Page 67 - Applied Probability

3. Newton’s Method and Scoring
50
where 1 {i=j} is the indicator function of the event {i = j}, and where ψ (s)
2
d
is the trigammaTrigamma function function
2 ln Γ(s) [9]. The digamma
ds
and trigamma functions appearing in the expressions (3.15) and (3.16)
should not be viewed as a barrier to computation since good software for
evaluating these transcendental functions does exist [1, 19].
Equation (3.16) for a single population can be summarized in matrix
form by
t
2
−d L(α)= D − c11 , (3.17)
where D is a diagonal matrix with ith diagonal entry

d i = ψ (α i ) − ψ (n i + α i ),
c is the constant ψ (α . ) − ψ (2n + α .), and 1 is a column vector of all 1’s.

Because the trigamma function is decreasing [9], d i > 0 when n i > 0. For
the same reason, c> 0. Since the representation (3.17) is preserved under
finite sums, it holds, in fact, for the entire sample.
The observed information matrix (3.17) is the sum of a diagonal matrix,
which is trivial to invert, plus a symmetric, rank-one perturbation. From
our discussion of Davidon’s symmetric, rank-one update, we know how to
correct the observed information when it fails to be positive definite. A
safeguarded Newton’s method can be successfully implemented using the
2
Sherman-Morrison formula to invert −d L(α) or its substitute.
TABLE 3.4. Allele Counts in Four Subpopulations
Allele White Black Chicano Asian
5 2 0 0 0
6 84 50 80 16
7 59 137 128 40
8 41 78 26 8
9 53 54 55 68
10 131 51 95 14
11 2 0 0 7
12 0 0 0 1
Total 2n 372 370 384 154

Example 3.7.1 Houston Data on the HUMTH01 Locus
The data of Edwards et al. [6] on the eight alleles of the HUMTH01
locus on chromosome 11 are reproduced in Table 3.4. The allele names for
this tandem repeat locus refer to numbers of repeat units. From the four
separate Houston subpopulations of whites, blacks, Chicanos, and Asians,
the eight α’s are estimated by maximum likelihood to be .11, 4.63, 7.33,