# Confidence limits for constrained penalized log likelihood model

I am estimating parameter $$\beta$$ as:

\begin{align} \hat \beta &= \mathop{\mathrm{arg\,max}}_\beta \;\; l(\beta;X,y) - \frac{\lambda}{2}\left(\tilde y-g(\beta,\tilde X)\right)^\prime C^\prime C\left(\tilde y-g(\beta,\tilde X)\right)\\ &s.t.\\ & \;\;\;\;\; A\beta\leq 0\\ &where\\ & \tilde y = (y^\prime,y^{\ast\,\prime})^\prime\\ &\tilde X = (X^\prime,X^{\ast\,\prime})^\prime\\ & g(p,Q) = \exp(Qp) \end{align}

How should I go about estimating the confidence limits on $$\hat \beta$$ theoretically?

## This question has an open bounty worth +50 reputation from A Gore ending in 20 hours.

Looking for an answer drawing from credible and/or official sources.

• $l(\beta;X,y) = ?$ – user158565 2 days ago
• Most software would supply an in-buildt method for getting a standard error e.g. using the delta-method. – Björn 9 hours ago

You can use bootstrap to estimate $$\beta$$ confidence intervals. Otherwise, you'll need to derive the variance of $$\beta$$.