Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 28K (Principles of Statistics)

DefineThe $Z_i$ are independent Rademacher random variables. Sendingto the machine therefore produces the required independent design points.

The Taylor theorem with Lagrange remainder supplies a point ${\xi }_i$ between $x_0$ and $X_i=x_0+h{Z}_i$ such thatSince $Z_i^2=1$ and $Y_i=g(X_i)+{\epsilon }_i$,

Ignoring the bounded second-order remainder, this is the linear regression modelThe ordinary least squares estimate through the known zero intercept isbecause $Z_i^2=1$. This is the randomized symmetric finite-difference derivative estimator.

Substituting part (b) givesThe curvature bound gives $|A|\le hM/2$. Conditional on the $Z_i$ and the evaluation points, $B$ has mean zero, so the cross term has expectation zero. Independence of the normal random variables ${\epsilon }_i$ givesConsequently

The two terms are balanced byAt this value both terms equal $\sigma M/(2\sqrt{N})$, and hence