代做MATH0099 Statistical Methods and Data Analytics 2018 Problem Sheet 7代做Python编程

日期：2024-08-23 04:35

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.