Consider the

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pressure dataset used in the Estimation procedure slide and also available in Dobson and Barnett $(2018,$ page $65).$

library$($dobson$)$

data$("$failure$")$

attach$($failure$)$

Assume that we are interested in fitting a Weibull distribution to the data with fixed shape

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$=$

$2$

$\backslash$lambda $=2.$

Implement the Newton$-$Raphson algorithm in

R

R to estimate the scale parameter of the Weibull parameter as a function of

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y $($data$),$

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$0$

$\backslash$theta

$0$

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$($starting value$),$

$$

$\backslash$lambda and the number of iterations. Run the algorithm with

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$0$

$\backslash$theta

$0$

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set to be the sample mean and reproduce Table $4\mathrm{.}2$ from Dobson and Barnett $(2018)$ without applying any rounding $($the results might be slightly different for that reason$).$

Plot the log$-$likelihood function as a function of the scale parameter

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$\backslash$theta $.$ Indicate the maximum likelihood estimator and the maximised value of the log$-$likelihood