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 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 =...
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 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,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 represent a random sample from a Rayleigh distribution with pdf
f(x; θ) = (x/θ) * e^x^2/(2θ) for x > 0
(a) It can be shown that E(X2 P ) = 2θ. Use this fact to construct an unbiased estimator of θ based on n i=1 X2 i .
(b) Estimate θ from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions:
16.88 10.23 4.59 6.66...
Let X1, X2, ..., Xn be a random sample of size from a
distribution with probability density function
f(x) = λxλ−1 , 0 < x < 1, λ > 0
a) Get the method of moments estimator of λ. Calculate the
estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3.
b) Get the maximum likelihood estimator of λ. Calculate the
estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3.
. Let X1, X2, ..., Xn be a random sample of size 75 from a
distribution whose probability distribution function is given by
f(x; θ) = ( 1 for 0 < x < 1, 0 otherwise. Use the central
limit theorem to approximate P(0.45 < X < 0.55)
Let X1, … , Xn. be a random sample from gamma (2, theta)
distribution.
a) Show that it is the regular case of the exponential class of
distributions.
b) Find a complete, sufficient statistic for theta.
c) Find the unique MVUE of theta. Justify each step.