Joshi and Nabar (1989) examine properties of linear estimators for the parameter in the so-called "Problem of the Nile," where (X, Y) has the joint density
f(x, y|θ) = exp{-(θx + y/θ)}, x > 0, y > 0.
a. For an iid sample of size n, show that the Fisher information is I(θ) = 2n/θ2.
b. For the estimators

show that
(i) The information in T alone is [2n/(2n + l)]/(θ);
(ii) The information in (T, U) is 1(θ);
(iii) (T, U) is jointly sufficient but not complete.

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Σ . X and U=/ΣΧ Υ, |T= VΣΥΣx and U =