that the bias of $\hat{{}\theta}$ is $$\label{bias}
  \expect_G(\hat{{}\theta} - \bar{{}\theta}) = O(n^{-1}) \, .$$ This provides some reassurance that, as the information about the parameter ${{}\theta}$ grows, the bias from $\bar{{}\theta}$ will converge to ${{}0}_p$. Nevertheless, the finite sample bias of $\hat{{}\theta}$ is typically not zero. Under the same assumptions, it is also possible to write down (\[bias\]) in the more explicit form $\expect_G(\hat{{}\theta} - \bar{{}\theta}) = {{}b}(\bar{{}\theta}) + O(n^{-3/2})$, where ${{}b}(\bar{{}\theta})$ depends on joint moments of estimating functions and their derivatives under $G$. It is tempting, then, to replace $\hat{{}\theta}$ with a new estimator $\hat{{}\theta} - {{}b}(\hat{{}\theta})$, on the basis that, under \[consistency\], ${{}b}(\hat{{}\theta}) \stackrel{p}{\longrightarrow} {{}b}(\bar{{}\theta})$, and hoping that the new estimator will have better bias properties.

The estimator $\hat{{}\theta} - {{}b}(\hat{{}\theta})$ has been shown to, indeed, have better bias properties when estimation is by maximum likelihood and the model is correctly specified [see, Section 10 @efron:1975 for a proof]. By the model being correctly specified we mean that the unknown joint distribution function $G(z_1, \ldots, z_m)$ is assumed to be a particular member of the family of distributions specified fully by $F_i({{}y}_i | {{}x}_i, {{}\theta})$ when forming (\[objective\_function\]). In that particular case, it is also possible to evaluate 