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MSc Financial Mathematics
Statistical Methods and Data Analytics 2018
MATH0099
Problem Sheet 7
Problem 1. Let X1, . . . , Xn be iid copies of a random variable X with pdf
Find a consistent estimator of θ.
Problem 2. Let X1, . . . , Xn be iid copies of a random variable X ∼ N(µ, σ2). Consider the sequence of estimators (δn)n∈N defined by
Show that
1. Var(δn) = ∞.
2. If µ ≠ 0 and we delete the interval (−δ, δ) from the sample space, then Var(δn) < ∞.
3. If µ ≠ 0, the probability content of the interval (−δ, δ) tends to zero.
If two sequences of estimaters (δn)n∈N and (
)n∈N satisfy
in distribution, the asymptotic relative efficiency (ARE) of δn with respect to 
is
Problem 3. Let X1, . . . , Xn be iid copies of a random variable X ∼ Poisson(λ). Find the best unbiased estimator of
1. e −λ, the probability that X = 0;
2. λe−λ, the probability that X = 1;
3. For the best unbiased estimators of parts 1. and 2., calculate the asymptotic relative efficiency with respect to the MLE.
