.
Recall the peppered moth analysis introduced in Example 4.2. In the field, it is quite difficult to distinguish the
insularia
and
typica
phenotypes due to variations in wing color and mottle. In addition to the 622 moths mentioned in the example, suppose the sample collected by the researchers actually included
nU = 578 more moths that were known to be
insularia
or
typical
but whose exact phenotypes could not be determined.
a.
Derive the EM algorithm for maximum likelihood estimation of
pC,
pI, and
pI for this modified problem having observed data
nC,
nI,
nT, and
nU as given above.
b.
Apply the algorithm to find the MLEs.
c.
Estimate the standard errors and pairwise correlations for ˆpC, ˆpI, and ˆpI using the SEM algorithm.
d.
Estimate the standard errors and pairwise correlations for ˆpC, ˆpI, and ˆpI by bootstrapping.
e.
Implement the EM gradient algorithm for these data. Experiment with step halving to ensure ascent and with other step scalings that may speed convergence.
f.
Implement Aitken accelerated EM for these data. Use step halving.
g.
Implement quasi-Newton EM for these data. Compare performance with and without step halving.
h.
Compare the effectiveness and efficiency of the standard EMalgorithm and the three variants in (e), (f), and (g). Use step halving to ensure ascent with the three variants. Base your comparison on a variety of starting points. Create a graph analogous.