A survey on multivariate chi-square distributions and their applications in testing multiple hypotheses

Statistics

Volumn 49, Issue 2, 2015, Pages 427-454

  Dickhaus, Thorsten ^a   Royen, Thomas ^b  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b UNIVERSITY OF APPLIED SCIENCES BINGEN   (Germany)

Author keywords

contingency tables; Kruskal Wallis test; multiple Wald tests; multivariate analysis; multivariate gamma distributions; statistical genetics

Indexed keywords

EID: 84926249663     PISSN: 02331888     EISSN: 10294910     Source Type: Journal    
DOI: 10.1080/02331888.2014.993639     Document Type: Article

References (39)

1. SS Wilks. The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann Math Stat. 1938;9:60–62. doi: 10.1214/aoms/1177732360
2. T Dickhaus. Simultaneous statistical inference with applications in the life sciences. Berlin: Springer; 2014.
3. J Stange, N Loginova, T Dickhaus. Computing and approximating multivariate chi-square probabilities. WIAS Preprint No. 2005. Berlin: Weierstrass Institute for Applied Analysis and Stochastics; 2014 Available from: http://www.wias-berlin.de/preprint/2005/wias_preprints_2005.pdf.
4. KR Gabriel. Simultaneous test procedures – some theory of multiple comparisons. Ann Math Stat. 1969;40:224–250. doi: 10.1214/aoms/1177697819
5. T Dickhaus, J Stange. Multiple point hypothesis test problems and effective numbers of tests for control of the family-wise error rate. Calcutta Statist Assoc Bull. 2013;65:123–144.
6. R Marcus, P Eric, KR Gabriel. On closed test procedures with special reference to ordered analysis of variance. Biometrika. 1976;63:655–660. doi: 10.1093/biomet/63.3.655
7. S Kotz, N Balakrishnan, NL Johnson. Continuous multivariate distributions, 2nd ed., Models and applications, Vol. 1. New York: Wiley, 2000.
8. NH Timm. Applied multivariate analysis. New York: Springer; 2002.
9. D Jensen. A generalization of the multivariate Rayleigh distribution. Sankhyā, Series A. 1970;32:193–208.
10. T Dickhaus. Simultaneous statistical inference in dynamic factor models. SFB 649 Discussion Paper 2012-033, Sonderforschungsbereich 649. Berlin: Humboldt Universität zu; 2012. Available from http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2012-033.pdf.
11. RJ Simes. An improved Bonferroni procedure for multiple tests of significance. Biometrika. 1986;73:751–754. doi: 10.1093/biomet/73.3.751
12. SK Sarkar, C-K Chang. The Simes method for multiple hypothesis testing with positively dependent test statistics. J Am Stat Assoc. 1997;92:1601–1608. doi: 10.1080/01621459.1997.10473682
13. TW Anderson. An introduction to multivariate statistical analysis. 3rd ed. Hoboken, NJ: Wiley; 2003.
14. Th Royen. Multiple comparisons of polynomial distributions. Biom J. 1984;26:319–332. doi: 10.1002/bimj.4710260318
15. T Royen. On some central and non-central multivariate chi-square distributions. Statist Sin. 1995;5:373–397.
16. T Royen. Multivariate gamma distributions. In: Kotz, S, Read CB, Banks DL, editors. Encyclopedia of statistical sciences, Update Volume 1. New York: Wiley-Interscience; 1997. p. 419–425.
17. T Royen. Generalized maximum range tests for pairwise comparisons of several populations. Biom J. 1989;31:905–929. doi: 10.1002/bimj.4710310804
18. T Royen. Tables for studentized multivariate maximum ranges and their application by maximum range tests. Biom J. 1990;32:643–680. doi: 10.1002/bimj.4710320602
19. ML Puri, PK Sen. Nonparametric methods in multivariate analysis, Wiley series in probability and mathematical statistics, New York: Wiley; 1971.
20. WH Kruskal, WA Wallis. Use of ranks in one-criterion variance analysis. J Am Stat Assoc. 1952;47:583–621. doi: 10.1080/01621459.1952.10483441
21. P Sen. On permutational central limit theorems for general multivariate linear rank statistics. Sankhyā, Ser A. 1983;45:141–149.
22. V Moskvina, KM Schmidt. On multiple-testing correction in genome-wide association studies. Genetic Epidemiol. 2008;32:567–573. doi: 10.1002/gepi.20331
23. HW Block, T Costigan, AR Sampson. Product-type probability bounds of higher order. Probab Eng Inf Sci. 1992;6:349–370. doi: 10.1017/S0269964800002588
24. EL Lehmann, JP Romano. Testing statistical hypotheses, 3rd ed. Springer texts in statistics. New York: Springer; 2005.
25. N Katayama. Portmanteau likelihood ratio tests for model selection. Discussion Paper Series 2008-1, Faculty of Economics, Kyushu University; 2008.
26. W Maurer, B Mellein. On new multiple tests based on independent p-values and the assessment of their power. In: Bauer P, Hommel G, Sonnemann E, editors. Multiple Hypothesenprüfung – multiple hypotheses testing. Symposium Gerolstein 1987. Berlin: Springer. Medizinische Informatik und Statistik; Vol. 70; 1988. p. 48–66.
27. T Royen. A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions. Far East J Theor Stat. 2014;48:139–145.
28. PH Westfall, SS Young. Resampling-based multiple testing: examples and methods for p-value adjustment, Wiley series in probability and mathematical statistics. Applied probability and statistics. New York: Wiley; 1993.
29. AS Krishnamoorthy, M Parthasarathy. A multivariate gamma-type distribution. Ann Math Stat. 1951;22:549–557. doi: 10.1214/aoms/1177729544
30. T Royen. Integral representations and approximations for multivariate gamma distributions. Ann Inst Stat Math. 2007;59:499–513. doi: 10.1007/s10463-006-0057-5
31. RC Griffiths. Characterization of infinitely divisible multivariate gamma distributions. J Multivariate Anal. 1984;15:13–20. doi: 10.1016/0047-259X(84)90064-2
32. R Bapat. Infinite divisibility of multivariate gamma distributions and M-matrices. Sankhyā, Ser A. 1989;51:73–78.
33. T Royen. Some representations for convolutions of multivariate gamma distributions. Far East J Theor Stat. 2013;45:21–49.
34. FWJ Olver, DW Lozier, RF Boisvert, CW Clark, editors. NIST handbook of mathematical functions. Cambridge: Cambridge University Press; 2010.
35. T Royen. Expansions for the multivariate chi-square distribution. J Multivariate Anal. 1991;38:213–232. doi: 10.1016/0047-259X(91)90041-Y
36. T Royen. On some multivariate gamma distributions connected with spanning trees. Ann Inst Stat Math. 1994;46:361–371. doi: 10.1007/BF01720592
37. LE Blumenson, KS Miller. Properties of generalized Rayleigh distributions. Ann Math Stat. 1963;34:903–910. doi: 10.1214/aoms/1177704013
38. J Glaz, BM Johnson. Probability inequalities for multivariate distributions with dependence structures. J Am Stat Assoc. 1984;79:436–440. doi: 10.1080/01621459.1984.10478068
39. T Royen. Some upper tail approximations for the distribution of the maximum of correlated chi-square or gamma random variables. Far East J Theor Stat. 2013;43:27–56.