## 2.5 Exercises

### Kernel density estimation

Exercise 2.1 The Epanechnikov kernel is given by $K(x) = \frac{3}{4}(1 - x^2)$ for $$x \in [-1, 1]$$ and 0 elsewhere. Show that this is a probability density with mean zero and compute $$\sigma_K^2$$ as well as $$\|K\|_2^2$$.

For the following exercises use the log(F12) variable as considered in Exercise A.4.

Exercise 2.2 Use density() to compute the kernel density estimate with the Epanechnikov kernel to the log(F12) data. Try different bandwidths.

Exercise 2.3 Implement kernel density estimation yourself using the Epanechnikov kernel. Test your implementation by comparing it to density() using the log(F12) data.

### Benchmarking

Exercise 2.4 Construct the following vector
x <- rnorm(2^13)

Then use microbenchmark to benchmark the computation of

density(x[1:k], 0.2)

for k ranging from $$2^5$$ to $$2^{13}$$. Summarize the benchmarking results.

Exercise 2.5 Benchmark your own implementation of kernel density estimation using the Epanechnikov kernel. Compare the results to those obtained for density().

Exercise 2.6 Experiment with different implementations of kernel evaluation in R using the Gaussian kernel and the Epanechnikov kernel. Use microbenchmark() to compare the different implementations.