---
title: "Maximum likelihood estimation and resampling techniques"
output:
  pdf_document: default
---


# Maximum likelihood estimation in linear regression

1. Write up the simple linear regression model with one explanatory variable. 

2. Write the log likelihood for `n` observations.

3. Show that the maximum likelihood estimator (MLE) is 
   indeed the estimator minimising the mean squared error (MSE).
   
4. Demonstrate the two estimators numerically on the `trees` dataset 
   with `Volume` as response variable and `Girth` as explanatory variable. 
   Use `optim()` for both MLE and MSE, and compare to the output of `summary(lm(...))`.

# Overfitting and cross validation

Use the `trees` dataset (`Volume` explained by `Girth`) in these exercises.

1. Use both non-parametric and parametric bootstrap 
   to estimate standard errors of the parameter estimates in the 
   simple linear regression model. Compare to those obtained from `summary()`.
   
2. Use cross validation to investigate polynomial and spline regression for the 
   `trees` dataset (`Volume` explained by `Girth`).