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June, 1946 The Effect on a Distribution Function of Small Changes in the Population Function
Burton H. Camp
Ann. Math. Statist. 17(2): 226-231 (June, 1946). DOI: 10.1214/aoms/1177730983


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 $\epsilon = (1 + \delta)^n - 1,\quad \text{and}\quad\mid\phi_1/\phi - 1 \mid < \delta.$


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Burton H. Camp. "The Effect on a Distribution Function of Small Changes in the Population Function." Ann. Math. Statist. 17 (2) 226 - 231, June, 1946.


Published: June, 1946
First available in Project Euclid: 28 April 2007

zbMATH: 0063.00699
MathSciNet: MR16621
Digital Object Identifier: 10.1214/aoms/1177730983

Rights: Copyright © 1946 Institute of Mathematical Statistics

Vol.17 • No. 2 • June, 1946
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