| Title:
|
The multisample version of the Lepage test (English) |
| Author:
|
Rublík, František |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
41 |
| Issue:
|
6 |
| Year:
|
2005 |
| Pages:
|
[713]-733 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The two-sample Lepage test, devised for testing equality of the location and scale parameters against the alternative that at least for one of the parameters the equality does not hold, is extended to the general case of $k>1$ sampled populations. It is shown that its limiting distribution is the chi-square distribution with $2(k-1)$ degrees of freedom. This $k$-sample statistic is shown to yield consistent test and a formula for its noncentrality parameter under Pitman alternatives is derived. For some particular alternatives, the power of the $k$-sample test is compared with the power of the Kruskal–Wallis test or with the power of the Ansari–Bradley test by means of simulation estimates. Multiple comparison methods for detecting differing populations, based on this multisample version of the Lepage test or on the multisample version of the Ansari–Bradley test, are also constructed. (English) |
| Keyword:
|
multisample rank test for location and scale |
| Keyword:
|
Lepage statistic |
| Keyword:
|
consistency |
| Keyword:
|
non-centrality parameter |
| Keyword:
|
multiple comparisons for location and scale parameters |
| MSC:
|
62E20 |
| MSC:
|
62G10 |
| MSC:
|
62J15 |
| MSC:
|
65C60 |
| idZBL:
|
Zbl 1245.62047 |
| idMR:
|
MR2193861 |
| . |
| Date available:
|
2009-09-24T20:12:39Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135688 |
| . |
| Reference:
|
[1] Ansari A. R., Bradley R. A.: Rank-sum test for dispersions.Ann. Math. Statist. 31 (1960), 1174–1189 MR 0117835, 10.1214/aoms/1177705688 |
| Reference:
|
[2] Chernoff H., Savage I. R.: Asymptotic normality and efficiency of certain non-parametric test statistics.Ann. Math. Statist. 29 (1958), 972–994 MR 0100322, 10.1214/aoms/1177706436 |
| Reference:
|
[3] Conover W. J.: Practical Nonparametric Statistics.Wiley, New York 1999 |
| Reference:
|
[4] Critchlow D. E., Fligner M. A.: On distribution-free multiple comparisons in the one-way analysis of variance.Commun. Statist. Theory Meth. 20 (1991), 127–139 MR 1114636, 10.1080/03610929108830487 |
| Reference:
|
[5] Goria M. N., Vorlíčková D.: On the asymptotic properties of rank statistics for the two-sample location and scale problem.Aplikace matematiky 30 (1985), 425–434 MR 0813531 |
| Reference:
|
[6] Govindajarulu Z., Cam, L. Le, Raghavachari M.: Generalizations of theorems of Chernoff and Savage on the asymptotic normality of test statistics.In: Proc. Fifth Berkeley Symposium on Math. Statist. and Probab., Vol. 1 (1966) (J. Neyman and L. Le Cam, eds.), Univ. of California Press, Berkeley 1967, pp. 609–638 MR 0214193 |
| Reference:
|
[7] Hájek J., Šidák Z.: Theory of Rank Tests.Academia, Prague 1967 Zbl 0944.62045, MR 0229351 |
| Reference:
|
[8] Harter H. L.: Tables of range and studentized range.Ann. Math. Statist. 31 (1960) 1122–1147 Zbl 0106.13602, MR 0123384, 10.1214/aoms/1177705684 |
| Reference:
|
[9] Hayter A. J.: A proof of the conjecture that the Tukey–Kramer multiple comparison procedure is conservative.Ann. Statist. 12 (1984), 61–75 MR 0733499, 10.1214/aos/1176346392 |
| Reference:
|
[10] Hollander M., Wolfe D. A.: Nonparametric Statistical Methods.Wiley, New York 1999 Zbl 0997.62511, MR 1666064 |
| Reference:
|
[11] Koziol J. A., Reid N.: On the asymptotic equivalence of two ranking methods for $k$-sample linear rank statistics.Ann. Statist. 5 (1977), 1099–1106 Zbl 0391.62053, MR 0518897, 10.1214/aos/1176343998 |
| Reference:
|
[12] Kruskal W. H.: A nonparametric test for the several sample problem.Ann. Math. Statist. 23 (1952), 525–540 Zbl 0048.36703, MR 0050850, 10.1214/aoms/1177729332 |
| Reference:
|
[13] Kruskal W. H., Wallis W. A.: Use of ranks in one-criterion variance analysis.J. Amer. Statist. Assoc. 47 (1952), 583–621 Zbl 0048.11703, 10.1080/01621459.1952.10483441 |
| Reference:
|
[14] Lepage Y.: A combination of Wilcoxon’s and Ansari–Bradley’s statistics.Biometrika 58 (1971), 213–217 Zbl 0218.62039, MR 0408101, 10.1093/biomet/58.1.213 |
| Reference:
|
[15] Lepage Y.: A table for a combined Wilcoxon Ansari–Bradley statistic.Biometrika 60 1973), 113–116 Zbl 0256.62041, MR 0331625, 10.1093/biomet/60.1.113 |
| Reference:
|
[16] Mann H. B., Whitney D. R.: On a test whether one of two random variables is stochastically larger than the other.Ann. Math. Statist. 18 (1947), 50–60 MR 0022058, 10.1214/aoms/1177730491 |
| Reference:
|
[17] Miller R. G., Jr.: Simultaneous Statistical Inference.Second edition. Springer–Verlag, New York – Heidelberg 1985 Zbl 0463.62002, MR 0612319 |
| Reference:
|
[18] Puri M. L.: On some tests of homogeneity of variances.Ann. Inst. Stat. Math. 17 (1965), 323–330 Zbl 0161.16202, MR 0196863, 10.1007/BF02868176 |
| Reference:
|
[19] Puri M. L., Sen P. K.: Nonparametric Methods in Multivariate Analysis.Wiley, New York 1971 Zbl 0237.62033, MR 0298844 |
| Reference:
|
[20] Rao C. R., Mitra S. K.: Generalised Inverse of Matrices and its Applications.Wiley, New York 1971 MR 0338013 |
| Reference:
|
[21] Rublík F.: On optimality of the LR tests in the sense of exact slopes.Part II. Application to individual distributions. Kybernetika 25 (1989), 117–135 Zbl 0692.62016, MR 0995954 |
| Reference:
|
[22] Rublík F.: Asymptotic distribution of the likelihood ratio test statistic in the multisample case.Math. Slovaca 49 (1999), 577–598 Zbl 0957.62011, MR 1746901 |
| Reference:
|
[23] Tsai W. S., Duran B. S., Lewis T. O.: Small-sample behavior of some multisample nonparametric tests for scale.J. Amer. Statist. Assoc. 70 (1975), 791–796 Zbl 0322.62048, 10.1080/01621459.1975.10480304 |
| Reference:
|
[24] Wilcoxon F.: Individual comparisons by ranking methods.Biometrics Bull. 1 (1945), 80–83 10.2307/3001968 |
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