The exact amount of t-ness that the normal model can tolerate
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918413560971264 |
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| author | Hjort, Nils Lid |
| author_facet | Hjort, Nils Lid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26862 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The exact amount of t-ness that the normal model can tolerate Hjort, Nils Lid Methodology Statistics Theory 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. |
| title | The exact amount of t-ness that the normal model can tolerate |
| topic | Methodology Statistics Theory |
| url | https://arxiv.org/abs/2603.26862 |