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

Received: 22 March 2010 / Accepted: 1 October 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

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*Corresponding author; email: syin.teh@gmail.com