ex-1: change ε to 0.88 and write the correct results

ex-1/bootstrap.c

@@ -150,7 +150,7 @@ double gauss_kde(double x, void * params) {
    * by the sample variance times a factor which
    * depends on the number of points and dimension.
    */
-  double bw = 0.4 * p.var * pow((double)p.n*3.0/4, -2.0/5);
+  double bw = 0.777 * p.var * pow((double)p.n*3.0/4, -2.0/5);
 
   double sum = 0;
   for (size_t i = 0; i < p.n; i++)

notes/sections/1.md

@@ -289,10 +289,10 @@ where $\mathcal{N}$ is the kernel and the parameter $\varepsilon$, called
 determined in several ways. For simplicity, it was chosen to use Silverman's
 rule of thumb [@silverman86], which gives:
 $$
-  \varepsilon = 0.63 \, S_N
+  \varepsilon = 0.88 \, S_N
     \left(\frac{d + 2}{4}N\right)^{-1/(d + 4)}
 $$
-where the $0.63$ factor was chosen to compensate for the distortion that
+where the $0.88$ factor was chosen to compensate for the distortion that
 systematically reduces the peaks height, which affects the estimation of the
 mode, and:
 
@@ -303,13 +303,13 @@ With the empirical density estimation at hand, the FWHM can be computed by the
 same numerical method described for the true PDF. Again this was bootstrapped
 to estimate the standard error giving:
 $$
-  \text{observed FWHM: } w_o = \num{4.06 \pm 0.08}
+  \text{observed FWHM: } w_o = \num{4.11 \pm 0.07}
 $$
 Applying the $t$-test to these two values gives
 
-  - $t=0.495$
-  - $p=0.620$
+  - $t=1.338$
+  - $p=0.181$
 
-which shows a very good agreement and proves the estimator is robust.
-For reference, the initial estimation based on an histogram gave a rather
-inadequate \si{4 \pm 2}.
+which shows a good agreement and proves the estimator is robust. For reference,
+the initial estimation based on an histogram gave a rather inadequate \si{4 \pm
+2}.