arXiv 19 Sep 2026 · Statistics — Methodology
arXiv:2609.23220 · PDF · Extracted main text
We provide sufficient conditions for the consistency of penalized least squares procedures that select the order (dimension) of a regression model from a sequence of nested classes, allowing for dependent, martingale-difference errors. The main contribution is to relax the classical identifiability requirement: parameters indexing classes larger than the true order need not be identified, provided the additional, excess directions admit a linear approximation to the truth in a neighbourhood of the true parameter. This relaxation lets the number of candidate models grow with the sample size, removing the usual need for a fixed upper bound. We verify the resulting high-level conditions for two classes of nonlinear regression models used in applied work: multiple-regime smooth transition regression and mixture-of-experts models with generic generalized linear model experts. Under a BIC-type penalty, the resulting order selection rule is consistent in either model class whenever the number of candidate models grows slower than the logarithm of the sample size.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Luukkonen, Ritva and Saikkonen, Pentti and Teräsvirta, Timo (1988) Testing linearity against smooth transition autoregressive models | 1.000 | 5 | 3 | 100% |
| 2 | Sara van de Geer (2000) Empirical Processes in M-Estimation | 0.941 | 6 | 3 | 83% |
| 3 | Gassiat, Élisabeth and van Handel, Ramon (2013) Consistent order estimation and minimal penalties | 0.843 | 3 | 3 | 100% |
| 4 | Dacunha-Castelle, Didier and Gassiat, Elisabeth (1999) Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes | 0.737 | 3 | 2 | 100% |
| 5 | Rynkiewicz, Joseph (2016) Asymptotics for Regression Models Under Loss of Identifiability | 0.737 | 3 | 2 | 100% |
| 6 | Davies, Robert B (1977) Hypothesis testing when a nuisance parameter is present only under the alternative | 0.644 | 2 | 2 | 100% |
| 7 | Davies, Robert B (1987) Hypothesis testing when a nuisance parameter is present only under the alternative | 0.644 | 2 | 2 | 100% |
| 8 | van Handel, Ramon (2011) On the minimal penalty for Markov order estimation | 0.644 | 2 | 2 | 100% |
| 9 | Csiszár, I. and Shields, P.C (2000) The consistency of the BIC Markov order estimator | 0.511 | 2 | 1 | 100% |
| 10 | Jacobs, R.A. and Jordan, M.I. and Nowlan, S.J. and Hinton, G.E (1991) Adaptive mixtures of local experts | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 29 scored citations.