CHARACTERIZATIONS OF SHIFT-INVARIANT DISTRIBUTIONS BASED ON SUMMATION MODULO ONE

Let X₁Y_1,…, Y_n be independent random variables. We characterize the distributions of X and Y_j satisfying the equation {X+Y₁++Y_n}=_dX, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, either X is uniformly distributed on [0,1), or Y_j has.a shifted lattice distribution and X is shift-invariant. We also give a characterization of shift-invariant distributions. Finally, we consider some special cases of this equation.     

A NOTE ON THE DISTRIBUTION OF THE DOUBLY NON-CENTRAL F DISTRIBUTION1

A three-moment central-F approximation is examined for its usefulness in giving accurate values of the probability integral of the doubly non-central F distribution.     

ON THE DISTRIBUTIONS OF ORDER STATISTICS1

Approximations to the distributions of order statistics based on x²t are obtained. These are easy to compute and provide reasonably accurate values for the percentage points and probability integrals of the distributions.     

BALANCED DESIGNS FOR ROW-AND-COLUMN EXPERIMENTS WITH TWO NON-INTERACTING SETS OF TREATMENTS, ONE SET NOT BEING APPLIED TO ALL THE ROWS1

This note gives and discusses balanced row-and-column designs for experiments with two non-interacting sets of treatments, one set not being applied to all the rows. These designs are potentially useful for occasions when trees that have tested one set of experimental treatments are needed for testing a second set before the residual effects of the first have become negligible, but when the experimenter wishes to apply the second set to only some of the rows. The designs are appropriate if the residual and new effects do not interact.     

ON THE CONVERGENCE OF EMPIRICAL PROCESSES OF MIXING VARIABLES1

Conditions are given for the weak convergence of (t—t²)L_N(a_N^-1( t )) to a Gaussian process where v<1/2, a _N is a cdf and L _N is the normalized weighted empirical cumulative distribution function (cdf) for an α-mixing sample of random variables in R which may be non-stationary with discontinuous marginals.     

A BIVARIATE BINOMIAL DISTRIBUTION AND SOME APPLICATIONS1

This study is concerned with the joint distribution of the total numbers of occurrences of binary characters A and B, given three independent samples in which both characters, A but not B, and B but not A, are observed. The distribution function is given; its conditional distributions and regression functions are found; bounds on certain joint probabilities are established; and conditions for bivariate Poisson and Gaussian limits are studied. An application yields the joint distribution of sign statistics for the pair-wise comparison of treatments with a control.     

THE THEORY OF COMPETING RISKS1

The theory of competing risks is motivated, certain aspects are reviewed critically, and some extensions are indicated. A unified formulation of the theory is given covering dependent as well as independent risks. The relations between various functions useful in the theory are made explicit. In a historical note a valuable early result is put into modern notation. The currently controversial subject of identifiability when risks are dependent is discussed and it is indicated under what conditions some of the difficulties raised can be overcome. Consequences of assuming proportional hazard rates are set out. New conditions are provided under which this assumption holds and it is shown how the assumption may be tested. Some concluding remarks deal with limitations of the theory and point out areas needing further work.     

ON THE BEHRENS–FISHER PROBLEM1

In this article the Behrens-Fisher problem is reformulated in terms of a structural model of inference. For this version of the problem a solution is obtained which is valid for arbitrary absolutely continuous error distributions. These results are further discussed for the standard normal distribution and for some other special cases with not normally distributed populations.     

A PROBABILISTIC DERIVATION OF THE NON-CENTRAL χ2 DISTRIBUTION

A simple derivation of the non-central χ² distribution is presented. This requires no advanced mathematical knowledge, and is suitable for use in elementary courses.
The non-central χ² distribution is of great importance in statistical theory, both in its own right, and as a step in the derivation of other distributions; however, it tends to be neglected in statistical courses largely, it seems, because the standard derivations are too difficult. A recent review of derivations by Guenther (1964) shows that most require some knowledge of n -dimensional geometry or the equivalent matrix theory. The alternative is the use of generating functions, which is straightforward, apart from the inversion back to the density function. An objection to such methods is that they give no insight into the probabilistic nature of the proble.     

A LINEAR MODEL WITH ERRORS LACKING A VARIANCE1

The problem discussed is that of estimating β= (β¹, …, β_k) in the model Y=βX +ε when X has a specified multivariate distribution and the error ε does not necessarily have a finite second moment, for example, ε symmetric stable. We construct a moment estimator based on the empirical characteristic function and establish asymptotic unbiassedness and normality. Most of the paper is concerned with the case when X is normal. Forms of the suggested estimator are given in (2.5), (4.6) and (5.5).     

ON POINT MULTISERIAL CORRELATION1

Abstract
We present a simple form for the estimator of the point multiserial correlation coefficient between a quantitative variate X and a qualitative variate 7. Given a bivariate sample grouped in the form of an r × c contingency table the estimator is based on finding the optimum Y -scores which maximize the correlation coefficient. The resulting estimator is equivalent to Das Gupta's (1960) for ungrouped X -values, with the advantage of simplicity in its calculation. Under the assumption of conditional normality, the significance of point multiserial correlation may be studied by an F -test.