The code and the report can be found in my github repo
For the first question, I used the Nelder_Meade (NM) routine in simplex.f90. For the second, I use BFGS (BFGS) and smooth the dependent variable as indicated in the assignment. I tried sampling once from \(U_I\) (\(s=1\)) and 100 times (\(s=100\)).
For the last question, I bootstrapped the data 100 times and used the unbiased bootstrap estimator. I used the Cholesky decomposition routine from Intel LAPACK95 called POTRI to invert the matrix. I found that using intrinsic FORTRAN command MATMUL, when estimating the weight matrix (WM) \(\Sigma^{-1}\), increases significantly the processing time when dealing with large matrices. I opted to use a forall command doing operations element by element, which significantly reduced time, and leaving matmul for only small matrices operations.
The results of the estimations for \(s=1\) are displayed in table 1. Table 2 displays the estimations when \(s=100\) times and table 2 displays the bootstrapped weight matrix.
| \(\alpha\) | \(\lambda\) | \(\gamma\) | |
|---|---|---|---|
| NM (Indicator) | 6.365253 | 0.1493535 | -3.899324 |
| BFGS (Smooth) | 6.815446 | 1.0479742 | -5.240383 |
| NM (Indicator) \(\Sigma\) | 4.111111 | 5.8888889 | -8.555556 |
| BFGS (Smooth) \(\Sigma\) | 6.933845 | 1.0184489 | -5.079732 |
| \(\alpha\) | \(\lambda\) | \(\gamma\) | |
|---|---|---|---|
| NM (Indicator) | 4.000000 | 4.0000000 | -2.000000 |
| BFGS (Smooth) | 5.961832 | 0.1328066 | -6.291928 |
| NM (Indicator) \(\Sigma\) | 6.390450 | 0.1197139 | -3.905597 |
| BFGS (Smooth) \(\Sigma\) | 6.581037 | 0.3434273 | -5.701555 |
| 353944.34 | 176915.3 | 602546.2 |
| 3530.64 | 117860.7 | 301111.9 |
| 12024.83 | 6009.2 | 1055124.1 |
I found that NM performs better than BFGS. It is more consistent and depends a little less on initial guess. BFGS is all over the place. In particular, NM does better when \(s=1\) without weight matrix, but bad when we use the optimal weight matrix. BFGS is consistent giving same initial guess with or without weight matrix. When \(s=100\), NM does worse than BFGS. However, NM is back in the game when using the WM, whereas BFGS is somewhat off.