Powerful Choices: Tuning Parameter Selection Based on Power
Kjell A. Doksum, Chad Schafer · 2006
We consider procedures which select the bandwidth in local linear regression by maximizing the limiting power for local Pitman alternatives to the hypothesis that µ(x) ≡ E(Y | X = x) is constant. The focus is on achieving high power near a covariate value x0 and we consider tests based on data with X restricted to an interval containing x0 with bandwidth h. The power optimal bandwidth is shown to tend to zero as sample size goes to infinity if and only if the sequence of Pitman alternatives is such that the length of the interval centered at x0 on which µ(x) = µn(x) is nonconstant converges to zero as n → ∞. We show that tests which are based on local linear fits over asymmetric intervals of the form [x0 − (1 − λ)h, x0 + (1 + λ)h], where −1 ≤ λ ≤ 1, rather than the symmetric intervals [x0−h, x0+h] will give better asymptotic power. A simple procedure for selecting h and λ consists of using order statistics intervals containing x0. Examples illustrate that the effect of these choices are not trivial: Power optimal bandwidth can give much higher power than bandwidth chosen to minimize mean squared error. Because we focus on power, rather than plotting estimates of µ(x) we plot a correlation curve �ρ(x) which indicates the strength of the dependence between Y and X near each X = x. Extensions to