Evaluation and improvement of the inference based on the ML method for nonregular models

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K00051
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Statistical science
• Research Institution: Hiroshima University
• Principal Investigator: Wakaki Hirofumi 広島大学, 先進理工系科学研究科(理), 教授 (90210856)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: ランダム効果 / 線形混合モデル / 一般化線形混合モデル / ラプラス近似 / 変数選択基準

Outline of Final Research Achievements

In normal linear mixed models and generalised linear mixed models, the AIC criterion is not an asymptotically unbiased estimator of risk because the so-called regularity condition does not hold. In this study, the bias correction for the AIC criterion in growth curve models when the intercept term is random was derived using the Laplace approximation technique. The maximum likelihood estimator of the variance-covariance matrix when several regression coefficients are random was derived, and the bias of the AIC criterion when there are two random coefficients was derived in some asymptotic frameworks.
Asymptotic properties of the solution of the approximate likelihood equation using the Laplace approximation for generalised linear mixed models based on exponential and Poisson distributions were derived in the large-sample and large-sample/high-dimensional frameworks.

Free Research Field

数理統計学

Academic Significance and Societal Importance of the Research Achievements

正規線形混合モデルや一般化線形混合モデルは広く用いられる解析手法であるが、実際の解析場面では、未知母数の最尤推定量漸近正規性を持つことを前提に、検定・推定・モデル選択を行われることが多く、実際の信頼性が期待するものと異なる危険がある。本研究によって、理論的に妥当な解析手法を提案することができる。
本研究においてラプラス近似を、被積分関数が積分区間の内点以外で最大となる場合に拡張することができたが、この結果は近似手法を用いる様々な分野に応用できる。