Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show
that for...
Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show
that for any α ∈ (0, 1), there is a cn,α, such that
[max(X1,...,Xn),cn,α max(X1,...,Xn)] is a 1−α confidence interval
of θ.
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. . . . Xn be i.i.d f(x; θ) = θ(1 − θ)^x x = 0.. Is there
a function of θ for which there exists an unbiased estimator of θ
whose variance achieves the CRLB? If so, find it
R simulation:
Let X1, . . . , Xn be i.i.d. random variables from a uniform
distribution on [0, 2]. Generate
and plot 10 paths of sample means from n = 1 to n = 40 in one
figure for each case. Give
some comments to empirically check the Law of Large Numbers.
(a) When n is large,
X1 + · · · + Xn/n converges to E[Xi].
(b) When n is large,
X1^2+ · · · + Xn^2/n converges to...
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 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.
6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1)
random variables. Let M = max(X1,...,Xn).
(a) Find the density function of M. (b) Find E[M] and V[M].
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?)
Suppose X1, ..., Xn are i.i.d. from an exponential distribution
with mean θ. If we are testing H0 : θ = θ0 vs
Ha : θ > θ0. Suppose we reject H0 when
( X¯n/ θ0) > 1 + (1.645/
√n)
(a) (10 points) Calculate the power function G(ζ). You may leave
your answer in terms of the standard normal cdf Φ(x).
(b) (5 points) Is this test consistent?
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...