Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 210 / 2 / Solution

A kernel for density estimation is an integrable function $K$ with $\int K=1$, usually also bounded and nonnegative, and its scaled version is $K_h(u)=h^{-1}K(u/h)$. For suitable $g_1,g_2$, their convolution is

Because the observations have length-biased density $g(u)=uf(u)/\mu$,Thus the exact bias is $(K_h\ast f)(x)-f(x)$, exactly the same as for the ordinary kernel density estimator based directly on observations from $f$.

Write $R(K)=\int K(u{)}^{2}du$. The estimator is the average of $Y_i(x)=\mu X_i^{-1}{K}_h(x-X_i)$, soIts mean is $K_h\ast f$. Integrating the pointwise variance therefore givesFinally, Cauchy-Schwarz inequality under density $f$ givesEquality would require $X$ to be constant almost surely, impossible for a density, so $\mu \bar{\mu }>1$. The ordinary KDE has the same negative term but leading integrated variance $R(K)/(nh)$; length-biased sampling strictly inflates it.