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Verification of Statistical Hypotheses on The Type of a Distribution Based on Small Samples

1956; Society for Industrial and Applied Mathematics; Volume: 1; Issue: 2 Linguagem: Inglês

10.1137/1101018

1095-7219

Anton Petrov,

Statistical and Computational Modeling

This paper gives a method of estimating the number of observations necessary to distinguish between two hypotheses on the type of a distribution. The difficulties encountered in distinguishing two types of distributions on the basis of small samples are related. In the introduction the problem of verifying an hypothesis on the type of a distribution is described. Section 1 gives the principal results of a general theory verifying statistical hypotheses which are connected with the most powerful distinction test. An approximate method of evaluating the number of observations necessary to distinguish between two rival hypotheses is established. Section 2 deals with the distribution of a random vector having coordinates \[ y_i = \frac{{x_i - \bar x}} {s};\,i = 1, \cdots ,n; \] in which $\bar x = {1 / h}\sum x_i $, $s^2 = \sum (x_i - \bar x)$ where $x_i $ are equally distributed independent random variables. The results of Section 2 allow methods of Section 1 to be applied to the problem of distinguishing between two hypotheses on the type of a distribution. An example is given in Section 3, where the problem of distinguishing between the normal type and the rectangular type is solved by taking samples of size $n = 3$, and the number of necessary observations is evaluated. Sections 4 and 5 relate the problem of distinguishing between the normal type of distribution and a type closely related to it with nonzero skewness of nonzero excess. It is established how the necessary number of observations increases as the size of the sample decreases. Section 6 gives a relationship between the number of observations necessary for distinguishing between two hypotheses based on small samples and the number of observations necessary if all the data are obtained from one large sample.