How can I find an interval estimate for the mean of a Weibull distribution?

I have a sample of n = 75 taken from a Weibull distribution and have computed mle estimates for the scale a and shape b parameter.
The mean of a Weibull distribution (2 parameter) is given as

u = a^(-1/b)*gamma(1+ 1/b)

In which case I can find an estimate for u by simply plugging in the mle estimate for a and b.

But I am interested in finding a confidence interval for u, not just a point estimate, but I have have no information about the distribution of u or its point estimate.
Could anyone please help me out here, point me to some textbook/paper etc. , I can't find any information about this anywhere?
Last edited:
You can refer the following papers and books

Yuan, F. (2018). Parameter estimation for bivariate Weibull distribution using generalized moment method for reliability evaluation. Quality and Reliability Engineering International, 34(4), 631-640.
Rinne, H. (2008). The Weibull distribution: a handbook. Chapman and Hall/CRC.
Lawless, J. F. (1978). Confidence interval estimation for the Weibull and extreme value distributions. Technometrics, 20(4), 355-364.
Thoman, D. R., Bain, L. J., & Antle, C. E. (1969). Inferences on the parameters of the Weibull distribution. Technometrics, 11(3), 445-460.