Combining independent one-sample tests of significance Article Received: 09 November 1965 DOI :
10.1007/BF02911681

Cite this article as: Puri, M.L. Ann Inst Stat Math (1967) 19: 285. doi:10.1007/BF02911681
13
Citations
56
Downloads
Summary The problem of combining independent one-sample tests of significance is considered using techniques developed by the author (1965). LetX _{i1} ,..., X_{im} _{i} be positive observations andY _{i1} ,..., Y_{in} _{i} , the absolute values of negative observations in a sampleZ _{i1} ,..., Z_{iN} _{i} ofN _{i} =m_{i} +n_{i} independent and identically distributed random variables from a population with continuous cumulative distribution function\(\Pi _{0_i } (z);i = 1, \cdots ,k\) .

Then for testing the hypothesis that each of the distributions\(\Pi _{0_i } (z)\) is symmetric with respect to the origin, linear combinations of several one-sample test statistics are considered. Under suitable assumptions, two sets of combination coefficients are derived. One of them yields a class of tests with a region of consistency that is independent of the proportion of sample sizes (design-free tests) and the other has asymptotically the maximum power (locally asymptotically most powerful tests). Finally, these tests are compared with respect to the asymptotic values of their power against Pitman's shift alternatives and Lehmann's distribution free alternatives.

The research was sponsored by the Office of Naval Research under Contract Number Nonr-285 (38). Reproduction in whole or in part is permitted for any purpose of the United States Government.

References [1]

Allan Birnbaum, “Combining independent tests of significance,”

J. Amer. Statist. Ass. , 49 (1954), 559–574.

MATH CrossRef MathSciNet Google Scholar [2]

H. Chernoff and I. R. Savage, “Asymptotic normality and efficiency of certain nonparametric test statistics,”

Ann. Math. Statist. , 29 (1958), 972–994.

CrossRef MathSciNet Google Scholar [3]

Ph. van Elteren, “On the combination of independent two sample tests of Wilcoxon,”

Bull. Inst. Inter. Statist. , 37 (1960), 351–361.

MATH Google Scholar [4]

R. A. Fisher,

Statistical Methods for Research Workers , Oliver and Boyd, Edinburgh and London, (4th ed.) 1932.

MATH Google Scholar [5]

Z. Govindarajulu, “Central limit theorems and asymptotic efficiency for one sample procedures,” Technical Report 11, University of Minnesota, 1960.

[6]

C. Van Eeden and J. Hemelrijk, “A test for the equality of probabilities against a class of specified alternative hypotheses including trend,”

Proc. Kon. Ned. Akad., Wetensch. A , 58 (1955), 191–198,

Indag. Math. , 17 (1955), 301–308.

Google Scholar [7]

J. L. Hodges, Jr., and E. L. Lehmann, “The efficiency of some non-parametric competitors of the

t -test,”

Ann. Math. Statist. , 27 (1956), 324–335.

MATH CrossRef MathSciNet Google Scholar [8]

J. L. Hodges, Jr., and E. L. Lehmann, “Comparison of the normal scores and Wilcoxon tests,”

Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , 1 (1961), 307–317.

MathSciNet Google Scholar [9]

J. L. Hodges, Jr. and E. L. Lehmann, “Rank methods for comparison of independent experiments in analysis of variance,”

Ann. Math. Statist. , 33 (1962), 482–497.

MATH CrossRef MathSciNet Google Scholar [10]

J. L. Hodges, Jr. and E. L. Lehmann, “Matching in paired comparisons,”

Ann. Math. Statist. , 25 (1954), 787–781.

MATH CrossRef MathSciNet Google Scholar [11]

E. L. Lehmann, “The power of rank tests,”

Ann. Math. Statist. , 29 (1953), 23–43.

CrossRef MathSciNet Google Scholar [12]

E. L. Lehmann,Testing Statistical Hypotheses , John Wiley, 1959.

[13]

E. L. Lehmann, “Consistency and unbiasedness of certain non-parametric tests,”

Ann. Math. Statist. , 22 (1951), 165–179.

MATH CrossRef MathSciNet Google Scholar [14]

A. M. Mood, “On the asymptotic efficiency of certain non-parametric two-sample tests,”

Ann. Math. Statist. , 25 (1954), 514–522.

MATH CrossRef MathSciNet Google Scholar [15]

E. J. G. Pitman, Lecture notes on non-parametric statistical inference, Columbia University, 1949.

[16]

M. L. Puri, “Asymptotic efficiency of a class of

c -sample tests,”

Ann. Math. Statist. , 35 (1964), 102–121.

CrossRef MathSciNet Google Scholar [17]

M. L. Puri, “On the combination of independent two sample tests of a general class,”

Inter. Statist. Inst. Review , 33 (1965), 229–241.

MATH CrossRef MathSciNet Google Scholar [18]

M. L. Puri, “On the Pitman and Lehmann efficiency of some non-parametric tests in one way analysis of variance,”Ann. Math. Statist. (to be submitted), 1965.

[19]

M. L. Puri and P. K. Sen, “On the asymptotic normality of one-sample Chernoff-Savage test statistics,”Courant Inst. of Math. Sciences , New York University, (1966), IMM 350.

[20]

M. L. Puri and P. K. Sen, “On a class of mutlivariate multi-sample rank order tests,”

Sankhyā , 28 (1966), 353–376.

MATH MathSciNet Google Scholar [21]

I. R. Savage, “Contribution to the theory of rank order statistics: the one sample case,”

Ann. Math. Statist. , 30 (1959), 1018–1023.

MATH CrossRef MathSciNet Google Scholar [22]

W. A. Wallis, “Compounding probabilities from independent significance tests,”

Econometrika , 10 (1942), 229–248.

MATH CrossRef MathSciNet Google Scholar © The Institute of Statistical Mathematics 1967

Authors and Affiliations 1. Courant Institute of Mathematical Sciences New York University New York USA