Local Semiparametric Efficiency Bounds under Shape Restrictions

Consider the model y 5 xb0 1 f *~z! 1 «, where « 5 d N~0,s0!+ We calculate the smallest asymptotic variance that n102 consistent regular ~n102CR! estimators of b0 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 n102CR estimators of b0, homogeneity of f * may lead to substantial asymptotic efficiency gains in estimating b0+ In contrast, at least asymptotically, concavity and monotonicity of f * do not help in estimating b0 more efficiently, at least for n102CR estimators of b0+