk
k=0