Let Y1, Y2, . . ., Yn be a
random sample from a uniform distribution on the interval (θ - 2,
θ).
a) Show that Ȳ is a biased estimator of θ. Calculate the
bias.
b) Calculate MSE( Ȳ).
c) Find an unbiased estimator of θ.
d) What is the mean square error of your unbiased estimator?
e) Is your unbiased estimator a consistent estimator of θ?
Let
Y1, Y2, ..., Yn be a random sample from an exponential distribution
with mean theta. We would like to test H0: theta = 3 against Ha:
theta = 5 based on this random sample.
(a) Find the form of the most powerful rejection region.
(b) Suppose n = 12. Find the MP rejection region of level
0.1.
(c) Is the rejection region in (b) the uniformly most powerful
rejection region of level 0.1 for testing H0: theta = 3...
Let Y1, Y2, . . . , Y20 be a random sample of size n = 20 from a
normal distribution with unknown mean µ and known variance σ 2 = 5.
We want to test H0; µ = 7 vs. Ha : µ > 7. (a) Find the uniformly
most powerful test with significance level 0.05. (b) For the test
in (a), find the power at each of the following alternative values
of µ: µa = 7.5, 8.0, 8.5,...
Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn)
where Y | X = x is modeled by Y=β0+β1x+ε, ε∼N(0,σ^2), where
β0,β1and σ^2 are unknown. Let β1 denote the mle of β1. Derive
V(βhat1).
Let Y1,...,Yn be a sample from the Uniform density on [0,2θ].
Show that θ =
max(Y1, . . . , Yn) is a sufficient statistic for θ. Find a MVUE
(Minimal Variance Unbiased Estimator) for θ.
Let Y1,Y2,...,Yn be a Bernoulli distributed random sample with
P(Yi = 1) = p and P(Yi = 0) = 1−p for all i.
(a) Prove that E(¯ Y ) = p and V (¯ Y ) = p(1−p)/n2, for the
sample mean ¯ Y of Y1,Y2,...,Yn, and find a sufficient statistic U
for p and show it is sufficient for p.
(b) Find MVUE for p and show it is unbiased for p.
Let
Y1, ... , Yn be a random sample that follows normal distribution
N(μ,2σ^2)
i)get the mle for σ^2
ii)prove using i) that it is an efficient estimator
Let Y = (Y1, Y2,..., Yn) and
let θ > 0. Let Y|θ ∼ Pois(θ). Derive the posterior density of θ
given Y assuming the prior distribution of θ is Gamma(a,b) where a
> 1. Then find the prior and posterior means and prior and
posterior modes of θ.
Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent
random samples, with the variables Xi normally distributed with
mean μ1 and variance σ12 and the variables Yi normally distributed
with mean μ2 and variance σ22. The difference between the sample
means, X − Y, is then a linear combination of m + n normally
distributed random variables and, by this theorem, is itself
normally distributed.
(a) Find E(X − Y).
(b) Find V(X − Y).
(c)...