By Hardle, Liang, Gao.

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It turns out that the bootstrap distribution and the asymptotic normal distribution approximate the true ones very well even when the sample size of n is only 30. 1 Introduction In this chapter, we will focus on deriving the asymptotic properties of an estimator of the unknown function g(·) for the case of fixed design points. We consider its consistency, weak convergence rate and asymptotic normality. 3) with nonstochastic regressors and heteroscedastic errors. e. β = 0). M¨ uller and Stadtm¨ uller (1987) proposed an estimate of the variance function by using a kernel smoother, and then proved that the estimate is uniformly consistent.

P, A−1 n kn (2) (γi − γi )Xi ξi i=1 j = oP (n−1/2 ) or equivalently kn (2) (γi − γi )xij ξi = oP (n1/2 ). 22) 32 2. ESTIMATION OF THE PARAMETRIC COMPONENT Let {δn } be a sequence of real numbers converging to zero but satisfying δn > n−1/4 . Then for any µ > 0 and j = 1, . . , p, kn (2) (2) (γi − γi )xij ξi I(|γi − γi | ≥ δn ) > µn1/2 P i=1 (2) ≤ P max |γi − γi | ≥ δn → 0. 6). Next we deal with the term kn (2) (2) (γi − γi )xij ξi I(|γi − γi | ≤ δn ) > µn1/2 P i=1 (2) using Chebyshev’s inequality.

From Chebyshev’s inequality, for any given ζ > 0, kn (1) (1) ui Yi(2) − Yi(1) I(Qi ≤τn ) > ζn1/2 P i=1 1 kn (1) (1) Eu2i E(Yi(2) − Yi(1) )2 2 nζ i=1 kn Eu2 sup |Gn (t) − G(t)|2 , ≤ nζ 2 1 t ≤ which converges to zero as n tends to infinite. Thus Jn21 is oP (n1/2 ). A combination of the above arguments yields that n−1/2 Jn2 converges to zero in probability. Next we show that n−1/2 Jn3 converges to zero in probability, which is equivalent to showing that the following sum converges to zero in probability, n−1/2 kn (1) (1) h(Ti )(Yi(2) − Yi(1) )I(Qi >τn ) + n−1/2 i=1 kn (1) (1) ui (Yi(2) − Yi(1) )I(Qi >τn ) .