For a class of simple M-estimator $$\widehat{\theta}=\arg\max_{\theta} \sum_{i=1}^n m_{\theta}(X_i)$$ with true value $$\theta^*=\arg\max_{\theta}\mathbb{E}\{m_{\theta}(X)\}.$$ There exists examples such that $\widehat{\theta}$ converges to $\theta^*$ in a non root-n rate. For example, let $m_{\theta} = \boldsymbol{1}_{[\theta-1,\theta+1]}$, it has been proved under centain conditions $\widehat{\theta}$ converges to $\theta^*$ with $n^{-1/3}.$
My question is, is there exists some example of Z-estimator, $\widehat{\theta}$ is the zero point of $$\sum_{i=1}^n m_{\theta}(X_i)$$ with $\theta^*$ is the zero point of $\mathbb{E}\{m_{\theta}(X)\}$, such that $\widehat{\theta}$ converges to $\theta^*$ in non root-n rate?