Page 264 - Applied Probability

11. Radiation Hybrid Mapping
m−1
m−1

=
(1 − θ i,i+1 )].
θ i,i+1 − [1 −
i=1
i=1
2. In a haploid radiation hybrid experiment with m loci, let X ij be the
observation at locus j of clone i. Assuming independence of the 251
clones and no missing data, show that
n m
1 1
= (11.23)
ˆ r n 1 {X ij =1}
n m
i=1 j=1
is a strongly consistent sequence of estimators of r. Let a jk be the
probability Pr(X ij = X ik )= 2r(1 − r)θ jk . Show that
n
1
=
ˆ a njk 1 {X ij =X ik }
n
i=1
is a strongly consistent sequence of estimators of a jk . Finally, prove
that
ˆ ˆ a njk
θ njk = (11.24)
2ˆ r n (1 − ˆ r n )
is a strongly consistent sequence of estimators of θ jk , the breakage
probability between loci j and k.
3. In addition to the assumptions of the last problem, suppose that there
are just m = 2 loci. Prove that the estimates (11.23) and (11.24)
of r and θ reduce to the maximum likelihood estimates described in
Sections 11.4 and 11.7 when inequality (11.12) is satisfied empirically.
ˆ
4. Let θ njk be any strongly consistent sequence of estimators of θ jk for
polyploid radiation hybrid data. Prove that minimizing the estimated
total distance
m−1
ˆ

D(σ) = − ln[1 − θ n,σ(i),σ(i+1) ]
i=1
between the first and last loci of an order σ provides a strongly con-
sistent criterion for choosing the true order.
5. Let L(γ) be the loglikelihood for the data X on a single, haploid clone
t
fully typed at m loci. Here γ =(θ 1 ,... ,θ m−1 ,r) is the parameter
vector. The expected information matrix J has entries

∂
2
=E − L(γ) .
J γ i γ j
∂γ i ∂γ j