Let $X_{1}, \ldots, X_{n}$ be i.i.d. random observations taking values in $[0,1]$ with a continuous distribution function $F$. Let $\hat{F}_{n}(x)=n^{-1} \sum_{i=1}^{n} \mathbf{1}_{\left\{X_{i} \leqslant x\right\}}$ for each $x \in[0,1]$.

(a) State the Kolmogorov-Smirnov theorem. Explain how this theorem may be used in a goodness-of-fit test for the null hypothesis $H_{0}: F=F_{0}$, with $F_{0}$ continuous.

(b) Suppose you do not have access to the quantiles of the sampling distribution of the Kolmogorov-Smirnov test statistic. However, you are given i.i.d. samples $Z_{1}, \ldots, Z_{n m}$ with distribution function $F_{0}$. Describe a test of $H_{0}: F=F_{0}$ with size exactly $1 /(m+1)$.

(c) Now suppose that $X_{1}, \ldots, X_{n}$ are i.i.d. taking values in $[0, \infty)$ with probability density function $f$, with $\sup _{x \geqslant 0}\left(|f(x)|+\left|f^{\prime}(x)\right|\right)<1$. Define the density estimator

$$\left.\hat{f}_{n}(x)=n^{-2 / 3} \sum_{i=1}^{n} \mathbf{1}_{\left\{X_{i}\right.}-\frac{1}{2 n^{1 / 3}} \leqslant x \leqslant X_{i}+\frac{1}{2 n^{1 / 3}}\right\}, \quad x \geqslant 0 .$$

Show that for all $x \geqslant 0$ and all $n \geqslant 1$,

$$\mathbb{E}\left[\left(\hat{f}_{n}(x)-f(x)\right)^{2}\right] \leqslant \frac{2}{n^{2 / 3}} .$$