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 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 3.

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 4.

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 5.

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 6.

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.