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

Fix a probability P. Let u(x) satisfy



u(x)dP(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))dx 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)dP(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 13 and 14. The last construction is given in van der Vaart

(1998).]