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No abstract

Many of the most frequently used applications of the theory of statistics, such for example as the methods of analysis of variance and covariance, the general test of multiple regression and the test of a regression coefficient, depend essentially on the joint distribution of several quadratic forms in a univariate normal system. The object of this paper is to prove the main-relevant results about this distribution. As an application of these results, the theory involved in the method of analysis of covariance will be investigated.

It is proved that if there exists a sufficient statistic for the estimation of an unknown parameter of a population, the frequency function of the population must be of a certain type.It is shown that some modification of previous theory of the intrinsic accuracy of statistics is necessary when the range of the population sampled is a function of the parameter to be estimated.Finally, the theory is extended to sufficient sets of statistics, i.e. sets of statistics which together contain all the information provided by a sample about an unknown parameter.

A definition of “closer” and “closest” as applied to estimates of statistical parameters is given, and it is shown that we can sometimes prove that estimates properly derived from sufficient statistics are the closest possible.The scaling of a gamma distribution, and the location and scaling of an exponential distribution and of a rectangular distribution are discussed in detail, and the closest estimates of the parameters obtained.