I am trying to find the asymptotic distribution of a maximum estimator loss function of the type: $$ \hat{\theta} = \arg \max_\tilde{\theta} M_n(\tilde{\theta},x) $$ $$ \theta = \arg \max_\tilde{\theta} M(\tilde{\theta},x) $$ Having settled that $\sqrt{n} (\hat{\theta} - \theta) \to N(0,V)$ where $\theta$ is the true value, my approach to determine the asymptotic distribution of the loss function is a simple taylor expansion of $M$ around $\theta$ evaluated at $\hat{\theta}$: $$ n ( M(\hat{\theta},x) - M(\theta,x) ) = \sqrt{n} \nabla_\theta M(\theta,x) \sqrt{n} (\hat{\theta} - \theta) + n R(\theta,\hat{\theta},x) $$ Here I have trouble since $\nabla_\theta M(\theta,x) = 0$ since its an interior maximizer. I suppose an alternative would be using a taylor expansion around $\hat{\theta}$ instead of $\theta$, which would yield: $$ n ( M(\hat{\theta},x) - M(\theta,x) ) = \sqrt{n} \nabla_\theta M(\hat{\theta},x) \sqrt{n} (\hat{\theta} - \theta) - n R(\theta,\hat{\theta},x) $$ which I suppose under suitable assumptions would provide that $\sqrt{n} \nabla_\theta M(\hat{\theta},x) \to N(0,V_M)$. But then again, I now have a product of normals. Also, what would be an appropriate remainder formulation to show $R \to 0$? Is this a productive path? Any hints or references are greatly appreciated. I am having difficulty in finding the latter. Many thanks. Login To add answer/comment