Exercise 6

Question 1.

The number of power failures in an electrical network per day is Poisson distributed with an unknown mean λ. During the last month (30 days), 5 power failures have been registered. Let p be the probability that there is no power failures during a day.
  1. Find the MLE for p.
  2. Derive 95% asymptotic confidence intervals for p using asymptotic normality of MLE and using the variance stabilizing transformation for Poisson data.
  3. Show that above asymptotic confidence intervals are first-order equivalent.

Question 2.

Let Y1,...,Yn be a random sample with finite mean μ and varaince σ2. Find the asymptotic distributions of mean(Y)2 when μ ≠ 0 and μ=0, and e-mean(Y)

Question 3.

Let Y1,...,Yn be a sample from a shifted exponential distribution with the rate one, that is, f(y)=exp(-(y-θ)) for y > θ and zero otherwise.
  1. Find the MLE for θ.
  2. Is the MLE consistent in MSE? Is it consistent (in probability)?
    (Hint: show that the MLE has also a shifted exponential distribution with the same shift θ but with the rate n, that is, its density g(u)=n exp(-n(u-θ)), u > θ)
  3. Show that n(MLE(θ)-θ) ~ exp(1)
  4. Is MLE(θ) a CAN estimator? If not, why does the asymptotic normality of MLE not hold in this case?

Question 4.

A director of a large bank has a monthly information from its L branches about the numbers of new clients joined the bank for each of the last n months. Assume that the the number of new clients joined a j-th branch each month is Pois(λj), there is no correlation neither between diffirent branches nor between different months, i.e. Yij ~ Pois(λj), i=1,...,n; j=1,...,L and all Yij's are independent.
  1. Find the MLEs for λ=(λ1,...,λL)' and the asymptotic distribution of n1/2(MLE(λ)-λ).
  2. The director is particularily interested in the percent of new clients joined a specific branch (say, the first), i.e. in p=λ1/(λ1+...+λL). Find the MLE for p and the asymptotic distribution of n1/2(MLE(p)-p).
  3. Derive a GLRT for testing the hypothesis that all the branches are equally successful in "hunting" after new clients.
  4. Perform the test for a bank that has three branches and the average numbers of new clients joined the branches over the last 15 months were 10, 12 and 15 respectively (explain approximations you have used if any).