Question

Let X_{1},…, X_{n} be a sample of iid
Exp(?_{1}, ?_{2}) random variables with common pdf
f (x; ?_{1}, ?_{2}) =
(1/?_{1})e^{−(x−?2)/?1} for x
> ?_{2} and Θ = ℝ × ℝ^{+}.

a) Show that S = (X_{(1)}, ∑^{n}_{i=1}
Xi ) is jointly sufficient for (?_{1}, ?_{2}).

b) Determine the pdf of X_{(1)}.

c) Determine E[X_{(1)}].

d) Determine E[X^{2}_{(1)} ].

e ) Is X_{(1)} an MSE-consistent estimator of
?_{2}?

f) Given S = (X_{(1)}, ∑^{n}_{i=1}
X_{i} )is a complete sufficient statistic
for(?_{1}, ?_{2}), determine the UMVUEs of
?_{1} and ?_{2}.

Answer #1

Let X1,…, Xn be a sample of iid random
variables with pdf f (x; ?) = 3x2 /(?3) on S
= (0, ?) with Θ = ℝ+. Determine
i) a sufficient statistic for ?.
ii) F(x).
iii) f(n)(x)

Let X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?

Let X1,…, Xn be a sample of iid random variables with
pdf f (x ∶ ?) = 1/? for x ∈ {1, 2,…, ?} and Θ = ℕ. Determine the
MLE of ?.

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
distribution with pdf as follows:
fX(x) = e^-(x-θ) , x > θ
0 otherwise.
Find the sufficient statistic for θ.
Find the maximum likelihood estimator of θ.
Find the MVUE of θ,θˆ
Is θˆ a consistent estimator of θ?

Let X1,…, Xn be a sample of iid N(0, ?)random variables with Θ =
ℝ.
a) Show that T = (1/?)∑ni=1
Xi2 is a pivotal quantity.
b) Determine an exact (1 − ?) × 100% confidence interval for ?
based on T.
c) Determine an exact (1 − ?) × 100% upper-bound confidence
interval for ? based on T.

Let X1,...,Xn be iid
exp(θ) rvs.
(a) Compute the pdf of Xmin.
I have the pdf
(b) Create an unbiased estimator for θ based on Xmin.
Compute the variance of the resulting estimator.
(c) Perform a Monte Carlo simulation of N= 10,0000 samples of
your unbiased estimator from part (b) using θ = 2 and n = 100 to
validate your answer. Include a histogram of the samples.
(d) Which is more efficient: your estimator from part (b) or the...

Let X1, X2, …, Xn be iid with pdf ?(?|?) = ? −(?−?)? −? −(?−?) ,
−∞ < ? < ∞. Find a C.S.S of θ

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,…, Xn be a sample of iid Exp(?) random variables. Use the
Delta Method to determine the approximate standard error of ?^2 =
Xbar^2

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