Bayesian hypothesis testing using a Pareto distribution
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I am having trouble with the following problem - the example asks to consider a Pareto distribution with scale $theta$ and shape $a$ to give the distribution
$$pi(theta)=begin{cases}frac{alphatheta_0^alpha}{theta^{alpha+1}}&,,thetagetheta_0\0&,,theta<theta_0end{cases}$$
where $theta >0$ and $alpha>0$.
When $alpha>2$ the mean is $frac{alphatheta_0}{alpha-1}$ and variance $frac{alphatheta_0^2}{(alpha-1)^2(alpha-2)}$.
We are then asked to consider $x_1, ldots , x_n$ observations from an uniform distribution on the interval $(0, theta)$. The prior distribution for $theta$ is Pareto$(alpha,theta_0)$ with $alpha$ known, and the corresponding
posterior distribution for $theta$ is a Pareto with scale $t = max(theta_0, max_{i=1,...,n} x_i)$ and shape $alpha + n$. To then set a prior such that the mean is $frac32$ and variance $frac34$ and finally test the following hypotheses using a Bayesian approach:
$H_0 : theta ≥ 3$ and $H_1 : theta >3$ with the following data $x_1 = 1.2, x_2 = 1.8, x_3 = 1.5, x_4 =2$.
My working so far:
Through equating the mean and variance to the values given in the example I have deduced the prior distribution is a Pareto with parameters $3$ and $1$. However I am now unsure where to go from here. This is my first time learning Bayesian modelling and cannot seem to find too much online around Uniform - Pareto hypothesis testing so any help would be greatly appreciated, thanks.
statistics statistical-inference bayesian hypothesis-testing
edited Nov 22 at 19:08
StubbornAtom
5,09511138
asked Nov 22 at 18:58
C. Jones
276
add a comment |
up vote
0
down vote
favorite
I am having trouble with the following problem - the example asks to consider a Pareto distribution with scale $theta$ and shape $a$ to give the distribution
$$pi(theta)=begin{cases}frac{alphatheta_0^alpha}{theta^{alpha+1}}&,,thetagetheta_0\0&,,theta<theta_0end{cases}$$
where $theta >0$ and $alpha>0$.
When $alpha>2$ the mean is $frac{alphatheta_0}{alpha-1}$ and variance $frac{alphatheta_0^2}{(alpha-1)^2(alpha-2)}$.
We are then asked to consider $x_1, ldots , x_n$ observations from an uniform distribution on the interval $(0, theta)$. The prior distribution for $theta$ is Pareto$(alpha,theta_0)$ with $alpha$ known, and the corresponding
posterior distribution for $theta$ is a Pareto with scale $t = max(theta_0, max_{i=1,...,n} x_i)$ and shape $alpha + n$. To then set a prior such that the mean is $frac32$ and variance $frac34$ and finally test the following hypotheses using a Bayesian approach:
$H_0 : theta ≥ 3$ and $H_1 : theta >3$ with the following data $x_1 = 1.2, x_2 = 1.8, x_3 = 1.5, x_4 =2$.
My working so far:
Through equating the mean and variance to the values given in the example I have deduced the prior distribution is a Pareto with parameters $3$ and $1$. However I am now unsure where to go from here. This is my first time learning Bayesian modelling and cannot seem to find too much online around Uniform - Pareto hypothesis testing so any help would be greatly appreciated, thanks.
statistics statistical-inference bayesian hypothesis-testing
edited Nov 22 at 19:08
StubbornAtom
5,09511138
asked Nov 22 at 18:58
C. Jones
276
Could try Cross-Validated for getting an answer.
– StubbornAtom
Nov 30 at 14:21
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am having trouble with the following problem - the example asks to consider a Pareto distribution with scale $theta$ and shape $a$ to give the distribution
$$pi(theta)=begin{cases}frac{alphatheta_0^alpha}{theta^{alpha+1}}&,,thetagetheta_0\0&,,theta<theta_0end{cases}$$
where $theta >0$ and $alpha>0$.
When $alpha>2$ the mean is $frac{alphatheta_0}{alpha-1}$ and variance $frac{alphatheta_0^2}{(alpha-1)^2(alpha-2)}$.
We are then asked to consider $x_1, ldots , x_n$ observations from an uniform distribution on the interval $(0, theta)$. The prior distribution for $theta$ is Pareto$(alpha,theta_0)$ with $alpha$ known, and the corresponding
posterior distribution for $theta$ is a Pareto with scale $t = max(theta_0, max_{i=1,...,n} x_i)$ and shape $alpha + n$. To then set a prior such that the mean is $frac32$ and variance $frac34$ and finally test the following hypotheses using a Bayesian approach:
$H_0 : theta ≥ 3$ and $H_1 : theta >3$ with the following data $x_1 = 1.2, x_2 = 1.8, x_3 = 1.5, x_4 =2$.
My working so far:
Through equating the mean and variance to the values given in the example I have deduced the prior distribution is a Pareto with parameters $3$ and $1$. However I am now unsure where to go from here. This is my first time learning Bayesian modelling and cannot seem to find too much online around Uniform - Pareto hypothesis testing so any help would be greatly appreciated, thanks.
statistics statistical-inference bayesian hypothesis-testing
edited Nov 22 at 19:08
StubbornAtom
5,09511138
asked Nov 22 at 18:58
C. Jones
276
I am having trouble with the following problem - the example asks to consider a Pareto distribution with scale $theta$ and shape $a$ to give the distribution
$$pi(theta)=begin{cases}frac{alphatheta_0^alpha}{theta^{alpha+1}}&,,thetagetheta_0\0&,,theta<theta_0end{cases}$$
where $theta >0$ and $alpha>0$.
When $alpha>2$ the mean is $frac{alphatheta_0}{alpha-1}$ and variance $frac{alphatheta_0^2}{(alpha-1)^2(alpha-2)}$.
We are then asked to consider $x_1, ldots , x_n$ observations from an uniform distribution on the interval $(0, theta)$. The prior distribution for $theta$ is Pareto$(alpha,theta_0)$ with $alpha$ known, and the corresponding
posterior distribution for $theta$ is a Pareto with scale $t = max(theta_0, max_{i=1,...,n} x_i)$ and shape $alpha + n$. To then set a prior such that the mean is $frac32$ and variance $frac34$ and finally test the following hypotheses using a Bayesian approach:
$H_0 : theta ≥ 3$ and $H_1 : theta >3$ with the following data $x_1 = 1.2, x_2 = 1.8, x_3 = 1.5, x_4 =2$.
My working so far:
Through equating the mean and variance to the values given in the example I have deduced the prior distribution is a Pareto with parameters $3$ and $1$. However I am now unsure where to go from here. This is my first time learning Bayesian modelling and cannot seem to find too much online around Uniform - Pareto hypothesis testing so any help would be greatly appreciated, thanks.
statistics statistical-inference bayesian hypothesis-testing
statistics statistical-inference bayesian hypothesis-testing
edited Nov 22 at 19:08
StubbornAtom
5,09511138
asked Nov 22 at 18:58
C. Jones
276
edited Nov 22 at 19:08
StubbornAtom
5,09511138
asked Nov 22 at 18:58
C. Jones
276
edited Nov 22 at 19:08
StubbornAtom
5,09511138
edited Nov 22 at 19:08
StubbornAtom
5,09511138
edited Nov 22 at 19:08
StubbornAtom
5,09511138
5,09511138
asked Nov 22 at 18:58
C. Jones
276
asked Nov 22 at 18:58
C. Jones
276
asked Nov 22 at 18:58
C. Jones
276
276
Could try Cross-Validated for getting an answer.
– StubbornAtom
Nov 30 at 14:21
add a comment |
Could try Cross-Validated for getting an answer.
– StubbornAtom
Nov 30 at 14:21
Could try Cross-Validated for getting an answer.
– StubbornAtom
Nov 30 at 14:21
Could try Cross-Validated for getting an answer.
– StubbornAtom
Nov 30 at 14:21
add a comment |
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Could try Cross-Validated for getting an answer.
– StubbornAtom
Nov 30 at 14:21