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I am currently working on a problem in which I have observations $y_{i}$ that are distributed,

$y_{i} \sim \textrm{Exponential}(\beta = ax_{i})\cdot T[b, \infty)$

where, $\beta$ is the rate parameter of the exponential distribution, $T$ describes that the Exponential distribution is left-truncated to the interval $[b, \infty)$, and I have observations of the independent variable $x_{i}$ associated with each observation $y_{i}$.

My goal is to estimate the parameters, $a$ and $b$. I can do this in a Bayesian framework and get credible intervals for both parameters easily. However, I'd like to also be able to fit this model in a maximum likelihood framework. My main issue is getting a confidence interval on $b$. As with most minimum/maximum problems, I know the maximum likelihood estimate for the minimum of the distribution, $b$, is likely to be the minimum observed $y_{i}$. Furthermore, I know that the minimum value of $y_{i}$ provides a maximum for $b$. If I fix $b$ to be the minimum $y_{i}$, I get sensible estimates and confidence intervals of $a$ using standard software for maximum likelihood estimation (the 'bbmle' package in R using the function 'mle2'). However, I'm having trouble coming up with a way to get a confidence interval or lower interval value for $b$. I've found a paper 'Some theorems relevant to life testing from an exponential distribution' by Epstein and Sobel (1954) in The Annals of Mathematical Statistics (https://projecteuclid.org/download/pdf_1/euclid.aoms/1177728793) that describes a lower interval estimate when samples are iid from a left-truncated or 'two-parameter' exponential distribution, but, in my case, the observations are not iid. Any help in constructing a confidence interval or lower interval estimate for $b$ would be greatly appreciated whether anyone has an analytical or computational solution.

Kcoblentz
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  • whuber has an excellent answer [here](https://stats.stackexchange.com/questions/48897/maximum-likelihood-estimators-for-a-truncated-distribution) – Demetri Pananos Jul 03 '20 at 21:04
  • @DemetriPananos AFAICS that talks about MLEs not about confidence intervals ... – Ben Bolker Jul 03 '20 at 22:05
  • Maybe I should have expanded and said: once you have MLEs, you can obtain confidence sets by inverting the information matrix. – Demetri Pananos Jul 03 '20 at 22:32
  • @DemitriPananos, I'm not sure that the Wald (inverse information matrix) confidence intervals work in this situation; because the MLE of the lower limit is equal to the minimum value of $x$, and the likelihood is 0 for $b>\min(x)$, the likelihood surface isn't locally quadratic ... – Ben Bolker Jul 04 '20 at 21:05
  • Thanks, @DemetriPananos. I'm also not sure of the relevance of the answer given in the link. Are they not assuming that the upper and lower limits of the distribution are known? This strategy may help for getting the estimate of a in my problem, but I'm not sure how it helps get an interval estimate on the lower bound of the distribution. – Kcoblentz Jul 05 '20 at 20:30

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