MATH235 Week 7 - Assessed problems (coursework)

In a coin throwing experiment the trials are independent and $\theta$ is the probability of a head on any throw.

1. (a)

   Write down the likelihood function of $\theta$ when the observed outcome is the ordered sequence

   • (i)

     THHHT

   • (ii)

     TTTTTHTHTT

   [marks: 1]

2. (b)

   The figure below shows both likelihoods on the same graph, scaled by the MLE so that both are on the same scale. Comment on the similarities and differences.

   Note: The plots are of the relative likelihood, which is defined as $R(\theta )=\frac{L(\theta )}{L(\widehat{\theta })}$.

   [marks: 2]

3. (c)

   Now suppose $n$ throws are performed and $r$ heads are observed. Write down the resulting likelihood and log-likelihood functions.

   [marks: 2]

4. (d)

   Find the maximum likelihood estimate (MLE) of $\theta$. What is the value of the MLE for situation (ii) above?

   [marks: 2]

5. (e)

   Given your chosen model for (c), show that the maximum likelihood estimator is unbiased.

   [marks: 1]

6. (f)

   Show that the maximum likelihood estimator is consistent. (Hint: an unbiased estimator is also consistent if its variance tends to zero as $n\to \mathrm{\infty }$).

   [marks: 2]

Unnumbered Figure: Link

Relative likelihoods for situations (i) and (ii) in CW7.2