Question

The random variable X is distributed with pdf

f_{X}(x, θ) = c*x*exp(-(x/θ)^{2}), where x>0
and θ>0. (Please note the equation includes the term
-(x/θ)^{2} )

a) What is the constant c?

b) We consider parameter θ is a number. What is MLE and MOM of θ? Assume you have an i.i.d. sample. Is MOM unbiased?

c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare the variance of MOM with Crameer-Rao Lower Bound (CRLB).

Answer #1

The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please
note the equation includes the term -(x/θ)2 )
Calculate the probability of X1 < X2,
i.e. P(X1 < X2, θ).

The random variable X is distributed with pdf fX(x,
θ) = (2/θ^2)*x*exp(-(x/θ)2), where x>0 and
θ>0. Please note the term within the exponential is
-(x/θ)^2 and the first term includes a θ^2.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
b) Find the best estimator, i.e. MVUE, of θ?

The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0.
Calculate the probability of X1 < X2,
i.e. P(X1 < X2, θ).

Let X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?

Suppose the random variable X follows the Poisson P(m) PDF, and
that you have a random sample X1, X2,...,Xn from it. (a)What is the
Cramer-Rao Lower Bound on the variance of any unbiased estimator of
the parameter m? (b) What is the maximum likelihood estimator
ofm?(c) Does the variance of the MLE achieve the CRLB for all
n?

Suppose X1,..., Xn are iid with pdf f(x;θ) = 2x / θ2,
0 < x ≤ θ. Find I(θ) and the Cramér-Rao lower bound for the
variance of an unbiased estimator for θ.

Let X1,...,Xn be iid
exp(θ) rvs.
(a) Compute the pdf of Xmin.
I have the pdf
(b) Create an unbiased estimator for θ based on Xmin.
Compute the variance of the resulting estimator.
(c) Perform a Monte Carlo simulation of N= 10,0000 samples of
your unbiased estimator from part (b) using θ = 2 and n = 100 to
validate your answer. Include a histogram of the samples.
(d) Which is more efficient: your estimator from part (b) or the...

Let X1,...,Xn be iid
exp(θ) rvs.
(a) Compute the pdf of Xmin.
(b) Create an unbiased estimator for θ based on Xmin.
Compute the variance of the resulting estimator.
(c) Perform a Monte Carlo simulation of N= 10,0000 samples of
your unbiased estimator from part (b) using θ = 2 and n = 100 to
validate your answer. Include a histogram of the samples.
(d) Which is more efficient: your estimator from part (b) or the
MLE for θ?
(e)...

Let X_1,…, X_n be a random sample from the Bernoulli
distribution, say P[X=1]=θ=1-P[X=0].
and
Cramer Rao Lower Bound of θ(1-θ)
=((1-2θ)^2 θ(1-θ))/n
Find the UMVUE of θ(1-θ) if such exists.
can you proof [part (b) ] using (Leehmann Scheffe
Theorem step by step solution) to proof
[∑X1-nXbar^2 ]/(n-1) is the umvue , I have the key
solution below
x is complete and sufficient.
S^2=∑ [X1-Xbar ]^2/(n-1) is unbiased estimator of θ(1-θ) since
the sample variance is an unbiased estimator of the...

3. Let X be a continuous random variable with PDF
fX(x) = c / x^1/2, 0 < x < 1.
(a) Find the value of c such that fX(x) is indeed a PDF. Is this
PDF bounded?
(b) Determine and sketch the graph of the CDF of X.
(c) Compute each of the following:
(i) P(X > 0.5).
(ii) P(X = 0).
(ii) The median of X.
(ii) The mean of X.

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