1. Summary
The main object of this paper is to find a criterion for comparison of two tests for non-parametric hypotheses, taking advantage of the qualitative information that may exist. After a detailed analysis of the problem and some earlier suggestins for its solution (sections 2–4), a criterion is suggested in section 5. In order to apply it to a concrete case, a location problem is specified in section 6. The rank tests to be compared are analyzed in section 7, and the comparison by way of the criterion is carried out in section 8. It turns out that sign tests, sometimes slightly modified, are very often optimal according to the criterion used.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blomqvist, N. (1950), On a measure of dependence between two random variables. AMS21, p. 593–600.
Cochran, W. G. (1937), The efficiencies of the binomial series tests of significance of a mean and of a correlation coefficient. JRSS100, p. 69–73.
Dixon, W. J. andMood, A. M. (1946), The statistical sign test. JASA41, p. 557–566.
Fisher, R. A. (1925), Statistical methods for research workers, Sec. 24, ex. 19. Edinburgh.
-- (1935), Design of experiments, Sec. 21. Edinburgh.
Hemelrijk, J. (1950a), A family of parameterfree tests for symmetry with respect to a given point. Proc. Kon. Ned. Akad. van Wetensch.53, p. 945–955, 1186–1198.
-- (1950b), Symmetritoetsen en andere toepassingen van de theorie van Neyman en Pearson - Amsterdam.
Hoeffding, W. (1951), “Optimum” nonparametric tests. 2nd Berkeley Symp. p. 83–92.
—— (1952), The large-sample power of tests based on permutations of observations. AMS23, p. 169–192.
Lehmann, E. L. (1947), On families of admissible tests. AMS18, p. 97–104.
—— (1953), The power of rank tests. AMS24, p. 23–43.
Lehmann, E. L. andStein, C. (1949), On the theory of some non-parametric hypotheses. AMS20, p. 28–45.
Lindley, D. V. (1953), Statistical inference. JRSS B15, p. 30–76.
Mann, H. B. andWhitney, D. R. (1947), On a test of whether one of two random variables is stochastically larger than the other. AMS18, p. 50–60.
Neyman, J. andPearson, E. S. (1936), Contributions to the theory of testing statistical hypotheses, Part I. Statistical Research Memoirs, Vol. I, London.
Pitman, E. J. G. (1948), Notes on non-parametric statistical inference. Mimeogr., Columbia Univ.
Sverdrup, E. (1952), Weight functions and minimax procedures in the theory of statistical inference. Arkiv for Mathematik og Naturvidenskap,LI No. 7. Oslo.
—— (1953), Similarity, unbiassedness, minimaxibility and admissibility of statistical tests. SA36, p. 64–86.
van der Vaart, H. R. (1950), Some remarks on the power function of Wilcoxon’s test for the problem of two samples. Proc. Kon. Ned. Akad. van Wetensch.53, p. 494–520.
Wald, A. (1942), On the principles of statistical inference. Notre Dame Mathematical Lectures, No. 1.
-- (1950), Statistical decision functions, New York.
Walsh, J. E. (1946), On the power function of the sign test for slippage of means. AMS27, p. 358–362.
—— (1949), Some significance tests for the median which are valid under very general conditions. AMS20, p. 64–81.
Wilcoxon, F. (1945), Individual comparisons by ranking methods. Biometrics1, p. 80–83.
Wolfowitz, J. (1942), Additive partition functions and a class of statistical hypotheses. AMS13, p. 247–279.
Rights and permissions
About this article
Cite this article
Ruist, E. Comparison of tests for non-parametric hypotheses. Ark. Mat. 3, 133–163 (1955). https://doi.org/10.1007/BF02589351
Issue Date:
DOI: https://doi.org/10.1007/BF02589351