Let X1,…, Xn be a sample of iid N(0, ?) random
variables with Θ=(0, ∞). Determine...
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 be a sample of iid random variables with pdf f (x;
?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ.
Determine
a) L(?1, ?2).
b) the MLE of ?⃗ = (?1, ?2).
c) E(? ̂ 2).
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 be iid Poisson random variables
with unknown mean µ
1. Find the maximum likelihood estimator of µ
2.Determine whether the maximum likelihood estimator is unbiased
for µ
Let X1,…, Xn 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) Determine Var[X(1)].
f ) Is X(1) an MSE-consistent estimator of ?2?
g) Given...
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 ∼ 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 be i.i.d random variables with the density
function f(x|θ) = e^(θ−x) , θ ≤ x. a. Find the Method of Moment
estimate of θ b. The MLE of θ (Hint: Think carefully before taking
derivative, do we have to take derivative?)
Let X1, X2, X3, . . . be independently random variables such
that Xn ∼ Bin(n, 0.5) for n ≥ 1. Let N ∼ Geo(0.5) and assume it is
independent of X1, X2, . . .. Further define T = XN
.
(a) Find E(T) and argue that T is short proper.
(b) Find the pgf of T.
(c) Use the pgf of T in (b) to find P(T = n) for n ≥ 0.
(d) Use the pgf of...