Deepening statistical inference for stochastic processes via locally non-Gaussian quasi-likelihood

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K05367
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyushu University
• Principal Investigator: Masuda Hiroki 九州大学, 数理学研究院, 教授 (10380669)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: 確率過程モデル / 擬似安定レヴィ過程 / 統計的漸近理論 / 非正規型擬似安定尤度解析

Outline of Final Research Achievements

Concerned with statistical inference for non-ergodic non-Gaussian models, we introduced a locally stable regression model and proved theoretical properties of the locally stable quasi-likelihood estimator (QMLE). Most notably, we have shown that a kind of heteroskedasticity of the scale coefficient makes it possible to sidestep the known annoying degeneracy problem of the asymptotic Fisher-information matrix. As a result, we have established how to jointly estimate the three characteristics of the locally stable model: the activity index, the trend structure, and the scale structure, followed by a way to construct an approximate confidence set. The results obtained in our studies extends the well-established Gaussian QMLE for diffusion models, hence are expected to serve as a series of standards for making inference for models having complex dependence structure as well as the non-Gaussianity quite frequently observed in reality.

Free Research Field

数理統計学，確率過程論

Academic Significance and Societal Importance of the Research Achievements

近年計算機能力の爆発的進歩にともない，複雑な従属性および強い非正規性や時間非一様性が宿る高頻度時系列データが，数多の応用分野で確保可能となってきた．本研究では，これらのデータを適切に扱うための新たな推測理論体系の構築に取り組んだ．特に顕著な研究成果は，本研究で提案された擬似尤度（推定手法）によってモデルの全パラメータの近似信頼集合が構成可能となり，モデルのトレンド構造およびスケール構造のモデル評価基盤が得られたことである．提案された全ての手法は理論的に保証されたものであり，その実行アルゴリズムは明快で再現性も容易に担保できるため，内部機構のわかる統計ソフトウェアでの実装へつながる．