Math330 Additional Exercises 1

This sheet gives additional exercises covering Chapters 1-3 of the course.

1.
Consider $n$ independent observations from a Poisson $(\theta)$ distribution, that is

$P(X_{i}=x_{i}|\theta)={\exp(-\theta)\,\theta^{x_{i}}\over x_{i}!}$

where $i=1,\ldots,n.$
- (a)
  
  Write down the expression for the log-likelihood function $l(\theta)$ , and use it to find $\hat{\theta}$ , the maximum likelihood estimate.
- (b)
  
  Give an expression for a 95% confidence interval for $\theta$ .
- (c)
  
  Let $\phi$ denote the probability that $X_{i}=1$ . Write $\phi$ as a function of $\theta$ . Find the maximum likelihood estimator of $\phi$ and its asymptotic distribution. (2004 A2)
2.

Let $X_{1},\ldots,X_{n}$ be an IID sample from the Beta $(\alpha,1)$ distribution with density

$f(x|\alpha)=\left\{\begin{array}[]{lll}\alpha\,x^{\alpha-1}&,&0\leq x\leq 1\\ 0&,&\mbox{otherwise}.\end{array}\right.$

Write down the likelihood and log-likelihood for the problem of estimating $\alpha$ .

Calculate the MLE, $\hat{\alpha}$ and its asymptotic distribution.

What is the asymptotic distribution of $\log\hat{\alpha}$ ? (2002 A2)
3.

The left panel of the figure below shows 25 observations drawn independently from a Normal( $\mu$ =5, $\sigma^{2}$ =9) distribution and a histogram of the data. It also shows 4 density functions corresponding to different Normal distributions. The right panel gives the contours of the log-likelihood function $l(\mu,\sigma)$ assuming a Normal( $\mu$ , $\sigma^{2}$ ) model for the data. Each of the 4 points (labelled “0” to “3”) corresponds to one of the densities (labelled “a” to “d”) in the left panel.

Figure 1: Link, Caption: None

Match the points in the right panel with the densities in the left panel, explaining clearly your reasoning.