Sains Malaysiana 40(5)(2011): 521–525

A Robust Test Based on Bootstrapping for the Two-Sample Scale Problem

(Suatu Ujian Teguh Berdasarkan Kaedah Butstrap untuk Masalah Skala bagi Dua-Sampel)

A.R. Padmanabhan^1,2, Abdul Rahman Othman^1,3 & Teh Sin Yin^1,4*

¹Robust Statistics Computational Laboratory, School of Distance Education

Universiti Sains Malaysia, 11800 Minden, Penang, Malaysia

²Monash University, Clayton, Victoria 3800, Australia

³Institute of Postgraduate Studies, Universiti Sains Malaysia

11800 Minden, Penang, Malaysia

⁴School of Mathematical Sciences, Universiti Sains Malaysia

11800 Minden, Penang, Malaysia

Diserahkan: 22 Mac 2010 / Diterima: 1 Oktober 2010

ABSTRACT

For testing the homogeneity of variances, modifications of well-known tests are known which combine rigorous theory with resampling (bootstrap). We propose versions of these tests, which are computationally simpler (although asymptotically equivalent). The earlier procedures used the smooth bootstrap with two thousand bootstrap replications per sample whereas our proposals use only the classical bootstrap (or percentile method) with just one thousand bootstrap replications per sample, and also required much less computing time. Our proposals cover the Ansari-Bradley-, Mood- and Klotz-tests. We explain their superiority over the existing methodologies available in textbooks and packages.

Keywords: Ansari-Bradley; Bootstrap; Klotz; Mood; test of scale

ABSTRAK

Pada umumnya, pengubahsuaian ujian-ujian terkenal yang menggabungkan teori rapi dengan pensampelan semula (kaedah butstrap) digunakan untuk mengkaji kesamaan varians. Kami mencadangkan versi ujian yang lebih mudah daripada segi pengiraan (meskipun ia setara secara asimtotik). Prosedur yang sebelum ini menggunakan kaedah butstrap licin dengan dua ribu replikasi butstrap setiap sampel. Kami pula mengusulkan penggunaan kaedah butstrap klasik (atau kaedah persentil) dengan hanya seribu replikasi butstrap setiap sampel. Maka masa pengiraan juga jauh lebih singkat. Usul kami merangkumi ujian Ansari-Bradley, Mood dan Klotz. Kami menjelaskan keunggulan ujian-ujian tersebut berbanding dengan kaedah yang tersedia dalam buku teks dan pakej perisian di pasaran.

Kata kunci: Ansari-Bradley; kaedah Butstrap; Klotz; Mood; ujian skala

RUJUKAN

Ansari, A.R. & Bradley, R.A. 1960. Rank-sum tests for dispersions. Ann. Math. Stat. 31: 1174-1189.

Bethea, R.M., Duran, B.S. & Boullion, T.L. 1975. Statistical Methods for Engineers and Scientists. New York: Marcel-Dekker.

Conover, W.J., Johnson, M.E. & Johnson, M.M. 1981. A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics 23: 351-361.

Fligner, M.A. & Hettmansperger, T.P. 1979. On the use of conditional asymptotic normality. J. Roy. Statistical Society Ser. B 41: 178-183.

Fligner, M.A. & Killen, T.J. 1976. Distribution-free two-sample tests for scale. J. Amer. Statistical Assoc. 71: 210-213.

Hald, A. 1967. Theory with Engineering Applications. (7th ed.). New York: Wiley.

Hall, P. & Padmanabhan, A.R. 1997. Adaptive inference for the two-sample scale problem. Technometrics 39(4): 412-422.

Keselman, H.J., Kowalchuk, R.K. & Lix, L.M. 1998a. Robust nonorthogonal analyses revisited: An update based on trimmed means. Psychometrika 63: 145–163.

Keselman, H.J., Lix, L.M. & Kowalchuk, R.K. 1998b. Multiple comparison procedures for trimmed means. Psychol. Methods 3(1): 123-141.

Keselman, H.J., Othman, A.R., Wilcox, R.R. & Fradette, K. 2004. The new and improved two-sample t-test. Psychol. Science 15: 47-51.

Keselman, H.J., Wilcox, R.R., Lix, L.M., Algina, J. & Fradette, K. 2007. Adaptive robust estimation and testing. British J. Math. Statist. Psych. 60: 267-293.

Klotz, J. 1962. Nonparametric tests for scale. Ann. Math. Stat. 32: 498-512.

Mendenhall, W., Wackerly, D.D. & Schaeffer, R.L. 1990. Mathematical Statistics with Applications. (4th ed.). Boston: PWS-Kent.

Mood, A. 1954. On the asymptotic efficiency of certain non-parametric tests. Ann. Math. Stat. 25: 514-522.

Nair, V.N. 1984. On the behavior of some estimators from probability plots. J. Amer. Statistical Assoc. 79: 823-830.

Othman, A.R., Keselman, H.J., Wilcox, R.R., Fradette, K. & Padmanabhan, A.R. 2002. A test of symmetry. J. Mod. Appl. Stat. Meth. 2: 310-315.

Siegel, S. & Tukey, J.W. 1960. A nonparametric sum of ranks procedure for relative spread in unpaired samples. Journal of American Statistical Association 55(291): 429-445.

Steland, A., Padmanabhan, P. & Akram, M. 2007. Resampling Methods for the Nonparametric and Generalized Behrens-Fisher Problem. Sankhya Ser. A (accepted).

*Pengarang untuk surat-menyurat; email: syin.teh@gmail.com