Suppose that X1, ..., Xn form a random sample from a uniform
distribution for on the...
Suppose that X1, ..., Xn form a random sample from a uniform
distribution for on the interval [0, θ]. Show that T = max(X1, ...,
Xn) is a sufficient statistic for θ.
Let X1, . . . , Xn be a random sample from a uniform
distribution on the interval [a, b]
(i) Find the moments estimators of a and b.
(ii) Find the MLEs of a and b.
Suppose that X1,X2,...,Xn form a random sample from a uniform
distribution on the interval [θ1,θ2], where (−∞ < θ1 < θ2
< ∞). Find MME for both θ1 and θ2.
Let X1,X2, . . . , Xn be a random sample from the uniform
distribution with pdf f(x; θ1, θ2) =
1/(2θ2), θ1 − θ2 < x <
θ1 + θ2, where −∞ < θ1 < ∞
and θ2 > 0, and the pdf is equal to zero
elsewhere.
(a) Show that Y1 = min(Xi) and Yn = max(Xi), the joint
sufficient statistics for θ1 and θ2, are
complete.
(b) Find the MVUEs of θ1 and θ2.
2. Let X1, ..., Xn be a random sample from a uniform
distribution on the interval (0, θ) where θ > 0 is a parameter.
The prior distribution of the parameter has the pdf f(t) =
βαβ/t^(β−1) for α < t < ∞ and zero elsewhere, where α > 0,
β > 0. Find the Bayes estimator for θ. Describe the usefulness
and the importance of Bayesian estimation.
We are assuming that theta = t, but we are unsure if...
Let
X1,X2,...Xn
be a random sample of size n form a uniform distribution
on the interval [θ1,θ2].
Let Y = min
(X1,X2,...,Xn).
(a) Find the density function for Y. (Hint: find the
cdf and then differentiate.)
(b) Compute the expectation of Y.
(c) Suppose θ1= 0. Use part (b) to give an
unbiased estimator for θ2.
Let X1,X2,...,Xn be a random sample from a uniform distribution
on the interval (0,a). Recall that the maximum likelihood estimator
(MLE) of a is ˆ a = max(Xi).
a) Let Y = max(Xi). Use the fact that Y ≤ y if and only if each Xi
≤ y to derive the cumulative distribution function of Y.
b) Find the probability density function of Y from cdf.
c) Use the obtained pdf to show that MLE for a (ˆ a =...
Suppose X1, . . . , Xn is a random sample from the Normal(μ, σ2)
distribution, where μ is unknown but σ2 is known, and it is of
interest to test H0: μ = μ0 versus H1: μ ̸= μ0 for some value μ0.
The R code below plots the power curve of the test
Reject H0 iff |√n(X ̄n − μ0)/σ| > zα/2
for user-selected values of μ0, n, σ, and α. For a sequence of
values of μ,...
2. Let X1, . . . , Xn be a random sample from the distribution
with pdf given by fX(x;β) = β 1(x ≥ 1).
xβ+1
(a) Show that T = ni=1 log Xi is a sufficient statistic for β.
Hint: Use
n1n1n=exp log=exp −logxi .i=1 xi i=1 xi i=1
(b) Find the pdf of Y = logX, where X ∼ fX(x;β).
(c) Find the distribution of T . Hint: Identify the distribution of
Y and use mgfs.
(d) Find...
Suppose X1, X2, ..., Xn is a random sample from a Poisson
distribution with unknown parameter µ.
a. What is the mean and variance of this distribution?
b. Is X1 + 2X6 − X8 an estimator of µ? Is it a good estimator?
Why or why not?
c. Find the moment estimator and MLE of µ.
d. Show the estimators in (c) are unbiased.
e. Find the MSE of the estimators in (c).
Given the frequency table below:
X 0...