Second-Order Asymptotic Minimax Estimation in the Presence of a Nuisance Parameter

We consider second-order minimax estimation of the structural parameter $\theta_1$ in the presence of a nuisance parameter $\theta_2$ as the number of observations $n\to\infty$. We show that the effect of the nuisance parameter is largely determined by a “nontraditional” object in mathematical statistics – ector field $X=\partial/\partial\theta_1+J_{12}/J_{11}\partial/\partial\theta_2$, where $J_{11}$ and $J_{12}$ are elements of the inverse Fisher information matrix.