Bias Reduction for Moment Estimates and MLEs

I give estimates of low bias for functions of moments. Let \( F(x) \)be a distribution on \( R^s \). Let \( F_n(x) \) be the empirical distribution of a random sample of size \( n \) from\( F(x) \). Given a functional \( F(x) \), \( E\ T(F_n) \)estimates \( T(F) \)with bias \( \sim n^{-1} \). (The bias is zero for a mean, but this is the exception.) The jackknife and bootstrap estimates only reduce this bias to \( \sim n^{-2} \), and are computationally intensive. I review the main two analytic methods to obtain an estimate of \( T(F) \) of bias \( \sim n^{-k} \)for \( k\leq 4 \)in terms of the functional derivatives of \( T(F) \). I give a chain rule for these derivatives when \( T(F)=g(U(F)) \) and \( g:R^q\rightarrow R \)is any given smooth function with finite partial derivatives at \( U(F)\in R^q \). I apply this to give an estimate of \( T(F) \) of bias \( \sim n^{-k} \)for \( k\leq 4 \), in terms of the derivatives of \( g \)and \( U(F) \). Examples include moment estimates and maximum likelihood estimates.