Properties of sufficiency and statistical tests

TLDR

Previous work on small‑sample sufficiency focused mainly on estimation theory. This paper investigates the structure of small‑sample tests, exploring their connections to estimation, fiducial distributions, and goodness‑of‑fit testing. The authors formalize conditioning with the notation a|b, introduce a quasi‑sufficient conditional statistic T|, and show that its distribution satisfies the sufficiency property and captures all information on the parameter θ.

Abstract

1—In a previous paper, dealing with the importance of properties of sufficiency in the statistical theory of small samples, attention was mainly confined to the theory of estimation. In the present paper the structure of small sample tests, whether these are related to problems of estimation and fiducial distributions, or are of the nature of tests of goodness of fit, is considered further. The notation a | b implies as before that the variate a is conditioned by a given value of b . The fixed variate b may be denoted by | b , and analogously if b is clear from the context, a | b may be written simply as a |. Corresponding to the idea of ancillary information introduced by Fisher for the case of a single unknown θ , where auxiliary statistics control the accuracy of our estimate, I have termed a conditional statistic of the form T |, quasi-sufficient, if its distribution satisfies the “sufficiency” property and contains all the information on θ . In the more general case of other unknowns, such a statistic may contain all the available information on θ .