MATH333: Assessment Week 17

Assessment Deadline: Week 18, Tuesday 5pm.

A continuous random variable $Y$ is said to have a gamma distribution with positive shape parameter $\alpha$ and positive rate parameter $\beta$ if it has a probability density function of the form:

$$f_Y(y)=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{y}^{\alpha -1}\exp \{-\beta y\},\mathrm{for}y>0,$$

where $\mathrm{\Gamma }(\alpha )={\int }_0^{\mathrm{\infty }}{x}^{\alpha -1}\exp \{-x\}𝑑x$ is the gamma function. In the following, assume that the shape parameter $\alpha$ is known.

1. By re-expressing the probability density function, show that the gamma distribution belongs to the exponential family with canonical parameter $\theta =1-\beta$. Carefully define the parameter space for $\theta$. [3]

2. Write the cumulant generating function for the gamma distribution and show that $𝔼[Y]=\frac{\alpha }{1-\theta }$ and $\mathrm{var}(Y)=\frac{\alpha }{{(1-\theta )}^2}$. [2]

3. An operator of a theme-park ride will only allow a kart to go once all of the seats are filled. Assume that the arrival time of the customers are independent of one another and that there are $\alpha$ seats in every kart. It follows that the waiting time of the kart follows a gamma distribution with shape parameter $\alpha$ and unknown rate. Let $y_1,\mathrm{\dots },y_n$ be independent waiting time measurements of the kart at the loading station.

   1. Using your answer to question 1 or otherwise, derive the maximum likelihood estimate for the canonical parameter $\theta$. [2]

   2. Derive an expression for the expected information for $\theta$ at the MLE. [2]

   3. The table below presents 20 independent waiting times in seconds of a kart containing 4 seats whilst at the loading station. Calculate the MLE and an approximate 95% confidence interval for $\theta$. [1]

      58	91	86	93	81	138	125	68	85	48
      164	53	43	114	111	153	67	198	64	51

      [Hint: You may use the fact that the MLE has an approximate Normal distribution, with mean $\theta$ and variance equal to the inverse of the expected information i.e. $\widehat{\theta }\stackrel{𝑎}{\sim }N(\theta ,1/I_E(\widehat{\theta }))$]