统计计算考试题目.doc

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统计计算
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Epidemiologists are interested in studying the sexual behavior of individuals at risk for HIV infection. Suppose 1500 gay men were surveyed and each was asked how many risky sexual encounters he had in the previous 30 days. Letdenote the number of respondents reporting i encounters, for i=1,...,16. Table summarizes the responses.
These data are poorly fitted by a Poisson model. It is more realistic to assume that the respondents comprise three groups. First, there is a group of people who,for whatever reason, report zero risky encounters even if this is not true. Suppose a respondent has probabilityof belonging to this group.
With probability, a respondent belongs to a second group representing typical behavior. Such people respond truthfully, and their numbers of risky encounters are assumed to follow a Poisson() distribution.
Finally, with probability , a respondent belongs to a high-risk group. Such people respond truthfully, and their numbers of risky encounters are assumed to follow a Poisson() distribution.
The parameters in the model theth iteration of EM, we useto denote the current parameter values. The likelihood of the observed data is given by
where
for i=1,...,16.
The observed data are. The complete data may be construed to be , and ,where ,denotes the number o