## Description

1. Suppose (�#, �%, … , �’) is a random sample from a Bernoulli distribution where

�~����(�) with probability mass function, pmf of Y

�(�1|�) = �45(1 − �)#845; �1 ∈ {0,1; 1 = �������}

i. Find maximum likelihood estimate of �. (5 marks)

ii. In five independent Bernoulli trials from above Bernoulli process, three successes

and two failures were observed. Calculate the maximum likelihood estimates of �

in this situation? (5 marks)

iii. Plot the log likelihood function for the data in part ii by performing a grid search

over a set of possible values of � parameter. Add a vertical line to the plot at the

value of � that maximizes the log-likelihood. (3 marks)

2. Consider the case of simple logistic regression where � is the binary dependent variable

and � is the predictor variable with sample data (�#, �#), (�%, �%), … . , (�’, �’) . Binary

outcomes are modeled as Bernoulli trials as in Question 1 above. Here

�(�1|�1) = �1

45(1 − �1)#845; �1 ∈ {0,1; 1 = ���}

�1 = �EFGEHI5

1 + �EFGEHI5

I. Derive the log-likelihood function, �(�) where � = (�M, �#). (3 marks)

II. Write a function, ll(), to calculate the log likelihood. (3 marks)

III. Using Default data in the book, ISLR, fit a logistic regression model to predict

default given balance as a predictor using glm(). (3 marks)

IV. Use optim() function together with your ll() and initial parameter estimates as

zero to calculate maximum likelihood estimates of regression coefficients in part

iii. (3 marks)

V. Comment on the maximum likelihood estimates obtained using your work and

using glm() in part iii. (1 mark)

VI. Calculate the standard errors of your estimates. Hint: Include the parameter option

‘hessian = TRUE’ in the function optim() when you call optim() in part iv. (3

marks)

VII. Comment on the standard error estimates obtained using your work and using

glm() in part iii. (1 mark)

Chapter 5, Q6 on page 199

3. We continue to consider the use of a logistic regression model to predict the probability

of default using income and balance on the Default data set. In particular, we will now

compute estimates for the standard errors of the income and balance logistic regression

coefficients in two different ways: (1) using the bootstrap, and (2) using the standard

formula for computing the standard errors in the glm() function. Do not forget to set a

random seed to 100 before beginning your analysis.

I. Using the summary() and glm() functions, determine the estimated standard errors

for the coefficients associated with income and balance in a multiple logistic

regression model that uses both predictors. (2 marks)

II. Write a function, boot.fn(), that takes as input the Default data set as well as an

index of the observations, and that outputs the coefficient estimates for income

and balance in the multiple logistic regression model. (3 marks)

III. Use the boot() function together with your boot.fn() function to estimate the

standard errors of the logistic regression coefficients for income and balance. (3

marks)

IV. Comment on the estimated standard errors obtained using the glm() function and

using your bootstrap function. (2 marks)