The exact amount of t-ness that the normal model can tolerate

2026-03-27

AI Summary

This study addresses the robustness of inference under model misspecification when data are generated from a t-distribution but analyzed using a normal model. Within a local asymptotic framework, it establishes a precise boundary for this robustness: when the degrees-of-freedom parameter satisfies \( m \geq 1.458\sqrt{n} \), the misspecified normal maximum likelihood estimator achieves higher estimation accuracy than the correctly specified yet more complex three-parameter t-model. By integrating “corner asymptotics” theory with scale-mixture modeling, the work proposes a compromise estimator that lies between purely normal and fully non-normal approaches. The results are extended to regression settings and general normal scale mixtures, revealing universal conditions under which model misspecification paradoxically yields superior precision.

Abstract

Suppose that the normal model is used for data $Y_1,\ldots,Y_n$, but that the true distribution is a t-distribution with location and scale parameters $ξ$ and $σ$ and $m$ degrees of freedom. The normal model corresponds to $m=\infty$. Using a local asymptotic framework where $m$ is allowed to increase with $n$ two classes of estimands are identified. One small class, which in particular contains the functions of $ξ$ alone, is only affected by t-ness to the second order, and maximum likelihood estimation in the two- or three-parameter models become equivalent. For all other estimands it is shown that if $m\ge1.458\sqrt{n}$, then maximum likelihood estimation using the incorrect normal model is still more precise than using the correct three-parameter model. This is furthermore shown to be true in regression models with t-distributed residuals. We also propose and analyse compromise estimators that in various ways interpolate between the normal and the nonnormal models. A separate section extends the t-ness results to general normal scale mixtures, in which case the tolerance radius around the normal error distribution takes the form of an upper bound $0.3429/\sqrt{n}$ for the variance of the scale mixture distribution. Proving our results requires somewhat nonstandard `corner asymptotics' since behaviour of estimators must be studied when the crucial parameter $γ=1/m$ is close to zero, which is not an inner point of the parameter space, and the maximum likelihood estimator of $m$ is equal to $\infty$ with positive probability.

Problem

• t-distribution
• normal model misspecification
• robustness
• asymptotic efficiency
• scale mixtures

Innovation

• t-ness tolerance
• local asymptotic framework
• maximum likelihood efficiency
• normal scale mixtures
• corner asymptotics