Search

Distributions are derived of the product and of the quotient of maximum values in samples from a population described by the frequency αx k , which may be called a power-function population. The distribution of the quotient is used to test whether two samples have come from the same triangular population. Estimators of the parameters of the distribution are derived.

(1951). The Distribution of the Quotient of Ranges in Samples from a Rectangular Population. Journal of the American Statistical Association: Vol. 46, No. 256, pp. 502-507.

Statistical distributions of the form C(l + [' x | )~ are considered. For k = 2, m = 1 this is the Cauchy distribution. For k = 2, m == ^(n -\1), % == n~&, it is the Student-Fisher distribution with n degrees of freedom. The Cauchy distribution is of interest because the variance of the mean is infinite while that of the median is finite. In this paper a study is made of the case m = 1. The value of k is determined for which the variance of the mean and that of the median are equal. The distribution of the mean for the case k = 4, m = 1 is derived. Several other special cases are studied.

The distributions of midrange, half-range, median and geometric mean of samples from a triangular population are derived. Mean, median, and midrange are compared as to their efficiency in estimating the single parameter of the population, which is the upper end of the range of the distribution. The best estimator of the parameter, however, is based on the maximum value in the sample, since it is derived from the likelihood function and is a sufficient statistic.

A study is made of the distribution of the midranges of samples from five different symmetric populations of limited range, and of the relative efficiency of midrange and mean in estimating the population midrange, or mean, or median. (For a symmetric distribution these three averages are of course identical.) It is found that the midrange is more efficient than the mean for all of the populations considered, and that this efficiency increases with decreasing α4(= μ4/[sgrave]4), the standardized fourth moment. The values of α4 for the five populations studied are 2.19, 2.14, 1.8, 1.19, and 1.

In a previous papert the writer has called attention to a number of the interesting properties of the figuratrix in the calculus of variations. It is the purpose of the present paper to define a figuratrix for double integrals and to develop some of its important characteristics. 1. A new form of the problem of double integrals. The theory of extrema of double integrals has been treated for integrals of the form

This paper gives a method of estimating the parameter of a Poisson distribution which has been truncated at the lower end. Application is made to a number of actual examples.

(1955). Truncated Binomial and Negative Binomial Distributions. Journal of the American Statistical Association: Vol. 50, No. 271, pp. 877-883.

Tables are given of the expected value of 1/X, the expected value of 1/X 2, and the standard deviation of 1/X, where X is a variable following a negative binomial distribution whose zero class is lacking.

Hadamard defines the figurative of the point (x, y) as the curve F(x', y') 1, where x' and y' are the current co6rdinates, and x and y are considered constant.t The polar reciprocal of the figurative with respect to the unit circle x'2 +y'2 = 1 is termed by Hadamard the figuratrix. The importance of the figuratrix and its space analogue seems to have been overlooked, and it is the desire of the writer to call attention to a number of their interesting properties which enable one to interpret geometrically some of the well known functions and theorems of the calculus of variations. 1. Figurative, figuratrix, and indicatrix. We shall prefer to consider an integral of the form

IT was the original purpose of this study to attempt to discover the effect upon the distribution in random samples, particularly in small samples, of the productmiioment coefficient of correlation, r, when the samples are drawn from a non-normal instead of a normal population. In Part I the results of sampling from certain populations which differ greatly from the normal are given, also the results of sampling from a normal population having a high degree of correlation. As the sampling was done experimentally, it was necessary to deal with discrete populations. This opened up the question of the effect of grouping upon the distribution of r, a question which is investigated in Part II.

A comparison is made of the exact variances of medians of small samples from several special populations with the values obtained by using the formula for asymptotic variance.

(1917). An Intrinsic Equation Solution of a Problem of Euler. The American Mathematical Monthly: Vol. 24, No. 9, pp. 420-422.

Explanations are given of what is meant by ranked data. Questions of rank correlation and concordance are discussed. Coefficients of correlation and concordance are defined and methods of testing them for significance are described and illustrated.

Exact values of the variances of the medians of small samples from a Cauchy distribution are given. They are compared with the values obtained by using the formula for asymptotic variance.

(1922). The November Meeting of the Missouri Section. The American Mathematical Monthly: Vol. 29, No. 1, pp. 1-3.