Question

Let X1, ..., Xn be a random sample from a probability density function: f(x; β) = βxβ−1 for 0 < x < 1 (where β > 0).

a) Obtain the moment estimator for β.

b) Obtain the maximum likelihood estimator for β.

c) Is each of the estimators of a) and b) a sufficient statistic?

Answer #1

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 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...

6. Let θ > 1 and let X1, X2, ..., Xn be a random sample from
the distribution with probability density function f(x; θ) =
1/(xlnθ) , 1 < x < θ.
a) Obtain the maximum likelihood estimator of θ, ˆθ.
b) Is ˆθ a consistent estimator of θ? Justify your answer.

Let B > 0 and let X1 , X2 , … , Xn be a random sample from
the distribution with probability density function.
f( x ; B ) = β/ (1 +x)^ (B+1), x > 0, zero otherwise.
(i) Obtain the maximum likelihood estimator for B, β ˆ .
(ii) Suppose n = 5, and x 1 = 0.3, x 2 = 0.4, x 3 = 1.0, x 4 =
2.0, x 5 = 4.0. Obtain the maximum likelihood...

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 the distribution,
f(x; θ) = (θ + 1)x^ −θ−2 , x > 1, θ > 0. Find the maximum
likelihood estimator of θ based on a random sample of size n
above

4. 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 β2.

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.

Let X1,..., Xn be a random sample from a
distribution with pdf as follows:
fX(x) = e^-(x-θ) , x > θ
0 otherwise.
Find the sufficient statistic for θ.
Find the maximum likelihood estimator of θ.
Find the MVUE of θ,θˆ
Is θˆ a consistent estimator of θ?

1. Let X1, X2, . . . , Xn be a random sample from a distribution
with pdf f(x, θ) = 1 3θ 4 x 3 e −x/θ , where 0 < x < ∞ and 0
< θ < ∞. Find the maximum likelihood estimator of ˆθ.

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