Applied statistics (DVML, MATØK)

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Likelihood and maximum likelihood estimation

Literature

[WMMY] 9.14

Lecture material

This lecture as: slideshow (html), Rmarkdown (Rmd), notes (pdf).

The entire module: notes (pdf).

Exercises

1. Logistic regression: The dataset Default also contains a variable balance giving the balance on the costumer’s account.

   1. Find a logistic regression model for the probability of default as a function of balance. Compute the maximum likelihood estimates numerically.

   2. Compare with the output from glm.

2. One may also consider a multiple logistic regression model, where the probability of default depends on both balance x₁ and income x₂ as follows:
   $$p(x_1,x_2)=\frac{1}{1+e^{-(\alpha + \beta_1 x_1 + \beta_2 x_2)}}.$$
   That is, if we fix the value of x₁, then p(x₁, x₂) is a logistic function of x₂ and vice versa.

   1. Use numerical maximum likelihood estimation to estimate the parameters α, β₁, β₂ (you can follow the example from the lecture, the only difference being that theta is a vector of three parameters and the formula for computing px is different).

3. Suppose that X has a binomial distribution Binom(n, p) (see Lecture 1.2).

   1. What is the maximum likelihood estimator (MLE) of p?

   2. Plot the log likelihood function for n = 10 and X = 5.

   3. Plot the log likelihood function for n = 100 and X = 50.

   4. Plot b) and c) in same figure. How and why are they different?

   5. Use the optim() function to find the MLE of p numerically in both cases.

   6. Compare the result from e) to the result using the theoretical formula from a)

4. Finish old exercises.