Local semiparametric efficiency bounds under shape restrictions

Consider the model y = x'beta (0) + f*(z) + epsilon, where epsilon =(d) N(0,sigma (2)(0)). We calculate the smallest asymptotic variance that n(1/2) consistent regular (n(1/2)CR) estimators of beta (0) can have when the only information we possess about f* is that it has a certain shape. We focus on three particular cases: (i) when f* is homogeneous of degree r, (ii) when f* is concave, (iii) when f* is decreasing. Our results show that in the class of all n(1/2)CR estimators of beta (0), homogeneity of f* may lead to substantial asymptotic efficiency gains in estimating beta (0). In contrast, at least asymptotically, concavity and monotonicity of f* do not help in estimating beta (0) more efficiently, at least for n(1/2)CR estimators of beta (0).