In a given population, n individuals are sampled randomly, with replacement, and each

Question

sampled individual is asked whether his/her salary is greater than some fixed threshold z.
Assume that the salary of a randomly chosen individual has the exponential distribution
with unknown parameter λ. Asking whether the salary overcomes a given threshold rathen
than directly asking for the salary increases the number people that are willing to answer
and decreases the number of mistakes in the collected answers. Denote by X1, . . . , Xn the
binary responses (Xi ∈ {0, 1}, i = 1, . . . , n) of the n sampled individuals.
1. What is the distribution of the Xi’s ?
2. Let X¯n be the proportion of sampled individuals whose response was 1 (corresponding to Yes). Prove that X¯n is asymptotically normal and compute the asymptotic
variance.
¯ 3. Find a function f such that f(Xn) is a consistent estimator of λ.

¯
4. Prove that f(Xn) is asymptotically normal and compute the asymptotic variance.
5. What equation must z satisfy in order to minimize the asymptotic variance computed in Question 4 ? Write this equation in the form gλ(z) = z, where gλ is a
function that depends on the unknown parameter λ.
6. Let Y1, . . . , Yn be the salaries of the n sampled people.
a) If one could actually observe Y1, . . . , Yn, what would be the statistical model
?
b) In that case, what would be the Fisher information (as a function of the
unknown parameter λ ? Denote it by IY (λ).
c) In the model where only the Xi’s are observed (with fixed threshold z), what
is the Fisher information ? Denote it by IX(λ).
d) Compare IY (λ) and IX(λ): Which one is the largest ? How do you interpret
this fact ?