Hi, i know this is a long question but help would be greatly appreciated as ive no idea where to start, here it is:

Let x1, x2, . . . , xn be an iid sample from a Gamma distribution with

known shape parameter α and scale parameter β = 1/σ.

.

(a) Derive an estimate of σ using method of maximum likelihood

(b) Compute the bias, variance and mean squared error (MSE) of your

estimate in (a).

(c) Compute the Cramer-Rao lower bound for estimating σ. Does

your estimate achieve the bound? Explain

I have got to the stage in part a where i'm at the log likelihood function but i've no idea how to differenciate it

Let x1, x2, . . . , xn be an iid sample from a Gamma distribution with

known shape parameter α and scale parameter β = 1/σ.

.

(a) Derive an estimate of σ using method of maximum likelihood

(b) Compute the bias, variance and mean squared error (MSE) of your

estimate in (a).

(c) Compute the Cramer-Rao lower bound for estimating σ. Does

your estimate achieve the bound? Explain

I have got to the stage in part a where i'm at the log likelihood function but i've no idea how to differenciate it

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