MATH235 Week 9 - Assessed problems (coursework)

Let $X_1,X_2,\mathrm{\dots },X_n$ be a random sample from $X$ which has pdf $f(x|\theta )$ depending on a parameter $\theta$ and

• (i)

  $f(x|\theta )={(2\pi )}^{-\frac{1}{2}}\exp \{-\frac{1}{2}{(x-\theta )}^2\}$

• (ii)

  $f(x|\theta )={(2\pi \theta )}^{-\frac{1}{2}}\exp \{-x^2/(2\theta )\}$

where . In both of these two cases

1. (a)

   write down the log-likelihood function and find a 1-dimensional sufficient statistic for $\theta$.

2. (b)

   find the score function and the maximum likelihood estimator of $\theta$;

3. (c)

   find the observed information and evaluate the Fisher information at $\theta =1$.

   [marks: 6]

Let $Y_1,Y_2,\mathrm{\dots },Y_n$ be a random sample from a $\text{Pois}(\theta )$ distribution.

1. (a)

   Find an expression for the deviance function $D(\theta )$.

2. (b)

   We observe data

   ⬇

   2 0 0 1 0 1 3 0.

   Plot the deviance function over the interval $(0.25,3)$ and hence obtain a 95% confidence interval for $\theta$.

   [marks: 4]