Fix a probability P. Let u(x) satisfy u(x)d P(x) = 0 . (i) Assume supx |u(x)| < , so that p(x) = [1 +

Fix a probability P. Let u(x) satisfy



u(x)d P(x) = 0 .

(i) Assume supx |u(x)| < ∞, so that pθ(x) = [1 + θu(x)]

defines a family of densities (with respect to P) for all small |θ|. Show this family is q.m.d. at θ = 0. Calculate the quadratic mean derivative, score function, and I(0).

(ii) Alternatively, if u is unbounded, define pθ(x) = C(θ) exp(θu(x)), assuming

 exp(θu(x))d P(x) exists for all small |θ|. For this family, argue the family is q.m.d.

at θ = 0, and calculate the score function and I(0).

(iii) Suppose  u2(x)d P(x) < ∞. Define pθ(x) = C(θ)2[1 + exp(−2θu(x))]
−1 .
Show this family is q.m.d. at θ = 0, and calculate the score function and I(0). [The constructions in this problem are important for nonparametric applications, used later in Chapters 15 and 16. The last construction is given in van der Vaart (1998).]