# How to calculate Weibull confidence interval using chi-square distribution?

I have a dataset, which I assume has a weibull distribution :

vec <- c(90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84,
44, 59, 29, 118, 25, 156, 310, 76, 26, 44, 23, 62,
130, 208, 70, 101, 208)


and want to know how to calculate the Confidence Interval for parameters scale and shape using a chi-squared test. I have calculated alpha(=scale). I also know how to just extract Confidence Interval from the packages such as fitdistrplus. I know the formula for Confidence Interval is

D(a) = 2[log L(a_hat) − log L(a)] ≤ χ2(α)


but I don't know how this formula works. Can somebody help?

• You are using $\alpha$ for two distinct things in: "2[log L(α.hat) − log L(α)] ≤ χ2(α)"... On the left of the inequality it's a parameter of the Weibull but on the right it represents an upper-tail quantile (the value such that the probability of the relevant chi-squared variable being to the left of it is at least $1-\alpha$). DON'T use the same symbol for two different things. Since $\alpha$ is so entrenched as a convention on the right hand side, I strongly suggest you choose a different symbol (something other than $\alpha$) for the scale parameter of the Weibull Commented Apr 20, 2022 at 4:15
• This will be necessary for an answer to be able to respond suitably while still using symbols in a similar way to you. Please fix. Commented Apr 20, 2022 at 4:38
• @Glen_b I fixed it. Thanks for catching that typo. I meant to type a Commented Apr 20, 2022 at 19:12
• Using the visually similar "$a$" and "$\alpha$" within the same equation is just asking for trouble...
– whuber
Commented Apr 20, 2022 at 19:48
• Please see stats.stackexchange.com/a/566564/919, which illustrates the basic concept and provides general-purpose R code.
– whuber
Commented Apr 20, 2022 at 19:55