Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 24J / Solution

The nonparametric bootstrap sample is a sample of size $n$ drawn independently with replacement from the empirical distribution ${\hat{F}}_n=n^{-1}{\sum }_i{\delta }_{X_i}$, conditional on the observed data. To simulate it, take independent uniform $U_j$ and set $X_j^b=X_{1+\lfloor n{U}_j\rfloor }$, assigning $U_j=1$ to the last index if it occurs in a finite-precision generator.

Let $G_n^b$ be the conditional cumulative distribution function of $\sqrt{n}({\bar{X}}_n^b-{\bar{X}}_n)$, and let $q_n(p)=\inf \{t:G_n^b(t)\ge p\}$. A basic bootstrap confidence interval isIn practice the conditional quantiles are estimated from many simulated bootstrap means; the asymptotic argument below describes exact bootstrap quantiles or a simulation count tending to infinity.

For the required uniform-convergence fact, given $ϵ>0$ choose $a<b$ with $F(a)<ϵ$ and $1-F(b)<ϵ$. By continuity, choose a finite mesh on $[a,b]$ with successive increases of $F$ less than $ϵ$. Pointwise convergence at the mesh points makes $|F_n-F|<ϵ$ there for large $n$. Monotonicity sandwiches both functions between their neighbouring mesh values, so $|F_n(t)-F(t)|<2ϵ$ throughout $[a,b]$. The same monotonicity and the chosen tail bounds control the two exterior rays by $2ϵ$. Hence ${\sup }_t|F_n(t)-F(t)|\to 0$.

The bootstrap consistency theorem for the sample mean states that, for iid observations with $0<{\sigma }^2<\infty$, the conditional law of $\sqrt{n}({\bar{X}}_n^b-{\bar{X}}_n)$ converges weakly, in probability, to $N(0,{\sigma }^2)$. By continuity of that limiting distribution and the preceding monotonicity argument, the convergence of conditional distribution functions is uniform in probability. Since the limit is strictly increasing, $q_n(p)\to \sigma {\Phi }^{-1}(p)$ in probability for $0<p<1$. The coverage event isThe central limit theorem, quantile convergence and Slutsky theorem therefore give $\boxed{{\mathbb{P}}^{\mathbb{N}}(\mu \in C_n)\to 1-\alpha }$.

Positive variance is needed for this exact coverage conclusion. If the finite variance allowed in the wording is zero, all observations equal $\mu$ almost surely and the interval collapses to $\{\mu \}$, with coverage one. The bootstrap distribution is still consistent with the degenerate limit, but a nonrandomized usual interval cannot then have limiting coverage $1-\alpha$ for $0<\alpha <1$.