Mises’ Theorem on the Asymptotic Behavior of Functionals of Empirical Distribution Functions and Its Statistical Applications

Let $\xi_1,\xi_2,\dots\xi_n,\dots$ be a sequence of independent identically distributed random variables with a distribution function $F(x)$, and let $F_n^*(x)$ be an empirical distribution function of $\xi_1,\xi_2,\dots\xi_n$, We study the asymptotic properties of a functional $T[F_n^*]$ when $n\to\infty$. The following result is obtained for a certain class of functionals, called Mises’ functionals: the limiting distribution of $n^{m/2}\{T[F_n^*]-T[F]\}$, when $n\to\infty$, coincides with that of the functional of the form
$$n^{m/2}T_{(m)}[{F_n^*}]=n^{m/2}\int_{-\infty}^{+\infty}\dots\int_{-\infty}^{+\infty}{\psi\left({x_1,\dots,x_m}\right)}\prod\limits_{i=1}^m{d[{F_n^*({x_i})- F( {x_i})}],}$$
where the number $m$ and the function $\psi (x_1,\dots x_m)$ are defined by the functional $T$ and the distribution function $F(x)$ (Mises’ theorem). The case $m=1$ may be regarded as the basic one; in this case we deal with sums of independent identically distributed random variables.
The main result is the proof of the Mises property for a certain class of functionals. This class apparently includes all differentiable (in Mises’ sense) functionals of mathematical statistics. Further it is proved that the limiting distribution of $n^{m/2}T_{(m)}[{F_n^*}]$, when $n\to\infty$, coincides with the distribution of the multiple stochastic integral of some function with respect to the conditional Wiener process $\beta(t),0\leq t\leq 1,[\beta(0)=\beta(1)=0]$. The characteristic functions of the corresponding stochastic integrals for $m=1,2$ may be calculated. The above mentioned results are applied to some definite functionals of mathematical statistics.