Prof. Elvezio Ronchetti Research Center for Statistics and GSEM
Ass : Edoardo Vignotto University of Geneva
S203031 Probability and Statistics II Fall 2018
EXAM January 22, 2019
— The exam lasts 3h.
— You may answer in English or in French.
— You are not allowed to leave the room during the exam.
— Open book exam.
— Justify your answers and do not forget to write your name on your copy.
Good luck !
Exercise 1 (6 points)
Suppose that the true parameter of a parametric model is θ = 4 and our estimator of θ equals either 3
or 5 with probability 1
2
each. Then our estimator is unbiased and its MSE is 1. Now, suppose we “shrink”
our estimator by a factor r (between 0 and 1), so it equals either 3r or 5r with probability 1
2
each.
1. Compute the bias, the variance, and the MSE of the new estimator as a function of r.
2. What is the value r0 which minimizes the MSE? What is the reduction (in %) in MSE obtained
by shrinking the original estimator by the factor r0 ?
Exercise 2 (9 points)
Let u1, . . . , un be a random sample from a random variable U describing an angle with uniform
density on the interval [−pi
2
, pi
2
]. We make the transformation Z = tg(U).
1. Show that Z follows a Cauchy distribution with density fZ(z) = 1pi
1
1+z2
.
2. What are the expectation and the variance of Z ? Can we apply the Central Limit Theorem to
obtain an approximation for the distribution of the mean Z¯n = 1n
∑n
i=1 Zi ?
3. Compute the exact distribution of Z¯n.
Hint : use the characteristic function.
Exercise 3 (9 points)
Let x1, . . . , xn be a random sample from a random variableX with density f(x|θ) = θ(1 +x)−(θ+1),
where x > 0 and θ > 0 is an unknown parameter.
1. Compute the maximum likelihood estimator γˆ for γ = 1/θ.
2. Compute the density of Y = log(1 +X). Is γˆ unbiased for γ ?
3. Establish if γˆ reaches the Cramer-Rao bound.
Exercise 4 (6 points)
A particular device may be in operating state or under repair. Let X and Y be the random variables
that describe the duration of the operating state and the duration of the repairing operations, respectively.
Assume that X and Y are independent exponential random variables with unknown expectations θ and
λ respectively. Moreover, let ψ = θ/(θ + λ) be a known quantity.
1. Show that ψ = Pr(Y ≤ X) and give an interpretation of ψ.
2. Compute the maximum likelihood estimator for θ and λ based on the random sample
(x1, y1), . . . , (xn, yn) that uses a known value for ψ.