The authors give the exact coefficient of 1/N in a saddlepoint approximation to the Wilcoxon‐Mann‐Whitney null‐distribution. This saddlepoint approximation is obtained from an Edgeworth approximation to the exponentially tilted distribution. Moreover, the rate of convergence of the relative error is uniformly of order O (1/N) in a large deviation interval as defined in Feller (1971). The proposed method for computing the coefficient of 1/N can be used to obtain the exact coefficients of 1/Ni, for any i. The exact formulas for the cumulant generating function and the cumulants, needed for these results, are those of van Dantzig (1947‐1950).
In the present paper, we consider four approximations to the null-distribution of the two-sample Wilcoxon-Mann-Whitney statistic, namely a normal, an Edgeworth and a saddlepoint approximation, as well as an approximation by the sum of independent uniform random variables. We make numerical comparisons of these approximations for moderate sample sizes, namely for m = 20 and 20 ≤ n ≤ 80. It turns out that the saddlepoint improves on the Edgeworth and the uniform approximations only very far in the tails, while the Edgeworth outperforms the other three for less extreme cases. We also discuss the practical importance of our results in the era of statistical packages.
This paper considers periodic regression functions, which are solutions to a planar system of differential equations. In particular, it introduces a simple stochastic model which describes the interaction between predator and prey populations. The regression functions are solutions to the classical Lotka-Volterra system of equations, which admits closed orbits. The proposed method of estimation can be applied whenever pairs of predator-prey data are available, and the prey is the main source of food of the predator. Canadian mink-muskrat data are analysed from this new viewpoint. The estimation method is based on the existence of closed trajectories that describe the relationship between the two population sizes, and the paper shows how it can be extended to other systems of differential equations which admit closed orbits (e.g. Hamiltonian systems).
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