高次元データにおける幾つかの検定統計量の漸近分布について

1. 高次元データにおける幾つかの検定統計量の漸近分布について高次元データにおける幾つかの検定統計量の漸近分布について 藤本翔太1, 狩野裕1, Muni.S.Srivastava2 1大阪大学基礎工学研究科 2Department of Statistics, University of Toronto 統計的推測方法の理論的展開とその応用@熊本大学

2. Introduction New Results Numerical Simulations Conclusion and Remark Contents • Introduction • Abstract • Statistics and Conditions • New Conditions • New Results • Asymptotic Dist. under New Condition • Examples • Numerical Simulations • Conclusion and Remark

3. Introduction New Results Numerical Simulations Conclusion and Remark Abstract • 平均ベクトルの検定問題： • 伝統的な方法： • 高次元データ(n<p)では定義されない • Dempster (1958), Bai and Saranadasa (1996), Fujikoshi (2004), Srivastava (2007), Srivastava and Du (2008)などによって，高次元データにも対応できる検定方法が提案 • 共分散行列または相関行列に非常に強い仮定 現実的な条件に先行研究の結果を拡張

4. Introduction New Results Numerical Simulations Conclusion and Remark Statistics and Conditions

5. Introduction New Results Numerical Simulations Conclusion and Remark About Conditions

6. Introduction New Results Numerical Simulations Conclusion and Remark About Conditions

7. Introduction New Results Numerical Simulations Conclusion and Remark About Conditions

8. Introduction New Results Numerical Simulations Conclusion and Remark New Results

9. Introduction New Results Numerical Simulations Conclusion and Remark Asymptotic dist. Under New Condition

10. Introduction New Results Numerical Simulations Conclusion and Remark Estimators

11. Introduction New Results Numerical Simulations Conclusion and Remark Asymptotic dist. Under New Condition

12. Introduction New Results Numerical Simulations Conclusion and Remark Asymptotic dist. Under New Condition

13. Introduction New Results Numerical Simulations Conclusion and Remark Proof

14. Introduction New Results Numerical Simulations Conclusion and Remark Example 1

15. Introduction New Results Numerical Simulations Conclusion and Remark 参考

16. Introduction New Results Numerical Simulations Conclusion and Remark Example 2

17. Introduction New Results Numerical Simulations Conclusion and Remark Example 3

18. Introduction New Results Numerical Simulations Conclusion and Remark PDF

19. Introduction New Results Numerical Simulations Conclusion and Remark Explanation

20. Introduction New Results Numerical Simulations Conclusion and Remark Numerical Simulations

21. Introduction New Results Numerical Simulations Conclusion and Remark Numerical Simulations • 目的 • 条件(A),(B),(C)は充たさず，条件(D)を充たすモデルに関して，従来の検定法(正規近似)と新しい検定法を比較 • 方法 • 各統計量に対する近似法のActual Error ProbabilityをMonte Carlo法で計算 • 各パラメータの設定

22. Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1 and 2 • Model for Simulation 1 and 2 • Simulation 1 • Simulation 2

23. Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1

24. Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1

25. Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1

26. Introduction New Results Numerical Simulations Conclusion and Remark Simulation 2

27. Introduction New Results Numerical Simulations Conclusion and Remark Simulation 3 • Model for Simulation 3 • Asymptotic distribution • Approximation of the asymptotic distribution

28. Introduction New Results Numerical Simulations Conclusion and Remark Estimator

29. Introduction New Results Numerical Simulations Conclusion and Remark Simulation 3

30. Introduction New Results Numerical Simulations Conclusion and Remark Conclusion and Remark • 高次元データにおける1標本問題 • 先行研究よりも現実的な条件を仮定 • 漸近分布が共分散行列または相関行列に依存 • Numerical Simulation • 提案した検定法の良さを確認 • 分散の構造を間違えると，検定結果が信頼できない • 平均の検定の前に分散の構造を検討すべき • 問題点と今後の課題 • 真の分散の構造が分かっていることが前提 • 分散の構造を仮定しない検定法の提案が今後の課題

31. Introduction New Results Numerical Simulations Conclusion and Remark Reference