One-step estimator

A one-step estimator starts with an initial estimator $\tilde{\theta }$ and applies one correction derived from an estimated influence function:The correction aims to remove the leading discrepancy between the initial answer and the desired parameter. For the mean, use $\hat{\psi }(x)=x-\tilde{\theta }$. Then the corrected answer is exactly the sample mean, whatever initial value was chosen.

For a smooth estimating equation with empirical average $U_n(\theta )$ and nonsingular Jacobian matrix $A_n(\theta )$, one Newton method step is $\tilde{\theta }-A_n(\tilde{\theta }{)}^{-1}{U}_n(\tilde{\theta })$. Its correction has the same form with $\hat{\psi }(x)=-A_n(\tilde{\theta }{)}^{-1}\psi (x,\tilde{\theta })$. An asymptotic linear representation follows only when the initial error, derivative approximation and remaining terms are sufficiently controlled. The formula alone is not a general guarantee of improved accuracy.