(Solved): Q 2. The probability mass function of binomial distribution is given below. \ ...

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Q 2. The probability mass function of binomial distribution is given below. \[ f\left(y_{i} ; \pi_{i}\right)=\left(\begin{array}{c} n_{i} \\ y_{i} \end{array}\right) \pi_{i}^{y}\left(1-\pi_{i}\right)^{n_{i}-y_{i}}, \quad y_{i}=0,1, \cdots, n_{i} \] denoted as \( y_{i} \sim \operatorname{bin}\left(n_{i}, \pi_{i}\right) \). Now, assume all \( y_{i} \mathrm{~s} \) are independent, and use the model \[ \log \left(\frac{\pi_{i}}{1-\pi_{i}}\right)=\beta_{0}+\beta_{1} x_{i} \quad i=1,2, \cdots, n . \] (a) Show that the log-likelihood function is \[ l\left(\beta_{0}, \beta_{1} ; \mathbf{y}\right)=\sum_{i=1}^{n} \log \left(\begin{array}{c} n_{i} \\ y_{i} \end{array}\right)+\sum_{i=1}^{n} y_{i}\left(\beta_{0}+\beta_{1} x_{i}\right)-\sum_{i=1}^{n} n_{i} \log \left[1+\exp \left(\beta_{0}+\beta_{1} x_{i}\right)\right] \] \( (6 \) marks) (b) Show that the maximum likelihood estimates of \( \beta_{0} \) and \( \beta_{1} \) could be obtained by solving \[ \left\{\begin{array}{l} \sum_{i=1}^{n}\left[y_{i}-n_{i} \frac{\exp \left(\beta_{0}+\beta_{1} x_{i}\right)}{1+\exp \left(\beta_{0}+\beta_{1} x_{i}\right)}\right]=0, \\ \sum_{i=1}^{n} x_{i}\left[y_{i}-n_{i} \frac{\exp \left(\beta_{0}+\beta_{1} x_{i}\right)}{1+\exp \left(\beta_{0}+\beta_{1} x_{i}\right)}\right]=0 . \end{array}\right. \] \( (4 \) marks) (c) Show that the deviance is \[ D=2 \sum_{i=1}^{n}\left[y_{i} \log \left(\frac{y_{i}}{\hat{y}_{i}}\right)+\left(n_{i}-y_{i}\right) \log \left(\frac{n_{i}-y_{i}}{n_{i}-\hat{y}_{i}}\right)\right] \] where \( \hat{y}_{i}=n_{i} \hat{\pi}_{i} \) is the fitted value. (5 marks) (15 marks)