This paper presents two definitions of maximum cluster, that have sometimes been used to test for non-random clustering. We compare the power of the testa based on these statistics, and show that when the clustering interval is sufficiently small, one of the tests is more powerful for a wide class of alternative hypotheses. We show that this test is the generalized likelihood ratio test for an alternative hypothesis related to a particular type of non-random clustering.

On Certain Probability Distributions Arising from Points on a Line

Power Comparison of Two Tests of Non-Random Clustering

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The Distribution Theory of Runs

/pdf/tables-for-testing-randomness-of-grouping-in-a-sequence-of-b806aqppyc.pdf

Tables for Testing Randomness of Grouping in a Sequence of Alternatives

In sample surveys the final estimate is prepared from information collected for sample units of definite size (area) located at random. Large-scale work involves journeys from one sample unit to another so that both cost and precision of the result depend on size (area) as well as the number (density per sq. mile) of sample units. The object of planning is to settle these two quantities in such a way that (a) the precision is a maximum for any assigned cost, or (b) the cost is a minimum for any assigned precision. The present paper discusses the solution for (1) uni-stage sampling (with randomization in one single stage) both in the abstract and in the concrete; and for (2) multi-stage sample (with randomization in more than one stage) mostly in the abstract. The whole area is considered here as a statistical field consisting of a large number of basic cells each having a definite value of the variate under study. These values (with suitable grouping) form an abstract frequency distribution corresponding to which there exists a set of associated space distributions (of which the observed field is but one) generated by allocating the variate values to different cells in different ways. This raises novel problems which are space generalizations of the classical theory of sampling distribution and estimation. On the applied side it also enables classification of the technique into two types: (a) 'individual' or (b) 'grid' sampling depending on whether each sample unit consists of only one or more than one basic cell. For most space distribution precision of the result is nearly equal for both types of sampling; these are called fields of random type. For certain fields (including those usually observed in nature) precision depends on sampling type; these are fields of non-random type. Application to estimating acreage under jute covering 60,000 sq. miles in Bengal in 1941-2 is described with numerical data. The margin of error of the sample estimate was about 2%, while cost was only a fifteenth of that of a complete census made in the same year by an official agency.

On large-scale sample surveys

On the power function of the longest run as a test for randomness in a sequence of alternatives

Bipartitional functions are arithmetical functions of two partitions of the same number, and arise primarily in the theory of the symmetric function generating functions. Analytical methods of evaluating the partitional functions and of studying them in relation to the theory of distributions are largely due to Macmahon (1915). The use of partitional notation has rendered his methods distinctly simpler than those of his predecessors, but, simplified as they are, his methods do not make the practical evaluation of these functions particularly expeditious. If his methods are actually put into practice, it is found that they become increasingly laborious and impracticable with high-order symmetric functions. An excellent example of the difficulties encountered in the use of algebraic methods, especially those involving the action of differential operators, is to be found in the enumeration of the 5 x 5 and 6 x 6 Latin Squares (Fisher and Yates 1934). In this connexion it is shown by Fisher and Yates that the direct enumeration by trial is a much simpler approach than the development of the differential operators of Macmahon’s algebraic solution.

On Certain Probability Distributions Arising from Points on a Line

Citations

Power Comparison of Two Tests of Non-Random Clustering

References

The Distribution Theory of Runs

Tables for Testing Randomness of Grouping in a Sequence of Alternatives

On large-scale sample surveys

On the power function of the longest run as a test for randomness in a sequence of alternatives

On Bipartitional Functions

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