## 3.8 Exercises

### Kernel estimators

Exercise 3.1 Consider a bivariate data set $$(x_1, y_1), \ldots, (x_n, y_n)$$ and let $$K$$ be a probability density with mean 0. Then $\hat{f}(x, y) = \frac{1}{n h^2} \sum_{i=1}^n K\left(\frac{x - x_i}{h}\right) K\left(\frac{y - y_i}{h}\right)$ is a bivariate kernel density estimator of the joint density of $$x$$ and $$y$$. Show that the kernel density estimator $\hat{f}_1(x) = \frac{1}{n h} \sum_{i=1}^n K\left(\frac{x - x_i}{h}\right)$ is also the marginal distribution of $$x$$ under $$\hat{f}$$, and that the Nadaraya-Watson kernel smoother is the conditional expectation of $$y$$ given $$x$$ under $$\hat{f}$$.

Exercise 3.2 Suppose that $$K$$ is a symmetric kernel and the $$x$$-s are equidistant. Implement a function that computes the smoother matrix using the toeplitz function and $$O(n)$$ kernel evaluations where $$n$$ is the number of data points. Implement also a function that computes the diagonal elements of the smoother matrix directly with run time $$O(n)$$. Hint: find inspiration in the implementation of the running mean.