Let L() be the loglikelihood for the data X on a single, haploid clone fully typed at

Let L(γ) be the loglikelihood for the data X on a single, haploid clone fully typed at m loci. Here γ = (θ1,...,θm−1, r)t is the parameter vector. The expected information matrix J has entries Jγiγj = E

− ∂2

∂γi∂γj L(γ)



Show that [9, 14]
Jθiθi = r(1 − r)(2 − θi)
(1 − θir)θi(1 − θi + θir)
Jθir = Jrθi = (1 − 2r)(1 − θi)
(1 − θir)(1 − θi + θir)
Jrr = 1 r(1 − r) +
m −1 i=1 θi 1 − r r(1 − θir) +
r (1 − r)(1 − θi + θir)

.
Prove that all other entries of J are 0.

Hints: Use the factorization property of the likelihood. In the case of two loci, denote the probability Pr(X1 = i, X2 = j) by pij for brevity. Then a typical entry Jαβ

of J is given by Jαβ =

1 i=0

1 j=0 1

pij

∂pij

∂α  ∂pij

∂β 

.