Exercise 2

Question 1.

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 2.

Let $Y_1,\ldots,Y_n$ be a sample from a shifted exponential distribution with the rate one, that is, $f(y)=e^{-(y-\theta)},\;y \geq \theta$.
  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 that minimizes $\rho(y,\theta)=\alpha(y-\theta)_++(1-\alpha)(\theta-y)_+$ for a given $0 < \alpha < 1$ is the sample $\alpha \cdot 100\%$-quantile $Y_{(\alpha)}$.
  2. Find the asymptotic distribution of $Y_{(\alpha)}$.

Question 4.

  1. Let $Y$ be a continuous one-dimensional random variable with c.d.f. $F_\theta(y)$. Define a random variable $Z=F_\theta(y)$. Show that $Z \sim U[0,1]$ for any $F_\theta$.
  2. Let $Y_1,\ldots,Y_n$ be a random sample from a distribution $F_\theta(y)$. Show that $h({\bf y},\theta)=-\sum_{i=1}^n \ln F_\theta(y_i)$ is a pivotal quantity.
  3. Find the distribution of $h({\bf Y},\theta)$ (hint: recall a distribution of a function of a random variable to find first the distribution of $-\ln F_\theta(Y)$ and use then the properties of $\chi^2$ distributions)

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.

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).

Question 7.

Given a sample ${\bf Y}_1,\ldots,{\bf Y}_n$ from a $p$-dimensional multinormal distribution with an uknown mean vector $\mu$ and a known variance-covariance matrix $V$, find
  1. a conservative confidence region for $\mu_j,\;j=1,\ldots,p$ using Bonferroni approach
  2. an exact confidence region for $\mu_j,\;j=1,\ldots,p$
    (hint: recall that if ${\bf Y} \sim N(\mu,V)$, then $({\bf Y}-\mu)^t V^{-1}({\bf Y}-\mu) \sim \chi^2_p\;$).