Question: Suppose that data \(y_{1}, y_{2}, \ldots, y_{n}\) are modelled as observed values of i.i.d. random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) with common pdf

where \(\Gamma(\alpha)=\int_{0}^{\infty} x^{\alpha-1} e^{-x} d x\) denotes the gamma function.

1. Write down the likelihood function. [2]
2. By taking logs and differentiating, write down the two likelihood equations for estimating \(\alpha\) and \(\beta\) (write \(\left.\psi(\alpha)=\frac{d}{d \alpha} \log \Gamma(\alpha)\right)\). [3]
3. By eliminating \(\beta\), write down a single estimating equation for \(\alpha\) only. [2]
4. Describe carefully the Newton-Raphson algorithm for finding an approximate solution to this single estimating equation for \(\alpha\), starting from an initial guess \(\alpha_{0}\).
5. Writing \(\hat{\alpha}\) for the estimate of \(\alpha\), write down a formula for the resulting estimate of \(\beta\).

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