MSc Financial Mathematics
Statistical Methods and Data Analytics 2018
MATH0099
Problem Sheet 4
Problem 1. Let Y1, . . . , Yn be random variables defined by
Yi = βxi + Ei
, i = 1, . . . , n,
where x = (x1, . . . , xn) is a vector of constants and E = (E1, . . . , En) a vector of iid N(0, σ2
) random variables. Here θ = (β, σ) is the vector of unknown parameters.
1. Find a two-dimensional sufficient statistic for (β, σ2
).
2. Find the MLE of β and show that it is an unbiased estimator of β.
3. Find the distribution of the MLE of β.
4. Show that Yi/xi
is also an unbiased estimator of β.
5. Calculate the exact variance of Yi/xi and compare it to the variance of the MLE.
Problem 2. Let X1 . . . , Xn be iid with pdf
Find, if one exists, a UMVU estimator of θ.
Problem 3. Let X1, . . . , Xn be iid Bernoulli(p) rvs. Show that the variance of attains the Cram´er-Rao lower bound and hence is the UMVU estimator of p.