Student Notes To Accompany MS4214: STATISTICAL INFERENCE

a0 b0 a ∗ b ∗ Steps Eigenvalues
1.8 74.5 1.924941 78.12213 4 all positive
2.36 72 1.924941 78.12213 5 all positive
2 70 1.924941 78.12213 5 all positive
1 2 1.924941 78.12213 15 all positive
80 1 1.924941 78.12213 387 all positive
2 100 4.292059 −34.591322 2 crashed
Table 2.6.1: Newton-Raphson estimates of the Weibull parameters.
Given initial values a0 and b0 equation (2.8) can be used to obtained updated estimates
via
�
anew
bnew
�
=
�
a0
b0
�
+
�
Iaa(a0, b0) Iab(a0, b0)
Iba(a0, b0) Ibb(a0, b0)
� −1 �
Sa(a0, b0)
Sb(a0, b0)
�
. (2.9)
Having solved for a and b we check that a maximum has been achieved by calculating
I(a ∗ , b ∗ ) and verifying that both of its eigenvalues are positive.
Suppose n = 10 and m = 8 items fail at times 17, 29, 32, 55, 61, 74, 77, 93 (in hours)
with the remaining two items surviving the 100 hour test. Table 2.6.1 shows estimates
(a ∗ , b ∗ ) obtained from equation (2.9) using various starting values (a0, b0) along with
the number of iteration steps taken until convergence using ɛ = 0.000001. Four of
the starting pairs considered converged to estimates a ∗ = 1.924941 and b ∗ = 78.12213
for a and b. The starting pairs (1, 2) and (80, 1) might misleadingly suggest that the
algorithm is robust to the choice of starting values. However, the starting pair (2, 100)
produced a negative estimate of b, and as ln(b) is required in the computation of the
score function, caused the algorithm to crash. Other starting pairs not reported failed
to yield estimates due to a non-invertible I(θ0) matrix being produced during the
running of the algorithm. The remainder of this section describes commonly applied
modifications of the Newton-Raphson method.
Initial values
The Newton-Raphson method depends on the initial guess being close to the true value.
If this requirement is not satisfied the procedure might convergence to a minimum
instead of a maximum, or just simply diverge and fail to produce any estimates at all.
Methods of finding good initial estimates depend very much on the problem at hand
and may require some ingenuity.
The distribution function of a Weibull random variable T can be linearized as
ln(− ln[1 − F (t)]) = −a ln(b) + a ln(t).
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