Asymptotic bias reduction of maximum likelihood estimates via penalized likelihoods with differential geometry

Abstract

A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands is developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and a location-scale family are presented.

Figures (4)

• Figure 1: An integral manifold $\phi=\phi_0$, which encodes the penalty function $\tilde{l}$ by implicitization, and a characteristic curve on it. The normal vector field to the integral manifold, ${\rm grad}^*\phi$, and the Monge axis $({\rm grad}f,-\Delta^{(-1)}f/2)$ at a point on the characteristic curve are orthogonal.
• Figure 2: The foliation of the integral manifold $\phi=\phi_0$ by the level surfaces of $\tilde{l}$ projected onto the model manifold $M$ constitute the foliation of $M$ by the level surfaces of $f$. The contours are the level surfaces.
• Figure 3: The foliation of a model manifold $M$ by the equidistant points from the point $\zeta\in M$. A geodesic $\gamma$ joining $\zeta$ and a point $\xi\in M$ is also shown.
• Figure 4: The geodesic distance between a point $\xi=(\mu,\sigma)$ and the point $(0,1)$, which corresponds to the standard density, is the length of the curve shown in the Poincaré metric. Our task is to find the estimate $\hat{\xi}$ such that $t^2(\hat{\xi})$ is an asymptotically unbiased estimate of squared geodesic distance $t^2(\xi)$.

Theorems & Definitions (34)

• Proposition 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 2.4
• Remark 2.5
• Theorem 2.6
• Corollary 2.7
• Example 1
• Remark 2.8
• Theorem 2.9