A Primer of Statistics by Mary Phipps, Malcolm Quine

By Mary Phipps, Malcolm Quine

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For simplicity it is stated for the fixed design model. The rate of convergence for the more complicated random design is the same. 1. , 1984a) Assume the fixed design model with a one-dimensional predictor variable X and define cK = dK = K 2 (u)du u2 K(u)du. (3) Take the kernel weights {Whi } and assume (A0) K has support [−1, 1] with K(−1) = K(1) = 0, (A1) m ∈ C 2, (A2) maxi |Xi − Xi−1 | = O(n−1 ), (A3) var(εi ) = σ 2 , i = 1, . . , n, (A4) n → ∞, h → 0, nh → ∞. Then dM (x, h) ≈ (nh)−1 σ 2 cK + h4 d2K [m (x)]2 /4.

Tapia and Thompson (1978) summarize this discussion in the related setting of density estimation. Fisher neatly side-stepped the question of what to do in case one did not know the functional form of the unknown density. He did this by separating the problem of determining the form of the unknown density (in Fisher’s terminology, the problem of “specification”) from the problem of determining the parameters which characterize a specified density (in Fisher’s terminology, the problem of “estimation”).

1 Assume the stochastic design model with a one-dimensional predictor variable X and (A1) |K(u)| du < ∞, (A2) lim|u|→∞ uK(u) = 0, (A3) EY 2 < ∞, (A4) n → ∞, hn → 0, nhn → ∞. Then, at every point of continuity of m(x), f (x) and σ 2 (x), with f (x) > 0, n n p −1 Whi (x)Yi → m(x). i=1 38 3 Smoothing techniques The proof of this proposition is in the Complements of this section. The above result states that the kernel smoother converges in probability to the true response curve m(x). It is natural to ask how fast this convergence is going to happen.

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