Question

Let X1, . . . , Xn ∼ iid Exp (θ). Find the UMP test for H0 : θ ≥ θ0 vs H1 : θ < θ0.

Answer #1

Let X1, . . . , Xn ∼ iid Beta(θ, 1). (a) Find the UMP test for
H0 : θ ≥ θ0 vs H1 : θ < θ0. (b) Find the corresponding Wald
test. (c) How do these tests compare and which would you
prefer?

Let X1, . . . , Xn ∼ iid N(θ, σ^2 ) for σ ^2 known. Find the UMP
size-α test for H0 : θ ≥ θ0 vs H1 : θ < θ0.

Let X1, …,Xn be a random sample from f(x;
θ) = θ exp(-xθ) , x>0. Use the likelihood ratio test to
determine test H0 θ=1 against H1 θ ≠ 1.

Problem 2 Let X1, · · · , Xn IID∼ N(θ, θ) with θ > 0. Find a
pivotal quantity and use it to construct a confidence interval for
θ.

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

Let X1, X2, · · · , Xn be a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

Let X1,...,Xn be iid
exp(θ) rvs.
(a) Compute the pdf of Xmin.
(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
MLE for θ?
(e)...

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,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ is
InvGamma(10,2). (a) Find the posterior distribution for θ. (b) If
n= 10 and ̄x= 18.2, find the Bayes estimate under
squared error loss. (c) The variance of the data distribution is φ=
3θ2. Find the Bayes estimator (under squared error loss)
for φ.Let X1,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ
is InvGamma(10,2). (a) Find the posterior distribution for θ. (b)
If n= 10 and ̄x= 18.2, find...

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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