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.


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


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.

  • 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 '15 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 '15 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 '15 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 '15 at 8:19

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