Math330 Additional Exercises 1

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

   $$P(X_i=x_i|\theta )=\frac{\exp (-\theta ){\theta }^{x_i}}{x_i!},$$

   where $i=1,\mathrm{\dots },n$.

   • Write down the expression for the log-likelihood function $l(\theta )$, and use it to find $\widehat{\theta }$, the maximum likelihood estimate.

   • Give an expression for a 95% confidence interval for $\theta$.

   • Let $\varphi$ denote the probability that $X_i=1$. Write $\varphi$ as a function of $\theta$. Find the maximum likelihood estimator of $\varphi$ and its asymptotic distribution. (2004 A2)

   
   

2. Let $X_1,\mathrm{\dots },X_n$ be an IID sample from the Beta$(\alpha ,1)$ distribution with density

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

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

   What is the asymptotic distribution of $\log \widehat{\alpha }$? (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.

   
   

4. Ten light bulbs are left on for a period of days. Assume the lifetime of a lightbulb can be modelled as an exponential with density

   $$f(x|\lambda )=\lambda \exp (-x\lambda ).$$

   • State a result about the asymptotic distribution of the MLE of $\lambda$, $\widehat{\lambda }$, giving any additional assumptions if appropriate.

   • Describe how this result can be used to construct an approximate confidence interval for $\lambda$.

   • Derive the likelihood of $\lambda$.

   • The total lifetimes of the lightbulbs is ${\sum }_{i=1}^{10}{x}_i=250$. Derive the MLE of $\lambda$, $\widehat{\lambda }$.

   • Compute an approximate 95% confidence interval for the model parameter $\lambda$.

   • Given $\lambda$, what is the probability of a lightbulb working more than 30 days? Give your answer as an expression involving $\lambda$.

   • State a result about the MLE of $\varphi =P(X>30)$, and compute an approximate 95% confidence interval for the probability in (f).

   • Describe another method for obtaining an approximate likelihood for a single parameter, defining any appropriate terms and stating any appropriate results. Comment on the advantages / disadvantages of each method.