PART CHomework Answers
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PART C Problem 3. Let Xi.X^...be i.d. sample from a Rayleigh distribution, with parameter > 0:...
Problem 2. Consider a random sample of size n from a two-parameter distribution with parameter 0...
Problem 2. Consider a random sample of size n from a two-parameter distribution with parameter 0 unknown and parameter η known. The population density function is (xi - T) (a) Find the likelihood function simplifying it as much as possible. Likelihood
Let X1, X2, ..., Xn be a random sample from X which has pdf depending on...
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Let X1, X2, ..., Xn be a random sample of size n from the distribution with...
Let X1, X2, ..., Xn be a random sample of size n from the
distribution with probability density function
To answer this question, enter you answer as a formula. In
addition to the usual guidelines, two more instructions for this
problem only : write
as single variable p and
as m. and these can be used as inputs of functions as usual
variables e.g log(p), m^2, exp(m) etc. Remember p represents the
product of
s only, but will not work...
Let X1,X2,...,Xn denote a random sample from the Rayleigh distribution given by f(x) = ...
Let X1,X2,...,Xn denote a random sample from the Rayleigh distribution given by f(x) = (2x θ)e−x2 θ x > 0; 0, elsewhere with unknown parameter θ > 0. (A) Find the maximum likelihood estimator ˆ θ of θ. (B) If we observer the values x1 = 0.5, x2 = 1.3, and x3 = 1.7, find the maximum likelihood estimate of θ.
Negative binomial probability function: is the mean is the dispersion parameter Let there be two groups...
Negative binomial probability function:
is the mean
is the dispersion
parameter
Let there be two groups with numbers and means of
1) Write down the log-likelihood for the full model
2) Calculate the likelihood equations and find the general form
of the MLE for and
3) Write down the likelihood function in the reduced model (ie.
assuming )
and derive the MLE for in general
terms
4) Using the above answers only, give the MLE and standard error
for where...
Let X1,..., X, be an i.i.d. sample from a Rayleigh distribution with parameter e > 0:...
Let X1,..., X, be an i.i.d. sample from a Rayleigh distribution with parameter e > 0: f(x\C) = e ==/(20?), x20 (This is an alternative parametrization of that of Example A in Section 3.6.2.) a. Find the method of moments estimate of e. b. Find the mle of C. Find the asymptotic variance of the mle.
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
A Pareto distribution is often used in economics to explain a distribution of wealth. Let a...
A Pareto distribution is often used in economics to explain a
distribution of wealth. Let a random variable X have a Pareto
distribution with parameter θ so that its probability distribution
function is
for
and 0 otherwise. The parameters and
are
known and fixed; is a constant to
be determined.
a) Assuming that
find the expected value and variance of ?
b) Show that for 3 ≥ θ > 2 the Pareto distribution has a
finite mean but infinite variance,...
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0...
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0...
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
Problem 2. Consider a random sample of size n from a two-parameter distribution with parameter 0 unknown and parameter η known. The population density function is (xi - T) (a) Find the likelihood function simplifying it as much as possible. Likelihood
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Let X1, X2, ..., Xn be a random sample of size n from the
distribution with probability density function
To answer this question, enter you answer as a formula. In
addition to the usual guidelines, two more instructions for this
problem only : write
as single variable p and
as m. and these can be used as inputs of functions as usual
variables e.g log(p), m^2, exp(m) etc. Remember p represents the
product of
s only, but will not work...
Negative binomial probability function:
is the mean
is the dispersion
parameter
Let there be two groups with numbers and means of
1) Write down the log-likelihood for the full model
2) Calculate the likelihood equations and find the general form
of the MLE for and
3) Write down the likelihood function in the reduced model (ie.
assuming )
and derive the MLE for in general
terms
4) Using the above answers only, give the MLE and standard error
for where...
Let X1,..., X, be an i.i.d. sample from a Rayleigh distribution with parameter e > 0: f(x\C) = e ==/(20?), x20 (This is an alternative parametrization of that of Example A in Section 3.6.2.) a. Find the method of moments estimate of e. b. Find the mle of C. Find the asymptotic variance of the mle.
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
A Pareto distribution is often used in economics to explain a
distribution of wealth. Let a random variable X have a Pareto
distribution with parameter θ so that its probability distribution
function is
for
and 0 otherwise. The parameters and
are
known and fixed; is a constant to
be determined.
a) Assuming that
find the expected value and variance of ?
b) Show that for 3 ≥ θ > 2 the Pareto distribution has a
finite mean but infinite variance,...
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
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