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?
6. Let X1; : : : ;Xn be i.i.d. samples from Uniform(0;theta
).
(a) Find cn such that Theta1 = cn min(X1; : : : ;Xn) is an unbiased
estimator of theta
(b) It is easy to show that theta2 = 2(X-bar) is also an unbiased
estimator of theta(you do not need to
show this). Compare theta1 and theta2. Which is a better estimator
of theta? Specify your criteria.
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.
For a fixed θ>0, let X1,X2,…,Xn be i.i.d., each with the beta
(1,θ) density.
i) Find θ^ that is the maximum likelihood
estimate of θ.
ii) Let X have the beta (1,θ) density. Find the
density of −log(1−X). Recognize this as one of the famous ones and
provide its name and parameters.
iii) Find f that is the density of the MLE θ^
in part (i).
Let X1, . . . , Xn ∼ iid Exp(θ) and consider the test for H0 : θ
≥ θ0 vs H1 : θ < θ0.
(a) Find the size-α LRT. Express the rejection region in the
form of R = {X > c ¯ } where c will depend on a value from the χ
2 2n distribution.
(b) Find the appropriate value of c.
(c) Find the formula for the P-value of this test.
(d) Compare this test...
Let X1,..., Xn be an i.i.d. sample from a geometric distribution
with parameter p.
U = ( 1, if X1 = 1, 0, if X1 > 1)
find a sufficient statistic T for p.
find E(U|T)
Let X1,X2,...,Xn be i.i.d. Gamma random variables with
parameters α and λ. The likelihood function is difficult to
differentiate because of the gamma function. Rather than finding the
maximum likelihood estimators, what are the method of moments
estimators of both parameters α and λ?
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then, find a two dimensional sufficient statistic for (a, b)
Let X1,...,Xn be i.i.d. N(θ,1), where θ ∈ R is the
unknown parameter.
(a) Find an unbiased estimator of θ^2 based on
(Xn)^2.
(b) Calculate it’s variance and compare it with the Cram
́er-Rao lower bound.