Prentice (2016, Biometrika) proposed a generalization of the Clayton-Oakes bivariate failure time model, but it is not clear that his generalized model gives a proper multidimensional survivor function. In this talk, we try to find conditions to ensure it, starting from the review of the original Clayton-Oakes model and the key concept of cross ratio.
We study the large sample properties of sparse M-estimators in the presence of pseudo-observations. Our framework covers a broad class of semi-parametric copula models, for which the marginal distributions are unknown and replaced by their empirical counterparts. It is well known that the latter modification significantly alters the limiting laws compared to usual M-estimation. We establish the consistency and the asymptotic normality of our sparse penalized M-estimator and we prove the asymptotic oracle property with pseudo-observations, including the case when the number of parameters is diverging. Our framework allows us to manage copula based loss functions that are potentially unbounded. As additional results, we state the weak limit of multivariate rank statistics and the weak convergence of the empirical copula process indexed by such maps. We apply our inference method to the sparse estimation of the covariance matrix for Gaussian copulas and to the sparse estimation of conditional copula models. This is a joint work with Jean-David Fermanian (ENSAE-CREST).
金融機関のリスク管理において，正規接合関数やt接合関数が用いられることが多いものの，正規接合関数は変量間の裾従属性が弱く，t接合関数はリスクファクターの変動に対して対称の従属関係になってしまい，現実的なモデリングにならない．
本報告では，まず，正規接合関数やt接合関数への非対称性の導入とその性質について，特に裾次数を含む裾従属性の観点から，理論的に整理する．そのうえで，最尤推定を行う際の技術的な問題を整理する．最後に，正規，t及びそれらに非対称性を導入した接合関数の中で，情報量規準で選択されやすい接合関数について議論する．
臨床試験や動物実験で得られる生存時間データから治療効果を推定・検定するために要因実験計画法を採用することがある．生存時間は，打ち切りデータであることや非正規なデータであるため，通常の正規分布に基づく分散分析が適用できない．近年，独立打ち切りの仮定の下，ペアワイズ効果に基づく要因実験解析法が開発された（Dobler and Pauly 2020)．しかしながら，独立打ち切りの仮定が満足されない場合，この手法はバイアスのある結果を与える．本講演では，コピュラモデルを用いて，このペアワイズ効果に基づく要因実験法を修正し，従属打ち切りの下でも適用可能にすること考える．我々の提案手法は，コピュラを正しく特定できれば，バイアスの無い治療効果の推定・検定が可能になる．
A joint mix is a random vector with a constant component-wise sum. It is usually regarded as a concept of extremal negative dependence, and is known to represent the minimizing dependence structure of some common objectives, such as variance, stop-loss premium and expected shortfall, under the marginal constraints. In this talk we explore the connection between the joint mix structure and one of the most popular notions of negative dependence in statistics, called negative orthant dependence. We show that a joint mix does not always have negative dependence, but some natural classes of joint mixes have. In particular, the Gaussian class is characterized as the only elliptical class which supports negatively dependent joint mixes of arbitrary dimension. We also show that a negatively dependent Gaussian joint mix solves a multi-marginal optimal transport problem under uncertainty on the number of components.
We propose a measure of asymmetry between the lower and upper tail probabilities of bivariate distributions. The expression for the proposed measure can be simplified if bivariate distribution functions are represented using copulas. With this representation, it is seen that the proposed measure possesses some desirable properties as a measure of asymmetry. The limit of the proposed measure as the index goes to the boundary of its domain can be expressed in a simple form under certain conditions on copulas. A sample analogue of the proposed measure for a sample from a copula is presented and its weak convergence to a Gaussian process is shown. Another sample analogue of the presented measure is given, which is based on a sample from the original bivariate distribution on the plane. Simple methods for interval estimation are presented. As an example, the presented measure is applied to stock daily returns of S&P500 and Nikkei225. (This study is a joint work with Toshinao Yoshiba and Shinto Eguchi.)
コピュラ（接合関数）とは複数の単変量確率分布の分布関数を組み合わせて，同時分布関数を構築する関数である．コピュラは多変量データがもつ従属性を表現できるため，さまざまな統計モデルで活用されている．しかし，多くのコピュラは変量に関して対称な従属性のみを表現できる一方で，現実のデータではしばしば非対称な従属性が観測される．この問題に対して，コピュラを組み合わせて，新たなコピュラを構築するファクターコピュラやヴァイン・コピュラが研究されている．本発表は，ファクターコピュラから着想を得て，非対称な従属性を表せるコピュラの構築手法とその性質を紹介する．そして，地盤工学における土壌データ分析への応用例を通して，そのコピュラが多様な従属性のモデリングに適していることを示す．（本研究は法政大学の木村光宏氏との共同研究である．）
I propose a class of copulas which utilize trigonometric functions. They are flexible enough to represent asymmetric and/or heteroskedastic dependence. A motivating example is civil war duration and outcome. Let X1 and X2 be length of civil war and the degree to which the rebel succeeds in damaging the government, respectively. Suppose also that X1 and X2|1 follow F1 and F2|1 respectively. According to some works on international relation, either the government or the rebel wins decisively in a short intrastate conflict, while neither side prevails in a war of attrition. On the one hand, when F1(x1) = u1 is small, F2(X2|x1) = U2|1 is more likely to be either small or large. On the other hand, when u1 is large, U2|1 tends to be neither small nor large. There are many other examples of such dependence in social sciences. The maximum dependence trigonometric copulas can represent is modest. Trigonometric copulas can be easily extended to multi-dimensional copulas as well. They are easy, and thus fast, to compute.
周期データとは，単位円周上の点によって表されるようなデータであり，風向などの方向データが代表的な例である．周期データは原点をどこに定めるかによって，同じデータでも表現が変わってしまう．本報告では，そのような周期データに対するコピュラを考察する．原点の定め方に依存しないよう，コピュラの同値類のようなものを考える．そのうえで，周期データ版のコピュラの上限・下限を与える．当日は簡単なシミュレーション結果も報告する．