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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.