Exercise 2

Question 1.

Let $Y_1,\ldots,Y_n$ be a sample from an exponential distribution $\exp(\theta)$ with $\mu=EY=1/\theta$.
  1. Find UMVUEs for $\theta$ and $\mu$, and check whether they achieve Cramer-Rao lower bound.
    (hint: recall that $Y_1+\cdots+ Y_n \sim \frac{1}{2\theta}\chi^2_{2n}=Gamma(\alpha=n,\beta=\theta)$)
  2. The survival function of the exponential distribution $S(t)$ is defined as $S(t)=P(Y \geq t)=e^{-\theta t}$. Verify that for a given $t$, a trivial unbiased estimator for $S(t)$ is $1$, if $Y_1 \geq t$ and $0$ otherwise. Using it and applying the Rao-Blackwellization, find the UMVUE for $S(t)$.
    (hint: derive first the conditional density of $Y_1$ given $Y_1+\ldots+Y_n$ via the joint density of $Y_1$ and $Y_1+\ldots+Y_n$)

Question 2.

Let $Y_1,\ldots,Y_n$ be a random sample from a distribution with a finite mean $\mu$ and varaince $\sigma^2$. Find the asymptotic distributions of $\bar{Y}^2$ when $\mu \neq 0$ and $\mu=0$, and of $e^{-\bar{Y}}$.

Question 3.

Let $Y_1,\ldots,Y_n$ be a sample from a shifted exponential distribution with the rate one, that is, $f_\theta(y)=e^{-(y-\theta)},\;y \geq \theta$.
  1. Find the MLE $\hat{\theta}$ for $\theta$.
  2. Is $\hat{\theta}$ consistent in MSE? Is it consistent (in probability)?
    (Hint: show that $\hat{\theta}$ has also a shifted exponential distribution with the same shift $\theta$ but with the rate $n$, that is, its density $g(u)=n e^{-n(u-\theta)},\; u \geq \theta$)
  3. Show that $n(\hat{\theta}-\theta) \sim exp(1)$.
  4. Is $\hat{\theta}$ a CAN estimator? If not, why does the asymptotic normality of MLE not hold in this case?

Question 4.

  1. Let $Y_1,\ldots, Y_n \sim f_\theta(y)$. Show that the M-estimator corresponding to $\rho(y,\theta)=\alpha(y-\theta)_++(1-\alpha)(\theta-y)_+$ for a given $0 < \alpha < 1$ is the sample $\alpha \cdot 100\%$-quantile $Y_{(\alpha)}$.
  2. Show that under the regurality conditions, $Y_{(\alpha)}$ is a consistent estimator of a $\alpha \cdot 100\%$-quantile of the distribution $f_\theta(y)$.
  3. Assuming the required regularity conditions, find the asymptotic distribution of $Y_{(\alpha)}$.

Question 5.

The number of power failures in an electrical network per day is Poisson distributed with an unknown mean $\lambda$. 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.

Question 6.

Let $Y_1,\ldots,Y_m \sim B(1,p)$.
  1. Derive the asymptotic $a$-level Wald, Rao/score and Wilks tests for testing $H_0:p=p_0$ vs. $H_1:p \neq p_0$.
  2. Show that all the three test-statistics are asymptotically close under the null.

Question 7.

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 $Pois(\lambda)$, there is no correlation neither between diffirent branches nor between different months, i.e. $Y_{ij} \sim Pois(\lambda_j),\; i=1,\ldots,n;~ j=1,\ldots,L$ and all $Y_{ij}$'s are independent.
  1. Find the MLE $\hat{\lambda}$ for the vector ${\bf \lambda}=(\lambda_1,\ldots,\lambda_L)^t$ and the asymptotic distribution of $\sqrt{n}(\hat{\lambda}-\lambda)$.
  2. The director is particularily interested in the proportion of new clients joined a specific branch (say, the first), i.e. in $p=\lambda_1/\sum_{j=1}^L \lambda_j$. Find the MLE and an asymptotic 100(1-α)% confidence interval for p.
  3. Derive the α-level 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).