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 November 19, 2010 統計的推測方法の理論的展開とその応用@熊本大学
Contents Introduction Abstract Statistics and Conditions 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 November 19, 2010
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 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 November 19, 2010
Statistics and Conditions Introduction New Results Numerical Simulations Conclusion and Remark Statistics and Conditions November 19, 2010
About Conditions November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark About Conditions November 19, 2010
About Conditions November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark About Conditions November 19, 2010
About Conditions November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark About Conditions {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
New Results November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark New Results {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Asymptotic dist. Under New Condition Introduction New Results Numerical Simulations Conclusion and Remark Asymptotic dist. Under New Condition {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Estimators November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Estimators {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Asymptotic dist. Under New Condition Introduction New Results Numerical Simulations Conclusion and Remark Asymptotic dist. Under New Condition {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Asymptotic dist. Under New Condition Introduction New Results Numerical Simulations Conclusion and Remark Asymptotic dist. Under New Condition {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Proof November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Proof {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Example 1 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Example 1 {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
参考 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark 参考 {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Example 2 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Example 2 {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Example 3 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Example 3 {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
PDF November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark PDF {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Explanation November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Explanation {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Numerical Simulations Introduction New Results Numerical Simulations Conclusion and Remark Numerical Simulations {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Numerical Simulations 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 November 19, 2010
Simulation 1 and 2 Model for Simulation 1 and 2 Simulation 1 Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1 and 2 Model for Simulation 1 and 2 Simulation 1 Simulation 2 November 19, 2010
Simulation 1 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1 November 19, 2010
Simulation 1 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1 November 19, 2010
Simulation 1 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Simulation 1 November 19, 2010
Simulation 2 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Simulation 2 November 19, 2010
Simulation 3 Model for Simulation 3 Asymptotic distribution Introduction New Results Numerical Simulations Conclusion and Remark Simulation 3 Model for Simulation 3 Asymptotic distribution Approximation of the asymptotic distribution November 19, 2010
Estimator November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Estimator November 19, 2010
Simulation 3 November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Simulation 3 November 19, 2010
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 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 {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010
Reference November 19, 2010 Introduction New Results Numerical Simulations Conclusion and Remark Reference {\rm D}\;(\bm{A}_{p})\;\;\; November 19, 2010