A random variable X has probability density function f(x)=(a-1)x^(-a),for x>=1.
(a) For independent observations x1,...,xn show that the log-likelihood is given by,
l(a;x1,...,xn)=nlog(a-1)-a
(b) Hence derive an expression for the maximum likelihood estimate
for ↵.
(c) Suppose we observe data such that n = 6 and 6 i=1 log(xi) =
12. Show that the associated maximum likelihood estimate for ↵ is
given by aˆ ↵ =1 .5.
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A random variable X has probability density function
f(x)=(a-1)x^(-a),for x>=1.
(a) For independent observations x1,...,xn show...
3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a)...
3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a) For independent observations In show that the log-likelihood is given by, (b) Hence derive an expression for the maximum likelihood estimate for α. (c) Suppose we observe data such that n 6 and Σ61 log(xi) 12. Show that the associated maximum likelihood estimate for α is given by α = 1.5.
. Suppose that x1, . . . , xn are a random sample having probability density...
. Suppose that x1, . . . , xn are a random sample having probability density function f(x; θ) = (θ + 1)x^θ , 0 < x < (1) Here the parameter θ > 0. (a) Determine the log-likelihood, l(θ), and a 1-dimensional sufficient statistic. (b) Show that P(Xi ≤ b; θ) = b θ+1 for f(x; θ) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of...
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 < x < 1, > 0 a) Find a sufficient statistic for . Is the statistic minimal sufficient? b) Find the MLE for and verify that it is a function of the statistic in a) c) Find IX() and hence give the CRLB for an unbiased estimator of . pdf means probability distribution function We were unable to transcribe this...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) =...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) = for x>0 (a) Use method of moments to find estimators for µ and µ^2 . (b) What is the log likelihood as a function of µ after observing X1 = x1, . . . , Xn = xn? (c) Find the MLEs for µ and µ^2 . Are they the same as those you find in part (a)? (d) According to the Central Limit...
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)...
Suppose that x1, . . . , xn are a random sample having probability density function...
Suppose that x1, . . . , xn are a random sample having probability density function f(x; θ) = (θ + 1)x^θ , 0 < x < 1. (1) Here the parameter θ > 0. (a) Show that P(Xi ≤ b; θ) = b^(θ+1) for f(x; θ) given in (1). (b) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the xi are observed. For the rest of the observations, it...
X1,...,Xn are IID with N(0,2). a) Determine the mean and variance for (X (subscript 1)^2) b)...
X1,...,Xn are IID with N(0,2).
a) Determine the mean and variance for (X (subscript 1)^2)
b) Show
sqrt(n)
* [ log ( 1/n ∑(from i=1 to n)
Xi2) − log(σ2 ) ] d → N(0, 2).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ...
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter...
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log likelibood, 10), and a 1- dimensional (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(XiS b;0) = +1 for f(x:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the r, are observed. For the rest of the observations,...
let x1.........xn be independent where xi is normally distributed with unknown mean u and unknown variance 0...
let x1.........xn be independent where xi is normally
distributed with unknown mean u and unknown
variance 0
find the UMP test for testing
=0 against 0
when it is assumed that
is known.=1
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a) For independent observations In show that the log-likelihood is given by, (b) Hence derive an expression for the maximum likelihood estimate for α. (c) Suppose we observe data such that n 6 and Σ61 log(xi) 12. Show that the associated maximum likelihood estimate for α is given by α = 1.5.
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)...
X1,...,Xn are IID with N(0,2).
a) Determine the mean and variance for (X (subscript 1)^2)
b) Show
sqrt(n)
* [ log ( 1/n ∑(from i=1 to n)
Xi2) − log(σ2 ) ] d → N(0, 2).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log likelibood, 10), and a 1- dimensional (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(XiS b;0) = +1 for f(x:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the r, are observed. For the rest of the observations,...
let x1.........xn be independent where xi is normally
distributed with unknown mean u and unknown
variance 0
find the UMP test for testing
=0 against 0
when it is assumed that
is known.=1
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
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