Math330 Exercises Week 4

• Consider independent identically distributed data $x_1,\mathrm{\dots },x_n$ taken from the Weibull model with pdf

  $$f(x|\alpha ,\beta )={\beta }^{\alpha }\alpha x^{\alpha -1}{e}^{-{(\beta x)}^{\alpha }}$$

  , where $x\ge 0$, with $\alpha >0$ and $\beta >0$. Calculate the log-likelihood function for these data.

  For fixed $\alpha$, find ${\widehat{\beta }}_{\alpha }$, the maximum likelihood $\beta$ value given $\alpha$.

  Compute the profile log-likelihood for $\alpha$.

  Write down an equation that $\widehat{\alpha }$, the MLE of $\alpha$, has to satisfy.

• The figure depicts the profile deviance of $\theta$ for a statistical model with two parameters, $(\theta ,\varphi )$.

  Figure 1: Link, Caption: None

  From the graph, approximately construct a $95\%$ confidence interval for $\theta$.

• A random variable $X$ is said to follow a Poisson$(\mu )$ distribution [denoted as $X\sim \mathrm{Poisson}(\mu )$] if it has mass function given by

  $$f(x|\mu )=\frac{\exp (-\mu ){\mu }^x}{x!},\mathrm{for}x=0,1,2\mathrm{\dots }$$

  In the sequel, consider $n$ independent random variables, where $X_i\sim \mathrm{Poisson}({\mu }_i)$, with ${\mu }_i=\exp (\alpha +\beta z_i)$, and where $z_i$ is a known explanatory variable corresponding to observation $i$.

  • Write down the log-likelihood function $l(\alpha ,\beta )$. (2 marks)

  • Consider a reparametrised version of this model, where ${\alpha }^{*}={\sum }_{i=1}^n{\mu }_i/n$ and ${\beta }^{*}=\beta$ are the new parameters. The log-likelihood function for $({\alpha }^{*},{\beta }^{*})$ is given by

    $$l({\alpha }^{*},{\beta }^{*})=-n{\alpha }^{*}+\log ({\alpha }^{*})\left(\sum _i{x}_i\right)+{\beta }^{*}\left(\sum _i{x}_i{z}_i\right)-\log \left\{\sum _i\exp ({\beta }^{*}{z}_i)\right\}\left(\sum _i{x}_i\right).$$

    From the expression of $l({\alpha }^{*},{\beta }^{*})$, and without performing any calculations, explain why it is immediate that the new parametrisation corresponds to parameter orthogonality. (3 marks)
    

    Using this knowledge, sketch contours of $l({\alpha }^{*},{\beta }^{*})$ and mention their most salient features. (2 marks)

  • Find the profile deviance for ${\alpha }^{*}$, $D^{*}({\alpha }^{*})$. (2 marks)

  • The following figure displays the profile deviance of ${\alpha }^{*}$ for a certain set of data.

    Figure 2: Link, Caption: None

    Find, by eye, the maximum likelihood estimate of ${\alpha }^{*}$ and a 90% confidence interval. (1 mark)