応用統計学

アブストラクト


累積和,2方向累積和,2重累積和統計量の理論と応用, 121-143,

広津千尋

要旨

累積和および2 重累積和に基づく,形状制約および変化点仮説への総合的接近法を提案する.形状制約は用量反応解析において,パラメトリックモデルを仮定するのが難しい場合に有用なアプローチを与える.変化点仮説はいろいろな場面で時系列の変化を検出するために有用であり,形状制約と密接な関係がある.本論で取り上げる形状制約は単調性,および凸性であり,それぞれ段差,およびスロープ変化点モデルが対応する.提案される統計量は最大対比型と累積χ2 型であり,よく知られた制約付き尤度比検定に比べ統計量の構造が簡単で,非正規モデルや2 元表データ等,多様な問題への拡張が容易である.とくに,2 元表データでは,行や列の水準に自然な順序が有る,無いといういろいろな場合が想定され,それらに応じて興味ある様々な手法が提案される.凸性検定に対して,新たに2 重累積和に基づく方法が提案され,2 階マルコフ性に基づく簡潔なアルゴリズムが展開される.最後に,3 元表データへの拡張について言及される.

英文要旨

Theory and Its Application of the Cumulative, Two-way Cumulative and Doubly Cumulative Sum Statistics

Chihiro Hirotsu

Use of the cumulative and doubly cumulative sum statistics are discussed as a unifying approach to the shape and change-point hypotheses. The shape hypotheses are useful for a dose response analysis where it is usually difficult to assume a rigid parametric model. They are closely related to the change-point hypotheses which are useful for detecting a change of a time series. The proposed statistics are the maximal contrast type and the cumulative chi-squared type. As compared with the well known restricted maximum likelihood approach called the isotonic regression the proposed method is computationally simple and easy to extend to various problems including the non-normal distributions and the two-way data. In particular there are several variations of the two-way data with or without natural ordering in the row and/or columns. Several interesting procedures are introduced according to those variations. The doubly cumulative sum statistics are newly introduced for the analysis under the convexity assumption and an elegant calculation of probability is developed by the second order Markov property of the basic variables. Finally an application to the three-way data is mentioned.


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