11.9 Rank-Deficient Case (11.9.1-11.9.4), 11.10 Computing the Hat Matrix Diagonals: 11.10.1 Using the Cholesky Factorization, 11.10.2 Using the Thin QR Decomposition, 11.11 Calculating Test Statistics, 11.12 Robust Regression Calculations: 11.12.1 Algorithms for L1 Regression, 11.12.2 Algorithms for M- and GM-Estimation, 11.12.3 Elemental Regressions, 11.12.4 Algorithms for High-Breakdown Methods

Suppose that we fit a least squares regression to a subset $J$ of the $n$ available cases. Let $\mathbf{X}_{J}$ be the submatrix of $\mathbf{X}$ corresponding to the cases in $J$, let $e_{i}$ be the $i$ th residual from this fit, $i = 1, \ldots , n$, and let $h_{i j} = \mathbf{x}_{i}^{T}\left(\mathbf{X}_{J}^{T} \mathbf{X}_{J}\right)^{-1} \mathbf{x}_{j}$. Show that if we drop case $i \in J$ and add case $j \notin J$, the change in the residual sum of squares is

Using the result in Exercise 1 above, write an R function implementing Hawkins's method of calculating an approximation to the LTS estimate.

Write an R function to implement Ruppert's algorithm to calculate an approximation to the LTS estimate. You will need to refer to Ruppert [1992] for the details of the algorithm. Devise a small simulation to compare Ruppert's algorithm to Hawkins's.