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1. Computer Generation of Gamma Random Variates with Non-integral Shape Parameters.

Author

Wallace, N. D.

Subjects
*COMPUTER simulation, *THEORY of distributions (Functional analysis), *RANDOM variables, *MULTIVARIATE analysis, *PROBABILITY theory, *CLUSTER analysis (Statistics)
Abstract

When the shape parameter, α, is integral, generating gamma random variables with a digital computer is straightforward. There is no simple method for generating gamma random variates with non-integral shape parameters. A common procedure is to approximately generate such random variables by use of the so-called probability switch method. Another procedure, which is exact, is due to Jöhnk. This paper presents a rejection method for exactly generating gamma random variables when α is greater than 1. The efficiency of the rejection method is shown to be better than the efficiency of Jöhnk's method. The paper concludes that when α is non-integral the following mix of procedures yields the best combination of accuracy and efficiency: (1) when α is less than 1, use Jöhnk's method; (2) when 1 is less than α and α is less than 5, use the rejection method; (3) when α is greater than 5, use the probability switch method. [ABSTRACT FROM AUTHOR]

Published
1974
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2. STATISTICAL PROBLEMS IN SCIENCE. THE SYMMETRIC TEST OF A COMPOSITE HYPOTHESIS.

Author

Neyman, Jerzy

Subjects
*PROBABILITY theory, *STATISTICS, *CLUSTER analysis (Statistics), *RANDOM variables, *THEORY, *ERROR, *PROBLEM solving
Abstract

The purpose of the paper is to illustrate a "give and take" type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science, on the other. Invariably, research in science involves some observations x and the unknown "true" state of nature sigma. Quite often, replication of the experiments reveals considerable variation in x, indicating that no mathematical treatment of the problem is possible without the assumption that x is a sample value of a random variable X, treated in terms of frequentist theory of probability. As to the true state of nature, sigma, situations vary. Indeed, there are cases where it appears natural to consider that sigma is selected at random out of a certain known set SIGMA, with either known or unknown probability law. Section 2 lists three typical examples in which the frequentist theory of probability can "give" something to science. In each example, however, the assumption of the randomness of sigma appears extraneous. Section 3 describes another example of research in science, again with a non-random sigma, which happened to "give" something to the frequentist theory of statistics. The motivation to reduce the frequency of erroneous conclusions in the general circumstances of the biological study had led to considering the possibility that a certain derivative may fail to exist. This created a problem of testing (Section 4) which does not appear to have been considered. A solution of the problem, in the form of optimal symmetric (C alpha) tests, is given in Section 5. [ABSTRACT FROM AUTHOR]

Published
1969
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