J. S. Rao

Incomplete Dirichlet integrals with applications to ordered uniform spacings

Sobel, Uppuluri, and Frankowski, (Selected Tables in Math. Statistics, Vol. IV. Amer. Math. Soc., Providence, R. I., 1977) consider an incomplete Dirichlet integral of type I with several interesting applications connected with the multinomial distribution and provide tables of this integral along with other useful tables. Two incomplete Dirichlet integrals are discussed here along with some useful recurrence relations, providing simple methods of deriving the distribution theory of ordered uniform spacings.

Mathematical techniques for paleocurrent analysis: Treatment of directional data

Statistical procedures for (1) sampling, (2) testing the existence of a preferred direction, and (3) testing homogeneity of twodimensional directional data, which have been developed by the authors for paleocurrent studies, are presented. It is well known that conventional methods of statistical analysis are not applicable to directional data (e.g., crossbedding and ripplemark directions, grain lineations, etc.) which are “circularly distributed” on a compass dial. A sampling technique for directional data has been developed using the circular measures of dispersion and approximate ANOVA of G. S. Watson. On the basis of a pilot survey, it is possible to compute the minimum sample size required for estimating, with a desired precision, the mean paleocurrent direction of a formation. The optimum allocation of sample size between and within outcrops also can be accomplished at a minimum cost. The procedure described for testing uniformity (or lack of preferred direction) is based on the arc lengths made by successive sample points and is simple to use if the sample size is moderate. A table of critical points and a numerical example are given after a description of the test procedure. Finally, the procedures for testing the homogeneity of directional data from several geological formations are described by (1) tests for equality of the resultant directions (polar vectors) and (2) tests for equality of dispersions. With these tests it is possible to determine whether the paleocurrent directions from different geological formations belong to significantly different populations.

An Optimum Hierarchical Sampling Procedure for Cross-Bedding Data

The commonly used techniques for hierarchical or multistage sampling of cross-bedding foreset azimuths (for paleocurrent study) are based on the conventional analysis of variance. It is now well known that the classical method of analysis of variance (ANOVA), which partitions the sum of squares of the observations, can not be indiscriminately applied for the analysis of circularly distributed directional data. An efficient and economical sampling technique for cross-bedding data has been developed using the circular measures of dispersion and the approximate ANOVA of Watson. The technique is illustrated here with the help of the pilot-survey data of tne flu via tile Kamthi formation. The following sampling problems have been solved: (1) the minimum sample size required for estimating, with a desired precision, tne mean direction of a formation, (2) the optimum allocation of the samples between and within the outcrops that would allow efficient sampling at minimum cost.

A characterization of Gaussian spaces

A characterization of Gaussian subspaces of L2 is given. It is shown that Gaussian subspaces are essentially characterized by the property that all linear isometries are induced by measure-preserving transformations.