Question

Let X_{1}, ..., X_{n} be a random sample of an
Exponential population with parameter p. That is,

f(x|p) = pe^{-px} , x > 0

Suppose we put a Gamma (c, d) prior on p.

Find the Bayes estimator of p if we use the loss function L(p,
a) = (p - a)^{2}.

Answer #1

Let X_{1}, ..., X_{n} be a random sample of an
Exponential population with parameter p, i.e.,

The likelihood function is

The prior distribution of p follows Gamma (c,d), i.e.,

Then, the posterior distribution is the product of likelihood and prior distribution

Under the squared loss function L(p, a) = (p - a)^{2} ,
Bayes estimator is

The integral is a gamma function and its density is equal to one when we multiple by some constant, consider

a = n+c+1

Then,

Let X1, ..., Xn be a sample from an exponential population with
parameter λ.
(a) Find the maximum likelihood estimator for λ. (NOT PI
FUNCTION)
(b) Is the estimator unbiased?
(c) Is the estimator consistent?

Let X1, X2,..., Xn be a random sample from a population with
probability density function f(x) = theta(1-x)^(theta-1), where
0<x<1, where theta is a positive unknown parameter
a) Find the method of moments estimator of theta
b) Find the maximum likelihood estimator of theta
c) Show that the log likelihood function is maximized at
theta(hat)

Let X¯ be the sample mean of a random sample X1, . . . , Xn from
the exponential distribution, Exp(θ), with density function f(x) =
(1/θ) exp{−x/θ}, x > 0. Show that X¯ is an unbiased point
estimator of θ.

Let X1,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ is
InvGamma(10,2). (a) Find the posterior distribution for θ. (b) If
n= 10 and ̄x= 18.2, find the Bayes estimate under
squared error loss. (c) The variance of the data distribution is φ=
3θ2. Find the Bayes estimator (under squared error loss)
for φ.Let X1,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ
is InvGamma(10,2). (a) Find the posterior distribution for θ. (b)
If n= 10 and ̄x= 18.2, find...

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

Let X1. ..., Xn, be a random sample from Exponential(β) with pdf
f(x) = 1/β(e^(-x/β)) I(0, ∞)(x), B > 0 where β is an unknown
parameter. Find the UMVUE of β^2.

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

Let X1, X2, · · · , Xn be a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function f(x; θ) = (θ 4/6)x 3 e −θx if 0 < x
< ∞ and 0 otherwise where θ > 0
. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
sufficient statistic for θ. b. Compute E(1/Y ) and find the
function of Y which is the unique minimum variance unbiased
estimator of θ.
b. Compute...

Let X1, ... , Xn be a sample of iid Gamma(?, 1) random variables
with ? ∈ (0, ∞).
a) Determine the likelihood function L(?).
b) Use the Fisher–Neyman factorization theorem to determine
a
sufficient statistic S for ?.

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