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

Let ξ ₁ ,ξ ₂ , ⋯ ξ ₙ , ⋯ be a sequence of independent identically distributed random variables with a distribution function $F(x)$, and let Fₙ^ * (x) be an empirical distribution function of ξ ₁ ,ξ ₂ , ⋯ ξ ₙ, We study the asymptotic properties of a functional T[Fₙ^ * ] when n → ∞. The following result is obtained for a certain class of functionals, called Mises’ functionals: the limiting distribution of nm/2 \ T[Fₙ^ * ] - T[F]\, when n → ∞, coincides with that of the functional of the form \[nm/2 T( m ) [ {F_n^ * } ] = nm/2 ∫- ∞+ ∞ ⋯ ∫- ∞+ ∞ {ψ ( {x_1 , ⋯ ,x_m } )} ∏i = 1^m {d[ {F_n^ * ( {x_i } ) - F( {x_i } )} ],} \] where the number m and the function ψ (x₁ , ⋯ xₘ ) 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 nm/2 T( m ) [ Fₙ^ * ], when n → ∞, coincides with the distribution of the multiple stochastic integral of some function with respect to the conditional Wiener process β (t),0 t 1,[β (0) = β (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.