MAST30027 Modern Applied Statistics 2017
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Semester 2 Assessment, 2017
MAST30027 Modern Applied Statistics
Question 1 (10 marks) Let X1 , . . . , Xn be independent samples from a Pareto distribution Par(1, κ) with pdf f (x|κ) = κ(1 + x)-K-1 , x > 0.
(a) What is the log-likelihood for this example?
(b) What is the Fisher information for this example?
(c) Find the MLE of κ and its asymptotic distribution.
Question 2 (8 marks) The gamma distribution with shape parameter ν > 0 and rate parameter λ > 0 has the probability density function
f (x; ν, λ) = xv -1 e-入北 , x ≥ 0.
The mean is 入(v) and the variance is 入2(v) .
(a) Show that the gamma distribution is an exponential family.
(b) Obtain the canonical link. Show your work.
(c) Obtain the variance function. Show your work.
Question 3 (22 marks) The wavesolder data set has 48 observations of y, the number of defects, and five predictor variables, prebake, flux, speed, cooling, and temp. The data is taken from Condra, Lloyd, Reliability Improvement with Design of Experiment, CRC Press, 2001.
Examine the R code and output below, and then answer the questions that follow.
Firstly we need to combine the three replicates into a single data set, and then have a look at the data.
> rm(list=ls())
> library(faraway)
> data(wavesolder)
> y <- c(wavesolder$y1, wavesolder$y2, wavesolder$y3)
> wavesolder <- rbind(wavesolder, wavesolder, wavesolder)
> wavesolder <- wavesolder[- (1:3)]
> wavesolder$y <- y
> par(mfrow=c(2,3), mar=c(4,4,1,1))
> plot(y ~ prebake, wavesolder)
> plot(y ~ flux, wavesolder)
> plot(y ~ speed, wavesolder)
> plot(y ~ cooling, wavesolder)
> plot(y ~ temp, wavesolder)
1 2
prebake
1 2
cooling
2
flux
1
temp
|
1 2
speed
> modelA <- glm(y ~ prebake + flux + speed + temp,
+ family=poisson, data=wavesolder)
> summary(modelA)
Call:
glm(formula = y ~ prebake + flux + speed + temp, family = poisson,
data = wavesolder)
Deviance Residuals:
Min 1Q Median 3Q Max
-8 .0503 -1 .9044 -0 .5489 1 .8995 12 .5918
Coefficients:
Estimate Std . Error z value Pr(>|z|)
(Intercept) 2 .80541 0 .06948 40 .38 <2e-16 ***
prebake2 0 .67287 0 .05374 12 .52 <2e-16 ***
flux2 -0 .52878 0 .05262 -10 .05 <2e-16 ***
speed2 1 .23048 0 .06076 20 .25 <2e-16 ***
temp2 -0 .69315 0 .05392 -12 .86 <2e-16 ***
---
Signif . codes: 0 *** 0 .001 ** 0 .01 * 0 .05 . 0 .1 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 1450 .52 on 47 degrees of freedom
Residual deviance: 513 .75 on 43 degrees of freedom
AIC: 754 .49
Number of Fisher Scoring iterations: 5
> modelB <- glm(y ~ prebake + flux + speed + cooling + temp,
+ family=poisson, data=wavesolder)
> summary(modelB)
Call:
glm(formula = y ~ prebake + flux + speed + cooling + temp, family = poisson, data = wavesolder)
Deviance Residuals:
Min 1Q Median 3Q Max
-7 .7230 -2 .0135 -0 .2761 1 .5991 13 .2687
Coefficients:
Estimate Std . Error z value Pr(>|z|)
(Intercept) 2 .88947 0 .07576 38 .142 < 2e-16 ***
prebake2 0 .64801 0 .05450 11 .891 < 2e-16 ***
flux2 -0 .52878 0 .05262 -10 .049 < 2e-16 ***
speed2 1 .21614 0 .06098 19 .943 < 2e-16 ***
cooling2 -0 .14222 0 .05279 -2 .694 0 .00706 **
temp2 -0 .66902 0 .05463 -12 .247 < 2e-16 ***
---
Signif . codes: 0 *** 0 .001 ** 0 .01 * 0 .05 . 0 .1 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 1450 .52 on 47 degrees of freedom
Residual deviance: 506 .48 on 42 degrees of freedom
AIC: 749 .22
Number of Fisher Scoring iterations: 5
> anova(modelA, modelB, test="Chisq")
Analysis of Deviance Table
Model 1: y ~ prebake + flux + speed + temp
Model 2: y ~ prebake + flux + speed + cooling + temp
Resid . Df Resid . Dev Df Deviance Pr(>Chi)
1 43 513.75
2 42 506 .48 1 7 .2729 0 .007 **
---
Signif . codes: 0 *** 0 .001 ** 0 .01 * 0 .05 . 0 .1 1
> (phi <- sum(residuals(modelB, type="pearson")^2/modelB$df .residual))
[1] 13 .93209
> modelC <- glm(y ~ prebake + flux + speed + temp, family=quasipoisson, data=wavesolder) > modelD <- glm(y ~ prebake + flux + speed + cooling + temp, family=quasipoisson, + data=wavesolder)
> anova(modelC, modelD, test="F")
Analysis of Deviance Table
Model 1: y ~ prebake + flux + speed + temp
Model 2: y ~ prebake + flux + speed + cooling + temp
Resid . Df Resid . Dev Df Deviance F Pr(>F)
1 43 513.75
2 42 506.48 1 7.2729 0.522 0.474
(a) For modelA, assuming Poisson responses, what is the log-likelihood of the fitted model, and the log-likelihood of the full (saturated) model?
(b) Assuming Poisson responses, which is better, modelA or modelB? Give two (quantitative) reasons for your answer.
(c) Give an estimate for the expected number of defects, for prebake = 1, flux = 2, speed = 2, cooling = 1, temp = 1, under modelB.
(d) Give a (quantitative) reason why modelB may suffer from overdispersion. (e) Briefly the difference between a Poisson and quasi-Poisson model.
(f) Give the std. error for cooling2 in the case where we allow for overdispersion. (g) Allowing for overdispersion, do you prefer modelC or modelD, and why?
(h) What formula has been used to calculate the F statistic in the second analysis of deviance? What are the degrees of freedom for the F statistic?
Question 4 (14 marks) The following three-way table refers to results of a case-control study about effects of cigarette smoking and coffee drinking on myocardial infarction (MI) or heart attack for a sample of men under 55 years of age.
Cigarettes per Day Cups Coffee 0 1-24 25-34 ≥ 35 per Day Cases Controls Cases Controls Cases Controls Cases Controls |
||||||
0 |
66 |
123 |
30 52 |
15 12 |
36 |
13 |
1-2 |
141 |
179 |
59 45 |
53 22 |
69 |
25 |
3-4 |
113 |
106 |
63 65 |
55 16 |
119 |
30 |
≥ 5 |
129 |
80 |
102 58 |
118 44 |
373 |
85 |
Eight log-linear models with Poisson error have been fitted, with the residual deviances
given in the following table.
Model
Residual
deviance
coffee + cigar + MI coffee + cigar*MI cigar + coffee*MI MI + coffee*cigar coffee*cigar + coffee*MI coffee*cigar + cigar*MI coffee*MI + cigar*MI coffee*cigar + coffee*MI |
+ |
cigar*MI |
607.25 394.43 484.70 271.40 148.81 58.55 271.88 11.17 |
You will find the following chi-squared percentage points useful for problems (c) and (d).
> qchisq(0 .95, df=5:10)
[1] 11 .07050 12 .59159 14 .06714 > qchisq(0 .95, df=11:15)
[1] 19 .67514 21 .02607 22 .36203 > qchisq(0 .95, df=16:20)
[1] 26 .29623 27 .58711 28 .86930
15 .50731 16 .91898 18 .30704
23 .68479 24 .99579
30 .14353 31 .41043
(a) What are the residual degrees of freedom (d.f.) for each of the three models: coffee +
cigar + MI, cigar + coffee*MI, and coffee*cigar + cigar*MI?
(b) Give an interpretation for each of the following models.
(i) coffee + cigar + MI
(ii) MI + coffee*cigar
(iii) coffee*cigar + coffee*MI
(c) Test the hypothesis that there is no association between coffee and MI when cigar level is given (at the 95% level).
(d) Test the hypothesis that the association between MI and cigar is the same for all coffee levels. That is, test that there is no three-way interaction (at the 95% level).
Question 5 (14 marks)
(a) Here is some R code for simulating a discrete random variable Y . What is the probability mass function (pmf) of Y , i.e., P (Y = y) for y ≥ 2?
Y .sim <- function() {
U <- runif(1)
Y <- 2
Y <- Y + 1
}
return(Y)
}
(b) Let a random number X be generated by the following algorithm:
1o Generate U from Unif(0, 1) and V from Unif(0, 1) independently. 2o If U + V < 1, then X = 1 _ U ; otherwise, go to 1o .
What is the probability density function of X, i.e., f(x) for 0 < x < 1?
Question 6 (18 marks) Consider a random sample X from a Bernoulli distribution with pdf f(x|θ) = θz (1 _ θ)1-z ; x = 0, 1. Let the prior distribution for θ be Uniform(0, 1), i.e., p(θ) = 1 for 0 < θ < 1. We use the squared error loss function.
(a) Find the posterior distribution of θ .
(b) Find the Bayes estimator of θ .
(c) Find the risk of the Bayes estimator of θ .
(d) Find the Bayes risk of the Bayes estimator of θ .
Question 7 (20 marks) We assume that x1 , . . . , xn1 and y1 , . . . , yn2 are independently normally distributed as follows.
xi ~ N (µ1 , σ 2 ), i = 1, . . . , n1
yi ~ N (µ2 , σ 2 ), i = 1, . . . , n2
We impose the following prior distributions on µ 1 , µ2 and τ = 1/σ2 .
p(µ1 ) x 1
p(µ2 ) x 1
p(τ ) x 1/τ
(a) Among µ 1 , µ2 and τ , which parameter(s) have an improper prior?
µ 1 |x, y, µ2 , τ
µ2 |x, y, µ1 , τ
τ |x, y, µ1 , µ2
Hence give the (conditional) distributions of these variables, including their parameters.
(c) Briefly describe a Gibbs sampling scheme for sampling (µ1 , µ2 , τ ).
(d) How would you check for convergence of the Gibbs sampler? Provide both informal (graphical/visual checks) and formal methods. Also, briefly provide details of methods.
Question 8 (14 marks) Briefly describe an algorithm to simulate samples from the posterior predictive distribution:
p(y˜|y) = p(y˜|θ)p(θ|y)dθ .
How would you estimate the mean of the posterior predictive distribution using the simulated samples?
2022-09-06