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題名 以無母數方法來檢測變異
A nonparametric test for detecting increasing variability
作者 鄭雅文
Cheng, Ya Wen
貢獻者 黃子銘
Huang, Tzee Ming
鄭雅文
Cheng, Ya Wen
關鍵詞 無母數檢定
變異
nonparametric test
variability
日期 2010
上傳時間 5-九月-2013 15:11:55 (UTC+8)
摘要 當我們探討的是兩組樣本的變異是否有所差異時，常見的方法有以ANOVA 為
基礎的檢定與秩檢定，傳統的秩檢定需要假設兩母體具有相同的中位數或知道
其差異。本研究採用Moses (1963) 提出的rank-like 檢定方法，此方法在處理兩組樣本的變異問題時，優點是不需要估計任何中心參數，也不需要假設母體中心參數相同，在資料偏態的情況下也表現得很穩健，我們試圖在樣本數極小的情況下對此方法作修正，將此檢定方法與以ANOVA 為基礎的檢定和秩檢定進行模擬比較，以能夠良好的控制型一誤差與檢定力作為評斷標準。由模擬的結果可得知，rank-like 檢定方法與修正後的方法在不同的分配下皆表現的穩健而修正後的方法特別適用於小樣本的情形。
We consider the problem of detecting variability change in the two-sample case.Several classical variability tests are investigated, including the ANOVA based tests and the rank tests. Traditional two-sample rank tests assume that the location parameters for both samples are identical or of known difference. In this thesis, a modified version of the distribution-free rank-like test proposed by Moses (1963) is proposed. Moses’s test has several advantages. It does not require location parameter estimation, is applicable without assuming that location parameter are identical, and is robust for skewed data. However, Moses’s test has no power when each of the two samples has size 5 or less. The modified version of Moses’s test proposed in this thesis has some power when the sample sizes are small. Comparative
simulation results are presented. According to these results, both Moses’s test and the proposed test are robust under all conditions, and the proposed test
works better when the sample sizes are small.
參考文獻 [1] 洪志真. 監控製程變異之SPC 方法(II). 2003.
[2] M.S. Bartlett. Properties of sufficiency and statistical tests. Proceedings of
the Royal Society of London. Series A-Mathematical and Physical Sciences,
160(901):268, 1937.
[3] R.C. Blair and G.L. Thompson. A distribution-free rank-like test for scale
with unequal population locations. Communications in Statistics: Simulation
and Computation, 21:353–371, 1992.
[4] G.E.P. Box. Non-normality and tests on variances. Biometrika, 40(3/4):318–
335, 1953.
[5] M.B. Brown and A.B. Forsythe. Robust tests for the equality of variances.
Journal of the American Statistical Association, 69:364–367, 1974.
[6] Y.L. Chen. A Test for Two-Sample Problem Based on Sample Spacings.
Tamsui Oxford Journal of Mathematical Sciences, 20(2):267–278, 2004.
[7] H.B. Mann and D.R. Whitney. On a test of whether one of two random
variables is stochastically larger than the other. The Annals of Mathematical
Statistics, 18(1):50–60, 1947.
[8] L.E. Moses. Rank tests of dispersion. The Annals of Mathematical Statistics,
34:973–983, 1963.
[9] R.G. O’Brien. A general ANOVA method for robust tests of additive models
for variances. Journal of the American Statistical Association, 74:877–880,
1979.
[10] S.F. Olejnik and J. Algina. Type I error rates and power estimates of selected
parametric and nonparametric tests of scale. Journal of Educational
Statistics, 12:45–61, 1987.
[11] P.H. Ramsey. Testing variances in psychological and educational research.
Journal of Educational Statistics, 19:23–42, 1994.
[12] P.H. Ramsey and P.P. Ramsey. Updated version of the critical values of the
standardized fourth moment. Journal of statistical computation and simulation,
44(3):231–241, 1993.
[13] P.H. Ramsey and P.P. Ramsey. Testing variability in the two-sample case.
Communications in Statistics: Simulation and Computation, 36(2):233–248,
2007.
[14] L.H. Shoemaker. Tests for differences in dispersion based on quantiles. The
American Statistician, 49(2):179–182, 1995.
[15] S. Siegel and J.W. Tukey. A nonparametric sum of ranks procedure for relative
spread in unpaired samples. Journal of the American Statistical Association,
pages 429–445, 1960.
[16] P. Sprent and N. C. Smeeton. Applied nonparametric statistical methods.
Chapman & Hall Ltd, fourth edition, 2007.
描述 碩士
國立政治大學
統計研究所
98354001
99
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0098354001
資料類型 thesis
dc.contributor.advisor 黃子銘zh_TW
dc.contributor.advisor Huang, Tzee Mingen_US
dc.contributor.author (作者) 鄭雅文zh_TW
dc.contributor.author (作者) Cheng, Ya Wenen_US
dc.creator (作者) 鄭雅文zh_TW
dc.creator (作者) Cheng, Ya Wenen_US
dc.date (日期) 2010en_US
dc.date.accessioned 5-九月-2013 15:11:55 (UTC+8)-
dc.date.available 5-九月-2013 15:11:55 (UTC+8)-
dc.date.issued (上傳時間) 5-九月-2013 15:11:55 (UTC+8)-
dc.identifier (其他 識別碼) G0098354001en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/60437-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計研究所zh_TW
dc.description (描述) 98354001zh_TW
dc.description (描述) 99zh_TW
dc.description.abstract (摘要) 當我們探討的是兩組樣本的變異是否有所差異時，常見的方法有以ANOVA 為
基礎的檢定與秩檢定，傳統的秩檢定需要假設兩母體具有相同的中位數或知道
其差異。本研究採用Moses (1963) 提出的rank-like 檢定方法，此方法在處理兩組樣本的變異問題時，優點是不需要估計任何中心參數，也不需要假設母體中心參數相同，在資料偏態的情況下也表現得很穩健，我們試圖在樣本數極小的情況下對此方法作修正，將此檢定方法與以ANOVA 為基礎的檢定和秩檢定進行模擬比較，以能夠良好的控制型一誤差與檢定力作為評斷標準。由模擬的結果可得知，rank-like 檢定方法與修正後的方法在不同的分配下皆表現的穩健而修正後的方法特別適用於小樣本的情形。		zh_TW
dc.description.abstract (摘要) We consider the problem of detecting variability change in the two-sample case.Several classical variability tests are investigated, including the ANOVA based tests and the rank tests. Traditional two-sample rank tests assume that the location parameters for both samples are identical or of known difference. In this thesis, a modified version of the distribution-free rank-like test proposed by Moses (1963) is proposed. Moses’s test has several advantages. It does not require location parameter estimation, is applicable without assuming that location parameter are identical, and is robust for skewed data. However, Moses’s test has no power when each of the two samples has size 5 or less. The modified version of Moses’s test proposed in this thesis has some power when the sample sizes are small. Comparative
simulation results are presented. According to these results, both Moses’s test and the proposed test are robust under all conditions, and the proposed test
works better when the sample sizes are small.		en_US
dc.description.tableofcontents 1 緒論 7
2 文獻回顧 9
3 研究方法 12
3.1 Moses rank-like 檢定............................. 13
3.2 Moses rank-like 檢定小樣本的改進.................. 14
3.3 Savage 檢定...................................... 15
3.4 Siegel-Tukey 檢定................................ 16
3.5 Conover Squared Rank 檢定........................ 17
3.6 以ANOVA 為基礎的檢定.............................. 17
3.6.1 Brown-Forsythe 檢定........................ 17
3.6.2 O’Brien 檢定.............................. 18
3.6.3 結合Brown-Forsythe 檢定與O’Brien 檢定...... 19
4 模擬分析與討論 20
4.1 模擬設定......................................... 20
4.2 模擬結果與分析.................................... 22
5 結論與建議 37		zh_TW
dc.format.extent 345946 bytes-
dc.format.mimetype application/pdf-
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0098354001en_US
dc.subject (關鍵詞) 無母數檢定zh_TW
dc.subject (關鍵詞) 變異zh_TW
dc.subject (關鍵詞) nonparametric testen_US
dc.subject (關鍵詞) variabilityen_US
dc.title (題名) 以無母數方法來檢測變異zh_TW
dc.title (題名) A nonparametric test for detecting increasing variabilityen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] 洪志真. 監控製程變異之SPC 方法(II). 2003.
[2] M.S. Bartlett. Properties of sufficiency and statistical tests. Proceedings of
the Royal Society of London. Series A-Mathematical and Physical Sciences,
160(901):268, 1937.
[3] R.C. Blair and G.L. Thompson. A distribution-free rank-like test for scale
with unequal population locations. Communications in Statistics: Simulation
and Computation, 21:353–371, 1992.
[4] G.E.P. Box. Non-normality and tests on variances. Biometrika, 40(3/4):318–
335, 1953.
[5] M.B. Brown and A.B. Forsythe. Robust tests for the equality of variances.
Journal of the American Statistical Association, 69:364–367, 1974.
[6] Y.L. Chen. A Test for Two-Sample Problem Based on Sample Spacings.
Tamsui Oxford Journal of Mathematical Sciences, 20(2):267–278, 2004.
[7] H.B. Mann and D.R. Whitney. On a test of whether one of two random
variables is stochastically larger than the other. The Annals of Mathematical
Statistics, 18(1):50–60, 1947.
[8] L.E. Moses. Rank tests of dispersion. The Annals of Mathematical Statistics,
34:973–983, 1963.
[9] R.G. O’Brien. A general ANOVA method for robust tests of additive models
for variances. Journal of the American Statistical Association, 74:877–880,
1979.
[10] S.F. Olejnik and J. Algina. Type I error rates and power estimates of selected
parametric and nonparametric tests of scale. Journal of Educational
Statistics, 12:45–61, 1987.
[11] P.H. Ramsey. Testing variances in psychological and educational research.
Journal of Educational Statistics, 19:23–42, 1994.
[12] P.H. Ramsey and P.P. Ramsey. Updated version of the critical values of the
standardized fourth moment. Journal of statistical computation and simulation,
44(3):231–241, 1993.
[13] P.H. Ramsey and P.P. Ramsey. Testing variability in the two-sample case.
Communications in Statistics: Simulation and Computation, 36(2):233–248,
2007.
[14] L.H. Shoemaker. Tests for differences in dispersion based on quantiles. The
American Statistician, 49(2):179–182, 1995.
[15] S. Siegel and J.W. Tukey. A nonparametric sum of ranks procedure for relative
spread in unpaired samples. Journal of the American Statistical Association,
pages 429–445, 1960.
[16] P. Sprent and N. C. Smeeton. Applied nonparametric statistical methods.
Chapman & Hall Ltd, fourth edition, 2007.		zh_TW