On the Existence of a Best Approximation of One Distribution Function by Another of a Given Type

Abstract

For well over two centuries mathematicians have considered the conditions under which it is possible to obtain a good approximation of one probability distribution by another of a given type. However, the conditions which assure the existence of a best approximation of a given type seem to have been virtually neglected. Because of their intrinsic interest and because of their relevance for an estimation problem which is discussed later, such conditions are examined here with respect to the following example: Suppose that $F$ and $G$ are distribution functions and that an ordered real number pair $(a, b)$, with $a > 0$, is desired such that $F(ax + b)$ is close to $G(x)$ for all real $x$. Is there a best pair? For instance, is there a pair $(a_0, b_0)$ satisfying \begin{equation*}\tag{1}\sup_{-\infty < x < \infty} |F(a_0 x + b_0) - G(x)| = \inf_{\substack{0 < a < \infty\\-\infty < b < \infty}} \sup_{-\infty < x < \infty} |F(ax + b) - G(x)|?\end{equation*} In this note we give an example in which a pair $(a_0, b_0)$ satisfying (1) does not exist. We then prove two theorems each giving a simple sufficient condition for the existence of such a pair. One or the other condition is almost always satisfied in practice. For example, the first requires, merely, that both of the sets $\{x \mid \frac{1}{3} \leqq F(x) \leqq \frac{2}{3}\}$ and $\{x \mid \frac{1}{3} \leqq G(x) \leqq \frac{2}{3}\}$ be nondegenerate. Next, we show that in any case if the set of minimizing pairs is nonempty then it is convex. This fact is used to obtain a fairly precise description of the set of minimizing pairs for the case $F$ is increasing and continuous. In this case, simple conditions on $G$ imply the uniqueness of a minimizing pair. Applications, especially to an estimation problem involving an unknown scale and location parameter, are then discussed. Throughout the paper, the right hand side of (1) is denoted by $M$. Also, $F$ and $G$ are understood to be continuous on the right.