Question

Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0

(a) Assume both α and β are unknown, find their momthod of moment
estimators: αˆMOM and βˆMOM. (b) Assume α is known and β is
unknown, find the maximum likelihood estimation for β.

Answer #1

let X1, . . . , Xn i.i.d. Gamma(α, β), β > 0 known and α >
0 unknown. (a) Find the sufficient statistic for α (b) Use the
sufficient statistic found in (a) to find the MVUE of α n .
b) Use the sufficient statistic found in (a) to find the MVUE of
α n

Let X1, X2, . . . Xn be iid
random variables from a gamma distribution with unknown α and
unknown β. Find the method of moments estimators for α and β

Let X1, X2, . . . , Xn be iid exponential random variables with
unknown mean β.
(1) Find the maximum likelihood estimator of β.
(2) Determine whether the maximum likelihood estimator is
unbiased for β.
(3) Find the mean squared error of the maximum likelihood
estimator of β.
(4) Find the Cramer-Rao lower bound for the variances of
unbiased estimators of β.
(5) What is the UMVUE (uniformly minimum variance unbiased
estimator) of β? What is your reason?
(6)...

Let X1. ..., Xn, be a random sample from Exponential(β) with pdf
f(x) = 1/β(e^(-x/β)) I(0, ∞)(x), B > 0 where β is an unknown
parameter. Find the UMVUE of β^2.

4. Let X1. ..., Xn, be a random sample from Exponential(β) with
pdf f(x) = 1/β(e^(-x/β)) I(0, ∞)(x), B > 0 where β is an unknown
parameter. Find the UMVUE of β2.

Let X1, ... , Xn be a sample of iid Gamma(?, 1) random variables
with ? ∈ (0, ∞).
a) Determine the likelihood function L(?).
b) Use the Fisher–Neyman factorization theorem to determine
a
sufficient statistic S for ?.

Let X1, X2, · · · , Xn (n ≥ 30)
be i.i.d observations from N(µ1,
σ12 ) and Y1, Y2, · · ·
, Yn be i.i.d observations from N(µ2,
σ22 ). Also assume that X's and Y's are
independent. Suppose that µ1, µ2,
σ12 ,
σ22 are unknown. Find an
approximate 95% confidence interval for
(µ1µ2).

Let B > 0 and let X1 , X2 , … , Xn be a random sample from
the distribution with probability density function.
f( x ; B ) = β/ (1 +x)^ (B+1), x > 0, zero otherwise.
(i) Obtain the maximum likelihood estimator for B, β ˆ .
(ii) Suppose n = 5, and x 1 = 0.3, x 2 = 0.4, x 3 = 1.0, x 4 =
2.0, x 5 = 4.0. Obtain the maximum likelihood...

Let X1, . . . , Xn be iid from a Poisson distribution with
unknown λ. Following the Bayesian paradigm, suppose we assume the
prior distribution for λ is Gamma(α, β).
(a) Find the posterior distribution of λ.
(b) Is Gamma a conjugate prior? Explain.
(c) Use software or tables to provide a 95% credible interval
for λ using the 2.5th percentile and 97.5th percentile in the case
where xi = 13 and n=10, assuming α = 1 andβ =...

Let X1,…, Xn be a sample of iid Gamma(?, ?) random
variables with ? known and Θ=(0, ∞). Determine
a) the MLE ? of ?.
b) E(? ̂).
c) Var(? ̂).
e) whether or not ? is a UMVUE of ?.

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