Question Two - 15 Marks

Describe and illstrate a concept from the Bootstrap Section in the notes using one of the following distributions the Beta, Beta prime, Log-Laplace, Weibull, or Kumaraswamy.

There is a 1 to 2 page limit.

Rubric

Criteria

Descriptor

Marks

Data/Model

Description and Reproducibility

/2

Format

Clarity

/3

Demostration

Depth, Relevance, Terminology Used

/5

Concept

Diculty, Correctness and Relevance

/5

Question Three - 8 Marks

In the context of logistic regression describe how to perform the parametric bootstrap.

There is a 1 to 2 pages limit.

Your answer could include an example but it is not a requirement.

Rubric

Criteria

Descriptor

Marks

Format

Clarity, Organization

/2

Writing

Grammar & Punctuation, Clarity

/2

Content

Correctness & Relevant Terminology Used

/4

2

Question Four - 6 Marks

In your words, provide a context and then illstrate each of the three mechanisms that lead to missing data.

See the "Missingness Example" in section "3.1 - Missing Data".

One page limit.

Rubric

Criteria

Descriptor

Marks

Context

Creativity and Relevance

/2

Writing

Clarity

/2

Content

Relevant Terminology Used

/2

Question Five - 20 Marks

In this question you will derive and implement an EM algorithm to fit a multivariate-normal distribution to Ozone (z) and Wind (x) from the air quality dataset. Make sure you define any notation that you introduce.

data(airquality)

head(airquality)

##Ozone Solar.R Wind Temp Month Day

##

1

41

190

7.4

67

5

1

##

2

36

118

8.0

72

5

2

##

3

12

149

12.6

74

5

3

##

4

18

313

11.5

62

5

4

## 5

NA

NA 14.3

56

5

5

## 6

28

NA 14.9

66

5

6

a)[1 Mark] What is the joint distribution of the missing data and the observed data?

b)[2 Marks] State the conditional distribution of the missing data given the observed data.

c)[2 Marks] State the complete data likelihood

d)[4 Marks] E-step: Summarize the derivation of the expected complete data log-likelihood.

e)[4 Marks] M-step: Summarize the derivation of the updates for the parameters.

f)[6 Marks] Implement the above EM in R for the air quality dataset. Use starting values based on the complete cases. Give the MLE and plot the observed log-likelihood evaluated at each iteration.

g)[1 Mark] Plot the imputed dataset.