Let X1, . . . , Xn be a random sample from a Bernoulli(θ)
distribution, θ...
Let X1, . . . , Xn be a random sample from a Bernoulli(θ)
distribution, θ ∈ [0, 1]. Find the MLE of the odds ratio, defined
by θ/(1 − θ) and derive its asymptotic distribution.
Let X1, X2, ·······, Xn be a random sample from the Bernoulli
distribution. Under the condition...
Let X1, X2, ·······, Xn be a random sample from the Bernoulli
distribution. Under the condition 1/2≤Θ≤1, find a
maximum-likelihood estimator of Θ.
Let X1, . . . , Xn be a random sample from a Poisson
distribution.
(a)...
Let X1, . . . , Xn be a random sample from a Poisson
distribution.
(a) Prove that Pn i=1 Xi is a sufficient statistic for λ.
(b) The MLE for λ in a Poisson distribution is X. Use this fact
and the result of part (a) to argue that the MLE is also a
sufficient statistic for λ.
Let X1,…, Xn be a sample of iid Gamma(?, ?) random
variables with ? known and...
Let X1,…, Xn be a sample of iid Gamma(?, ?) random
variables with ? known and Θ=(0, ∞). Determine
a) the MLE ? of ?.
b) E(? ̂).
c) Var(? ̂).
e) whether or not ? is a UMVUE of ?.
Let X1,X2,...,Xn be a random sample from a geometric random
variable with parameter p. What is...
Let X1,X2,...,Xn be a random sample from a geometric random
variable with parameter p. What is the density function ofU =
min({X1,X2,...,Xn})
6. Let X1, X2, ..., Xn be a random sample of a random variable X
from...
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,...,Xn be a random sample from
Poisson(θ).
Use the factorization theorem to find the sufficient...
Let X1,...,Xn be a random sample from
Poisson(θ).
Use the factorization theorem to find the sufficient statistic T
for θ.
Let X1, X2, . . . , Xn be a random sample of size n from...
Let X1, X2, . . . , Xn be a random sample of size n from a
distribution with variance σ^2. Let S^2 be the sample variance.
Show that E(S^2)=σ^2.
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) =...
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then,
Show the density of the statistic T = X(n) is given by
FX(n) (x) = n/ab * (x/a)^{n/(b-1}} for 0 <= x <=
a ; otherwise zero.
# using the following
P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x
),
Then assume...
Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function...
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...