Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 206 / 5 / a / Solution

At a target $x$, approximate the unknown mean locally by $a+b(X_i-x)$ and solve the local linear regression weighted least-squares problemwhere $K$ is a nonnegative regression kernel and $h>0$ a smoothing bandwidth. A common factor $1/h$ in all weights makes no difference. Estimate $f(x)$ by the fitted intercept ${\hat{a}}_x$, and optionally $f^{\prime }(x)$ by ${\hat{b}}_x$. Repeat at each target rather than fitting a single global straight line.

For an explicit expression put $S_k={\sum }_i{w}_i(X_i-x{)}^k$ and $T_k={\sum }_i{w}_i(X_i-x{)}^k{Y}_i$. The normal equations yieldThe denominator is positive provided there are at least two distinct positively weighted predictor values. The weights sum to one and their first centered moment is zero, so local linear regression exactly reproduces affine functions and reduces boundary bias compared with a local constant fit. Smaller $h$ uses more local detail and increases variance; larger $h$ smooths more and can increase bias. Choose the bandwidth by cross-validation or another justified criterion. A nearest-neighbour span provides an adaptive bandwidth, as in the local fits used later.