# Limiting distribution of MLE when true value is on the boundary of the parameter space

We know the nice properties and consistency results of the MLE estimator don't hold when the true value is on the boundary of the parameter space, for eg. if the parameter space is $[0, \inf)$, so while we can still find $\hat{\theta}_{MLE}$, how do you go about finding $\hat{\theta}_{MLE} -\theta$ if $\theta$ might be zero?

The usual Taylor expansion of first derivative around true value is no longer equal to zero, so how does one proceed instead?

https://stats.stackexchange.com/a/240513/21465 : For some context.

• What do you mean by "proceed?" In any minimization problem, either the minimum value exists or it doesn't. If it does, it's either on the boundary or somewhere in between. So for example, you could numerically verify which of your points is the minimum by comparing their values. – Alex R. Oct 17 '16 at 22:11
• Here is a paper trying to address to your Q: ecommons.cornell.edu/bitstream/handle/1813/31683/… and another one: tandfonline.com/doi/pdf/10.1198/… – kjetil b halvorsen Dec 30 '18 at 10:57