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

40 Actuarial Modelling of Claim Counts
1.5.3 Computing the Maximum Likelihood Estimators with the
Newton–Raphson Algorithm
Calculation of the maximum likelihood estimators often requires iterative procedures. Let
H denote the Hessian (or matrix of second derivatives) of the log-likelihood function, with
elements
H ij =
2
i j
L
=
i
U j (1.50)
k∑
max
2
=− f k ln pk
i j
k=0
for i j = 1dim. For ⋆ close enough to ̂, a first-order Taylor expansion gives
0 = Û ≈ U ⋆ + H ⋆
(̂ −
⋆)
yielding
̂ ≈ ⋆ − H −1 ⋆ U ⋆
Starting from an appropriate initial value 0 , the Newton–Raphson algorithm is based on
the recurrence relation
̂r+1 = ̂
)
r −1
− H
(̂r U
(̂r) (1.51)
This result provides the basis for an iterative approach for computing the maximum
likelihood estimator known as the Newton–Raphson technique. Given a trial value, we use
(1.51) to obtain an improved estimate and repeat the process until the elements of the vector
of first derivatives are sufficiently close to zero.
This procedure tends to converge quickly if the log-likelihood is well-behaved in a
neighbourhood of the maximum and if the starting value is reasonably close to the maximum
likelihood estimator.
Remark 1.2 (Fisher Scoring) Noting that =−EH, an alternative procedure is
to replace minus the Hessian by its expected value, i.e. minus the Fisher information matrix.
The resulting procedure takes as an improved estimate
̂r+1 ≈ ̂
)
r −1
+
(̂r U
(̂r)
and is known as Fisher Scoring.