Math330 Exercises Week 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 $\log \widehat{\alpha }$?

  
  

• A random variable, $R$, which follows a Geometric distribution with parameter $\theta (0\le \theta \le 1)$, has probability mass function

  $$f(r;\theta )={\theta }^r(1-\theta )$$

  for $r=0,1,2,\mathrm{\dots }$ and $E(R)=\theta /(1-\theta )$. Suppose $R_1,\mathrm{\dots },R_n$ are IID random variables with the same distribution as $R$. For $R_1,\mathrm{\dots },R_n$

  • find the maximum likelihood estimator, $\widehat{\theta }$, of $\theta$;

  • show that the expected information on $\theta$ is $n/[\theta {(1-\theta )}^2]$. Hence, or otherwise, find the asymptotic variance of $\widehat{\theta }$;

  • find the maximum likelihood estimator of $\mathrm{Pr}(R\ge 5)$. Show that the asymptotic standard error of this estimator is $5{\theta }^{4.5}(1-\theta )/\sqrt{n}.$

  
  

• This question will be relevant to the online quiz on Moodle

  The Health Survey for England (HSE) in 2009 contains 3395 valid BMI measurements on adults. A sub-sample of 100 of these is in the data file bmi2009.csv (you will need this for the quiz, but not for this workshop).

  Recall that BMI$>30$ means ‘obese’. 26 of the 100 sampled people are thus classed as obese. Suppose the true population proportion $\theta$, of obese persons, is of interest. Construct an appropriate likelihood function for $\theta$, calculate the MLE, its variance, and a 95% confidence interval for $\theta$ based on the asymptotic distribution of the MLE.

  It is also of interest to model actual BMI values as Normal$(\mu ,{\sigma }^2)$. Given that ${\sum }_{i=1}^n{x}_i=2719.253$ and ${\sum }_{i=1}^n{x}_i^2=76629.74$, find the MLEs of $\mu$ and ${\sigma }^2$. Hence give the MLE of $\varphi =\frac{\mu }{\sigma }$.

• Is the likelihood ratio test the unique most powerful test for comparing two simple hypotheses

  $$H_0:{\theta }_0=a$$

  against

  $$H_1:{\theta }_0=b.$$

  Discuss. State on which set another most powerful test can differ from the likelihood ratio test if the most powerful test is not unique.

• For this question you will need to download the data cloth.rda and read it into R. Recall this is done using the read.table function.

  Description
  
  The data comprise lengths of cloth samples
  and the numbers of flaws found in them.
  35 observations on 2 variables:
  
  x    The length of the roll of cloth.
  y    The number of flaws found in the roll.
  

  1. Explore the data: produce appropriate graphical summaries to describe the relationship between $x$ and $y$. (You need to submit (readable) plot(s) to get full marks) (2 marks)

  2. Suppose that we decide to model the $Y_i$s as independent, but not identical, Poisson distributed random variables, i.e.

     $$Y_i\mid x_i,\lambda \sim \text{Pois}(x_i\lambda )$$

     where $i=1,\mathrm{\dots }32$.

     Explain why this is a reasonable model. (1 marks)

  3. Derive an expression for the log-likelihood for $\lambda$. (2 marks)

  4. Write a function that finds the log likelihood and plot this function. (Submit a plot to get full marks) (2 marks)

  5. Find the MLE of $\lambda$. (1 mark)

  6. Find a deviance based 95 % CI for $\lambda$ (you could do this numerically using uniroot, or use R to plot the deviance and do by eye). (2 marks)