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The Exact Confidence Interval for an MLE of a Gamma Distribution

Confidence interval of the parameter of $exp$ and normal distribution from MLE?Monte Carlo Confidence Interval for Standard DeviationCalculating this confidence Interval and making assumptionsMLE, Confidence Interval, and Asymptotic DistributionsConfidence interval for MLE estimate for non-regular likelihood functions95% Confidence Interval for $lambda$Constructing a 95% Confidence Interval (using output from r-studio)constructing a 95% confidence interval - manipulating inequalitiesProportion Confidence IntervalNormal approximation of MLE of Poisson distribution and confidence interval

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Above here is the information I've been given for one of my seminar questions, so far I have calculated the fisher information and from there I computed the asymptotic distribution for $hatlambda$ is:

$$lambda_n = Nleft(lambda,frac1nI(lambda)right)
= Nleft(lambda, fraclambda^2dnright)$$

After that I derived the 90% confidence interval for λ as:

$$T_1,2=hatlambda pm fracz_alpha/2sqrtnI(lambda)= fracdnx pm 1.64 fraclambdasqrtnd$$

And from here I need to find the exact 90% confidence interval but this is where I'm stuck. Can anyone provide any assistance?

confidence-interval gamma-distribution

share|cite|improve this question

asked Mar 29 at 1:52

kingking

425

$endgroup$

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  $begingroup$
  You cannot find the exact CI from an asymptotic CI; you have to start from scratch. You are given a hint on what to do in the last line of your question. That is pretty much the answer.
  $endgroup$
  – StubbornAtom
  Mar 29 at 6:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So how exactly would I do this, would I need to calculate the likelihood of this new function and go from there? @StubbornAtom
  $endgroup$
  – king
  Mar 29 at 10:20
  
  

  
  
  
  


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  $begingroup$
  Can you verify that $2lambdasum X_i$ has a chi-square distribution? If you can, then this is your pivotal quantity from which the CI follows.
  $endgroup$
  – StubbornAtom
  Mar 29 at 10:25
  

  
  
  
  

add a comment | 

0

$begingroup$

Above here is the information I've been given for one of my seminar questions, so far I have calculated the fisher information and from there I computed the asymptotic distribution for $hatlambda$ is:

$$lambda_n = Nleft(lambda,frac1nI(lambda)right)
= Nleft(lambda, fraclambda^2dnright)$$

After that I derived the 90% confidence interval for λ as:

$$T_1,2=hatlambda pm fracz_alpha/2sqrtnI(lambda)= fracdnx pm 1.64 fraclambdasqrtnd$$

And from here I need to find the exact 90% confidence interval but this is where I'm stuck. Can anyone provide any assistance?

confidence-interval gamma-distribution

share|cite|improve this question

asked Mar 29 at 1:52

kingking

425

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You cannot find the exact CI from an asymptotic CI; you have to start from scratch. You are given a hint on what to do in the last line of your question. That is pretty much the answer.
  $endgroup$
  – StubbornAtom
  Mar 29 at 6:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So how exactly would I do this, would I need to calculate the likelihood of this new function and go from there? @StubbornAtom
  $endgroup$
  – king
  Mar 29 at 10:20
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Can you verify that $2lambdasum X_i$ has a chi-square distribution? If you can, then this is your pivotal quantity from which the CI follows.
  $endgroup$
  – StubbornAtom
  Mar 29 at 10:25
  

  
  
  
  

add a comment | 

$begingroup$

Above here is the information I've been given for one of my seminar questions, so far I have calculated the fisher information and from there I computed the asymptotic distribution for $hatlambda$ is:

$$lambda_n = Nleft(lambda,frac1nI(lambda)right)
= Nleft(lambda, fraclambda^2dnright)$$

After that I derived the 90% confidence interval for λ as:

$$T_1,2=hatlambda pm fracz_alpha/2sqrtnI(lambda)= fracdnx pm 1.64 fraclambdasqrtnd$$

And from here I need to find the exact 90% confidence interval but this is where I'm stuck. Can anyone provide any assistance?

confidence-interval gamma-distribution

share|cite|improve this question

asked Mar 29 at 1:52

kingking

425

$endgroup$

share|cite|improve this question

asked Mar 29 at 1:52

kingking

425

share|cite|improve this question

asked Mar 29 at 1:52

kingking

425

asked Mar 29 at 1:52

kingking

425

asked Mar 29 at 1:52

kingking

425