統(tǒng)計計算考試題目

1、統(tǒng)計計算題目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

2、=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 respond

3、ent 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 belong

4、s 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 , w

5、here 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 Definefor i=0,.,16. These corr

6、espond to probabilities that respondents with i risky encounters belong to the various groups.a. 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 esti

7、mates, using any available method.解：（1）則有其中表示不同組，表示危險性行為。即得證；（2） 下證EM算法更新推導(dǎo)過程：計算E步的Q函數(shù)：(3)計算M步的Q函數(shù)求極值過程：（i）由于要使Q函數(shù)達到最大，同時參數(shù)必須滿足，運用拉格朗日乘法可得從而有；(ii）由于要使Q函數(shù)達到最大，即對求偏導(dǎo)。 從而有； 從而有；即得證。算法：（1）首先將混合正態(tài)模型的參數(shù)初始化為；（2）E步：通過混合正態(tài)分布進行隨機模擬得到n個樣本，計算完全數(shù)據(jù)對數(shù)似然關(guān)于數(shù)據(jù)的期望值，對數(shù)似然函數(shù)的期望（3）M步：最優(yōu)化期望值，即通過迭代找到的最大值；即題目7.2 Simulating f

8、rom the mixture distribution in Equation (7.6) is straightforward see part (a) of Problem 7.1. However, using the MetropolisHastings algorithm to simulate realizations from this distribution is useful for exploring the role of the proposal distribution.a. Implement a MetropolisHastings algorithm to

9、simulate from Equation (7.6) with ,usingas the proposal distribution. For each of three starting values,x(0) =0,7,and 15, run the chain for 10,000 iterations. Plot the sample path of the output from each chain. If only one of the sample paths was available, what would you conclude about the chain? F

10、or each of the simulations, create a histogram of the realizations with the true density superimposed on the histogram. Based on your output from all three chains, what can you say about the behavior of the chain?b. Now change the proposal distribution to improve the convergence properties of the ch

11、ain. Using the new proposal distribution, repeat part (a).算法：1. 從兩個正態(tài)總體里分別以0.7和0.3的概率產(chǎn)生100個隨機模擬樣本2.選取一個建議分布，從建議分布中抽取一個候選值。3.計算Metropolis-Hastings比率（通常實際中用貝葉斯推斷得到的一個比率。4.以等于R的概率接受，如果接受,則,如果沒有接受,則。5.增加t,重復(fù)上述過程，直到收斂。題目7.5 A clinical trial was conducted to determine whether a hormone treatment benefits

12、women who were treated previously for breast cancer. Each subject entered the clinical trial when she had a recurrence. She was then treated by irradiation and assigned to either a hormone therapy group or a control group. The observation of interest is the time until a second recurrence, which may

13、be assumed to follow an exponential distribution with parameter(hormone therapy group) or(control group). Many of the women did not have a second recurrence before the clinical trial was concluded, so that their recurrence times are censored. In Table 7.2, a censoring time M means that a woman was o

14、bserved for M months and did not have a recurrence during that time period, so that her recurrence time is known to exceed M months. For example, 15 women who received the hormone treatment suffered recurrences, and the total of their recurrence times is 280 months. Letbe the data for theth person i

15、n the hormone group, whereis the time andequals 1 ifis a recurrence time and 0 if a censored time. The data for the control group can be written similarly. The likelihood is then. Youve been hired by the drug company to analyze their data. They want to know if the hormone treatment works, so the tas

16、k is to find the marginal posterior distribution ofusing the Gibbs sampler. In a Bayesian analysis of these data, use the conjugate prior Physicians who have worked extensively with this hormone treatment have indicatedthat reasonable values for the hyperparameters are (a, b, c, d) = (3, 1, 60, 120)

17、.a. Summarize and plot the data as appropriate.b. Derive the conditional distributions necessary to implement the Gibbs sampler.c. Program and run your Gibbs sampler. Use a suite of convergence diagnostics to evaluate the convergence and mixing of your sampler. Interpret the diagnostics.d.Compute su

18、mmary statistics of the estimated joint posterior distribution, including marginal means, standard deviations, and 95% probability intervals for each parameter. Make a table of these results.e. Create a graph which shows the prior and estimated posterior distribution forsuperimposed on the same scal

19、e.f. Interpret your results for the drug company. Specifically, what does your estimate ofmean for the clinical trial? Are the recurrence times for the hormone group significantly different from those for the control group?g. A common criticism of Bayesian analyses is that the results are highly dependent on the priors. Investigate this issue by repeating your Gibbs sampler for values of the hyperparameters that are half and double the original hyperparameter values.Provide a table of summary statistics to compare your resul