统计计算考试题目.docVIP

统计计算 题目4.2. 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 4.2 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 areand.At 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 of respondents in group reporting risky encounters and and correspond to the zero, typical,and promiscuous groups, respectively. Thus,andfor i=1,...,16.Let Define for i=0,...,16. These correspond to probabilities that respondents with i risky encounters belong to the various groups. Show that the EM algorithm provides the following updates: b. Estimate the parameters of the model, using the observed data. c. Estimate the standard errors and pairwise correlations of your parameter estimates, using any available method. 解： （1） 则有 其中表示不同组，表示危险性行为。即得证； 下证EM算法更新推导过程： 计算E步的Q函数： (3)计算M步的Q函数求极值过程： （i）由于要使Q函数达到最大，同时参数必须满足，