MAST30027 Modern Applied Statistics 2018
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Semester 2 Assessment, 2018
MAST30027 Modern Applied Statistics
Question 1 (10 marks) The Poisson distribution has the probability density function (pdf)
λze −入
,
(a) Show that the Poisson distribution is an exponential family, identifying the parameters θ
and φ as well as the functions b(θ) and a(φ).
(b) Obtain the canonical link. Show your work.
(c) Obtain the variance function. Show your work.
Question 2 (8 marks) Let X1 , . . . , Xn be independent random variables from a Poisson distri- bution with the pdf given in Question 1.
(a) What is the log-likelihood for this example?
(b) What is the Fisher information for this example?
(c) Find the MLE of λ and its asymptotic distribution.
Question 3 (18 marks) The Fiji Fertility Survey gives data on the number of children ever born to married women of the Indian race classified by duration since their first marriage (grouped in six categories), type of place of residence (Suva, other urban and rural), and educational level (classified in four categories: none, lower primary, upper primary, and secondary or higher). The dataset has 70 rows representing 70 groups of families. Each row has 5 entries giving:
· duration: marriage duration of mothers in the group (years),
· residence: residence of families in the group (Suva, urban, rural),
· education: education of mothers in the group (none, lower primary, upper primary, sec-
ondary+),
· nChildren: number of children ever born in the group (e.g. 4), and
· nMother: number of mothers in the group (e.g. 8).
Examine the R code and output below, and then answer the questions that follow. First, we can summarise data as a table as follows.
> dat <- read .table("examData .dat", header=TRUE)
> dat$duration <- factor(dat$duration, levels=c("0-4","5-9","10-14","15-19","20-24", "25-29"), ordered=TRUE)
> dat$residence <- factor(dat$residence, levels=c("Suva", "urban", "rural")) > dat$education <- factor(dat$education, levels=c("none", "lower", "upper", "sec+")) > ftable(xtabs(cbind(nChildren,nMother) ~ duration + residence + education, dat))
nChildren nMother
duration residence education
|
0-4 |
Suva |
none lower upper sec+ |
4 24 38 37 |
8 21 42 51 |
|
|
urban |
none lower upper sec+ |
14 23 41 35 |
12 27 39 51 |
|
|
rural |
none lower upper sec+ |
60 98 104 35 |
62 102 107 47 |
|
5-9 |
Suva |
none lower upper sec+ |
31 80 49 38 |
10 30 24 22 |
|
|
urban |
none lower upper sec+ |
59 98 118 48 |
13 37 44 21 |
|
|
rural |
none lower upper sec+ |
171 317 200 47 |
70 117 81 21 |
|
10-14 |
Suva |
none |
49 |
12 |
|
|
|
lower upper sec+ |
99 58 24 |
27 20 12 |
|
|
urban |
none lower upper sec+ |
75 143 105 50 |
18 43 29 15 |
|
|
rural |
none lower upper sec+ |
364 546 197 30 |
88 132 50 9 |
|
15-19 |
Suva |
none lower upper sec+ |
59 153 41 11 |
14 31 13 4 |
|
|
urban |
none lower upper sec+ |
108 225 92 19 |
23 42 20 5 |
|
|
rural |
none lower upper sec+ |
577 481 135 2 |
114 86 30 1 |
|
|
||||
|
47 |
2022-09-06