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

(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).

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

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).(ii) Using the result in (i), show that the likelihood equation can be ex-pressed as ˆθ = ¯x{1 − exp(−ˆθ)} (1)and hence that it yields the same estimate of θ as the method of mo-ments.(iii) Verify (1) by direct differentiation of the log likelihood function for θ.(iv) Verify that in accordance with this distribution belonging to the regularexponential family the observed information matrix I(ˆθ) is equal to theestimated Fisher information I (ˆθ

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

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