Consider the data from the London Times [15] for the years 1910 to 1912 reproduced in Table 2.6. The two columns labeled “Deaths i”

refer to the number of deaths of women 80 years and older reported by day. The columns labeled “Frequency ni” refer to the number of days with i deaths. A Poisson distribution gives a poor fit to these data, possibly because of different patterns of deaths in winter and summer. A mixture of two Poissons provides a much better fit. Under the Poisson admixture model, the likelihood of the observed data is



9 i=0

αe−µ1 µi 1

i! + (1 − α)e−µ2 µi 2

i!

ni

, where α is the admixture parameter and µ1 and µ2 are the means of the two Poisson distributions.

From the initial estimates α0 = .3, µ01 = 1.0, and µ02 = 2.5, compute via the EM algorithm the maximum likelihood estimates ˆα = .3599, µˆ1 = 1.2561, and ˆµ2 = 2.6634. Note how slowly the EM algorithm converges in this example.