1. Suppose X = (X1, . . . , Xn) is a random sample from the Gamma distribution withshape α and rate θ, i.e.,fX (x) = θαΓ(α)xα−1e−xθ, x > 0, α > 0, θ > 0.Each Xi has expectation E(X) = α/θ, variance Var(X) = α/θ2, and momentgenerating function MX (t) = (1 − t/θ)−α.(a) Assuming both α and θ to be unknown, write down the log likelihood functionℓ(θ, α; X) and the corresponding score functions ∂ℓ∂θ and ∂ℓ∂α .(b) Verify that each score function has zero expectation.(c) Assuming α to be known:(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of anunbiased estimator of θ.(ii) Is there any unbiased estimator of θ whose variance attain the CRLB?(iii) Show thatS = α − 1nnXi=1 1Xiis an unbiased estimator for θ. What is the MVU estimator for θ?(iv) Identify a change of parameter η = η(θ) for which there exists an unbiasedestimator with variance attained the CRLB.(v) For the parameter in the previous part, identify the MVU estimator whosevariance attains the CRLB. Compute this variance

Let X1, . . . , Xn denote a random sample from a N (µ, σ2) distribution,where the mean µ and the variance σ2 are both unknown so that the param-eter vector is given byθ = (µ, σ2)T .(i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameterregular exponential family, stating what the canonical parameter isequal to in terms of θ.

Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T . (i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameter regular exponential family, stating what the canonical parameter is equal to in terms of θ. (ii) Use the result in (i) to derive the maximum likelihood (ML) estimate of θ, θˆ = (μˆ, σˆ2)T , without the need to differentiate the log likelihood function with respect to θ.

the moment estimator of θ;• the MLE of θ

(a)The Poisson distribution truncated at zero has probability function givenbyf (x; θ) = exp(−θ)θxA(θ)x! , x = 1, 2, ... (θ > 0),whereA(θ) = 1 − exp(−θ).Let X1, . . . , Xn be a random sample of size n from this truncated Poissondistribution.(i) Show that this distribution belongs to the regular exponential distri-bution and give its canonical parameter c(θ) and the correspondingsufficient statistic T (X1, ..., Xn).

For each of the following pdfs let X1, . . . , Xn be iid observations. Find a complete sufficient statistic,or show that one does not exist.(a) f (x; θ) = 2xθ2 , 0 < x < θ, θ > 0(b) f (x; θ) = e−(x−θ) exp (−e−(x−θ)) , −∞ < x < ∞, −∞ < θ < ∞(c) f (x; θ) = (2x)θx(1 − θ)2−x, x = 0, 1, 2, 0 ≤ θ ≤ 1