Let X1, . . . , Xn ∼ iid N(θ, σ2 ), with one-sided hypotheses H0...
Let X1, . . . , Xn ∼ iid N(θ, σ2 ), with one-sided hypotheses H0
: θ ≤ θ0 vs H1 : θ > θ0. (a) If σ^2 is known, we can use the UMP
size-α test. Find the formula for the P-value of this test.
Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ).
Consider the hypotheses H0 : µ = µ0 and H1 : µ (not equal)= µ0
and the test statistic (X bar − µ0)/ (S/√ n). Note that S has been
used as σ is unknown.
a. What is the distribution of the test statistic when H0 is
true?
b. What is the type I error of an α−level test of this type?
Prove it.
c. What is...
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 a sample of iid N(0, ?) random
variables with Θ=(0, ∞). Determine
a) the MLE ? of ?.
b) E(? ̂).
c) the asymptotic variance of the MLE of ?.
d) the MLE of SD(Xi ) = √ ?.
Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Is this family MLR in Y
= X(n)? (b) Find the UMP size-α test for H0 : θ ≤ θ0 vs H1 : θ >
θ0. (c) Find the UMP size-α test for H0 : θ ≥ θ0 vs H1 : θ < θ0.
(d) Letting R1 be the rejection region for the test in part (b) and
R2 be the rejection region for the test in part...
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?
3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ)
with the two samples X1...Xn, and Y1...Xn independent of each
other. Assume that the common population SD σ is known but the two
means are not. Consider testing the hypothesis null: μx = μy vs
alternative: μx ≠ μy.
d. Assume σ=1 and n=20. How large must δ be for the size
0.01 test to have power at least 0.99?
e. Assume σ=1and δ=0.2. How large must n be for...
3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ)
with the two samples X1...Xn, and Y1...Xn independent of each
other. Assume that the common population SD σ is known but the two
means are not. Consider testing the hypothesis null: μx = μy vs
alternative: μx ≠ μy.
a. Find the likelihood ratio test statistic Λ. Specify
which MLEs you are using and how you plug them in.
Let X1, X2, · · · , Xn be iid samples from density:
f(x) = {θx^(θ−1), if 0 ≤ x ≤ 1}
0 otherwise
Find the maximum likelihood estimate for θ. Explain, using
Strong Law of Large Numbers, that this maximum likelihood estimate
is consistent.
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.
) Let X1, . . . , Xn be iid from the distribution with parameter
η and probability density function: f(x; η) = e ^(−x+η) , x > η,
and zero otherwise. 1. Find the MLE of η. 2. Show that X_1:n is
sufficient and complete for η. 3. Find the UMVUE of η.