Question

Suppose that X is Poisson, with unknown mean λ, and let X1, ..., Xn be a random sample from X.

a. Find the CRLB for the variance of estimators based on X1, ..., Xn. ̂̅̂

b. Verify that ? = X is an unbiased estimator for λ, and then show that ? is an efficient estimator for λ.

Answer #1

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 sample from an exponential population with
parameter λ.
(a) Find the maximum likelihood estimator for λ. (NOT PI
FUNCTION)
(b) Is the estimator unbiased?
(c) Is the estimator consistent?

Let X1,...,Xn be a random sample from a normal
distribution where the variance is known and the mean is
unknown.
Find the minimum variance unbiased estimator of the
mean. Justify all your steps.

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 X¯ be the sample mean of a random sample X1, . . . , Xn from
the exponential distribution, Exp(θ), with density function f(x) =
(1/θ) exp{−x/θ}, x > 0. Show that X¯ is an unbiased point
estimator of θ.

Let X1, . . . , Xn be i.i.d from pmf f(x|λ) where f(x) =
(e^(−λ)*(λ^x))/x!, λ > 0, x = 0, 1, 2
a) Find MoM (Method of Moments) estimator for λ
b) Show that MoM estimator you found in (a) is minimal
sufficient for λ
c) Now we split the sample into two parts, X1, . . . , Xm and
Xm+1, . . . , Xn. Show that ( Sum of Xi from 1 to m, Sum...

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

Let X1, . . . , Xn be a random sample from a Poisson
distribution.
(a) Prove that Pn i=1 Xi is a sufficient statistic for λ.
(b) The MLE for λ in a Poisson distribution is X. Use this fact
and the result of part (a) to argue that the MLE is also a
sufficient statistic for λ.

Suppose X1, . . . , Xn are a random sample from a N(0, σ^2)
distribution. Find the MLE of σ^2
and show that it is an unbiased efficient estimator.

Let Y1,Y2,Y3,...,Yn denote a random sample from a Poisson
probability distribution.
(a) Show that sample mean, ˆ θ1 = ¯ Y and the sample variance, ˆ
θ2 = S2 are both unbiased estimators of θ.
(b) Calculate relative eﬃciency of the two estimators in (a).
Based on your calculation, Which of the two estimators in 3a would
you select as a better estimator?

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