Analytic perturbations and systematic bias in statistical modeling and inference

TL;DR

This paper investigates how small analytic perturbations in design and covariance matrices can cause significant biases in statistical inference within linear and related models, highlighting potential impacts on solutions and applications.

Contribution

It provides a formal analysis of perturbation effects in statistical models, an area previously unexplored, especially regarding their influence on inference and bias.

Findings

Small perturbations can lead to large biases in estimates.

Perturbation effects are significant in nonlinear models with a single nonlinearity parameter.

Potential applications include various nonlinear statistical modeling scenarios.

Abstract

In this paper we provide a comprehensive study of statistical inference in linear and allied models which exhibit some analytic perturbations in their design and covariance matrices. We also indicate a few potential applications. In the theory of perturbations of linear operators it has been known for a long time that the so-called ``singular perturbations'' can have a big impact on solutions of equations involving these operators even when their size is small. It appears that so far the question of whether such undesirable phenomena can also occur in statistical models and their solutions has not been formally studied. The models considered in this article arise in the context of nonlinear models where a single parameter accounts for the nonlinearity.