A find the likelihood and log likelihood functions

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(a) Find the likelihood and log likelihood functions for the parameter vector θ = ( α, β ) 0 . (b) Obtain the expressions for score vector and information matrix. (Note: you can use Γ 0 ( α ) to represent ∂α Γ( α ) and φ ( α ) = Γ 0 ( α ) / Γ( α ) in your expression) (c) Find the maximum likelihood estimates for the two parameters using Newton Raphson algorithm. (Note: two R functions digamma and trigamma can be used to calculate φ ( α ) = Γ 0 ( α ) / Γ( α ) and φ 0 ( α ) = ∂α φ ( α )) (d) Now assume the shape parameter α = 1 is known, there is only one unknown parameter β . Find the maximum likelihood estimator of the expected claim size i.e. E ( Y i ). (e) Construct Wald based and LR (likelihood ratio) based 95% confidence intervals for the expected claim size in (d). State which one you prefer and why. 1
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2. The negative binomial distribution can be used to model the number of failures (denoted Y ) in a sequence of Bernoulli trials before a specified number of successes (denoted r ) occur in general. If we denote the success probability of a Bernoulli trial as π , the probability mass function of Y can be written as f ( y ; π, r ) = y + r - 1 r - 1 π r (1 - π ) y , y = 0 , 1 , 2 , . . . where r > 0 and 0 < π < 1. Note that when r = 1, it becomes a geometric distribution.
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