On the construction of efficient estimators in semiparametric models

On efficient estimation of linear functionals of a bivariate distribution with known marginals

“…These stronger conditions are verified in Schick (1993) and Schick (1994) in homoscedastic and heteroscedastic regression models. See also Forrester at al (2003) for weaker conditions under additional structural assumptions. As all the above constructions call for appropriate estimators of the efficient influence function, they are easier to implement when the influence function is available in closed form.…”

Moment estimation in a semiparametric generalized linear model

“…The proof of uniform convergence in Theorem 2 and Theorem 3 below is similar to Forrester, et al(2003), their proof is essentially in the spirit of Härdle and Luckhaus (1984) and Härdle, et al(1988). We omit the details of the proof.…”

“…The weak consistency is the basis upon which rates of convergence and asymptotic normality can further be established. Forrester, et al(2003) proposed their estimator of the nonparametric part in partially linear models. We shall follow the idea of the latter to construct our estimator and give conditions which guarantee the existence and uniform consistence.…”

“…Assumption 7 (i) and (ii) appeared in Forrester, et al(2003) and are Condition A2 (a) and (b) of Bhattacharya and Zhao (1997). Severini and Wong(1992) imposed, in Lemma 8, assumptions on high order differentiability of the marginal density of Z and the conditional density, the uniform boundedness, and high order derivatives, see their (c) and (d), page 1787.…”

“…These estimators and the partial derivatives converge at certain rates (see (18) below) under smoothness assumptions, involving high order partial derivatives of the densities. Forrester, et al(2003) presented their estimator of the nonparametric part in partially linear model under weaker smoothness assumptions. Sparked from their work, we propose our estimators and obtain faster rates under weaker assumptions.…”