Description
1. Consider a random sample X1, · · · , Xn from a Weibull distribution
f(x| θ, β) = (β/θ)x
β−1
exp(−x
β
/θ), x > 0
where β > 0 is known and θ is a parameter.
(i) Find the method of moment estimator ˜θ and the MLE ˆθ of θ;
(ii) Are ˜θ and ˆθ unbiased estimators?
(iii) Find the Cramer-Rao lower bound;
(iv) Is ˆθ UMVUE?
2. Let X1, X2, · · · , Xn be a random sample from a uniform distribution on the interval
(θ − 1, θ + 1).
(i) Find the method of moments estimator for θ;
(ii) Is your estimator in part (i) an unbiased estimator of θ?
(iii) Given the following 5 observations,
6.61, 7.70, 6.98, 8.36, 7.26
give a point estimate of θ;
(iv) The method of moments estimator has greater variance than the estimator
(1/2)(X(1) + X(n)). Compute the value of this estimator for the 5 observations in (iii).
3. Let X1, · · · , Xn be a random sample from the density function
f(x|θ) = θ xθ−1
, 0 < x < 1, θ > 0.
(i) Find the MLE ˆθn of θ and show that ˆθn is a consistent estimator of θ;
(ii) Find the method of moments estimator of θ.
4. Let X1, …, Xn be a random sample from distribution with the following probability density
function
f(x; θ) = 1
2θ
3
x
2
e
−x/θ
, 0 < x < ∞, 0 < θ < ∞
(i) Find the MLE ˆθ for θ;
(ii) Find the MLE for τ (θ) = 1
θ
;
(iii) Find the C-R inequality for τ (θ);
(iv) Is ˆθ unbiased? Is τ (
ˆθ) unbiased?
(v) Find the asymptotic distribution of √
n(τ (
ˆθ) − τ (θ)).
5. Let X1, · · · , Xn be a random sample from the following distribution
f(x|θ) = x
2
e
−x/θ
2θ
3
, x > 0
Show that the UMVU estimator of θ is
T =
1
3n
Xn
i=1
Xi
.
6. Let X1, …, Xn be a random sample from N(µ, σ2
). Prove that (Pn
i=1 Xi
,
Pn
i=1(Xi − X¯)
2
) is
minimal jointly sufficient statistics.
