von Mises Calculus

Let be the empirical distribution of a random sample X1, … , Xn from a distribution P. Many statistics can be written in the form whereis a function that maps every distribution of interest into some space, which for simplicity is taken equal to the real line. Because the observations can be regained from completely (unless there are ties), any statistic can be expressed in the empirical distribution. The special structure assumed here is that the statistic can be written as a fixed functionof , independent of n, a strong assumption.

Because converges to P astends to infinity, we may hope to find the asymptotic behavior of through a differential analysis ofin a neighborhood of P. A first-order analysis would have the form

where is a “derivative” and the remainder is hopefully negligible. The simplest approach towards defining a derivative is to consider the function for a fixed perturbation H and as a function of the real-valued argument t. Iftakes its values in JR, then this function is just a function from the reals to the reals.