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?

  • 1
    $\begingroup$ 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 $\endgroup$
    – Glen_b
    Apr 20, 2022 at 4:15
  • $\begingroup$ 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. $\endgroup$
    – Glen_b
    Apr 20, 2022 at 4:38
  • $\begingroup$ @Glen_b I fixed it. Thanks for catching that typo. I meant to type a $\endgroup$
    – Pie-ton
    Apr 20, 2022 at 19:12
  • 3
    $\begingroup$ Using the visually similar "$a$" and "$\alpha$" within the same equation is just asking for trouble... $\endgroup$
    – whuber
    Apr 20, 2022 at 19:48
  • 1
    $\begingroup$ Please see stats.stackexchange.com/a/566564/919, which illustrates the basic concept and provides general-purpose R code. $\endgroup$
    – whuber
    Apr 20, 2022 at 19:55

1 Answer 1


All your entries in your dataset contribute to the likelihood function. If you maximize your likelihood function, then you will get the most likely values for your shape and scale parameters.

For that maximum value, the shape and scale can only have one specific value. Any other value will produce a lower value of the likelihood function.

Let's say you lower your likelihood function by a Chi-squared distribution, with 2 degrees of freedom, and a confidence interval of 90% (approx 1.35), then the shape and scale parameters have more "room" to take on a different value.

Basically, by lowering your likelihood function, you enable the shape and scale parameters to be other values, within a certain range.

See also: https://reliawiki.org/index.php/Confidence_Bounds#Likelihood_Ratio_Confidence_Bounds


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