The Effect on a Distribution Function of Small Changes in the Population Function

Abstract

It is generally assumed in the application of distribution theory that, if the actual population function is not very different from the one used in the theory, then the true sampling distribution of a statistic will not be very different from the one obtained in the theory. But elsewhere in mathematics we do not assert that a conclusion will be only slightly modified by a small deviation in the hypothesis. This paper presents some theorems which are useful in determining the maximum effect on a sampling distribution of certain kinds of small changes in the population function. In particular, if the population is denoted by the function $\phi(t)$, if a sample of $n$ independent measurements $(t_1, \cdots, t_n)$ is taken from this population, if a statistic $x = g(t_1, \cdots, t_n)$ is formed from the sample, and if $D(x)$ denotes the distribution of this statistic; then, when $\phi (t)$ is changed by a small proportionate amount to $\phi_1(t), D(x)$ will be changed to $D_1(x)$, and the relation between $D$ and $D_1$ will be subject to the inequality: $\| \int^b_a (D - D_1)dx\| \leqq \epsilon \int^b_a D(x)dx,$ where $\epsilon = (1 + \delta)^n - 1,\quad \text{and}\quad\mid\phi_1/\phi - 1 \mid < \delta.$