1
$\begingroup$

Problem:

I estimated the shape $\alpha$ and scale $\lambda$ parameter of the Weibull distribution using Bayesian methods. That gave me a marginal posterior distributions for both parameters. Calculating the highest posterior density (HPD) I got two credibility intervals and a point estimate for both paramters.

Using the point estimates I can calculate the probability that a Weibull distributed variable $X$ will yield a value smaller than $x$.

$F(x) = 1 - exp(- (x/\lambda)^\alpha)$

Now I also want to create a credibility interval for that probability.

Question:

How can I calculate the credibility interval for $F(x)$?

Example

Lets say that using the HPD I got credibility intervals for $\alpha$ and $\lambda$ that are [0.52 ; 0.74] and [156.3 ; 187.8] respectively. The point estomates are 0.61 and 175.5.

Using the point estimates and $x = 25$ one gets $F(25) = 0.262584$

What I did so far:

I used the credibility intervals of the parameters and calculated $F(25)$ for all possible combinations of the interval values and point estimates.

  • 0.52 & 156.3 $\rightarrow$ $F(25) = 0.3199171$
  • 0.52 & 175.5 $\rightarrow$ $F(25) = 0.3044124$
  • 0.52 & 178.8 $\rightarrow$ $F(25) = 0.3019741$
  • ...
  • 0.74 & 178.8 $\rightarrow$ $F(25) = 0.2080028$

I then used the minimum of these values as the lower probability interval and the maximum as the upper probability interval. But I don't know if that is a valid thing to do.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ What you did is definitely not a valid approach. I answered the question here: stats.stackexchange.com/questions/88515/… in a different context. $\endgroup$
    – jaradniemi
    Feb 20, 2015 at 17:13
  • $\begingroup$ You cannot use point estimates as pluggins in a Bayesian approach but need to infer the posterior on $F(25)$ from the posterior on $\alpha$ and $\lambda$. $\endgroup$
    – Xi'an
    Feb 20, 2015 at 18:44
  • $\begingroup$ Thanks for your input. Because I have to simulate the posterior using the R package LaplacesDemon I will add the estimation of F(x) for each iteration of $\alpha$ and $\lambda$. That should give me a distribution for the F(x) value from which I can get the credibility interval. $\endgroup$
    – elevendollar
    Feb 22, 2015 at 12:42
  • 1
    $\begingroup$ I just realized that the approach is not feasible as I have to estimate $F(X)$ for 3650 values of $X$ and also calculate the credibility interval for each value of $X$. That will take way to long considering the time it already takes to simulate the posterior of $\alpha$ and $\lambda$. @Xi'an: Can you give me a hint on how to infer the posterior of $F(25)$ from the posterior of $\alpha$ and $\lambda$? $\endgroup$
    – elevendollar
    Feb 23, 2015 at 8:19

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.