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PreprintCC BY 4.0arXiv

Consistent Order Selection under Non-Identifiability and Dependence

  • Eduardo Fonseca Mendes
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Abstract

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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Cite

BibTeXRIS
BibTeX
@misc{mendes2026consistent,
  title = {{Consistent Order Selection under Non-Identifiability and Dependence}},
  author = {Eduardo Fonseca Mendes},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23220},
  eprint = {2609.23220},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23220},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Consistent Order Selection under Non-Identifiability and Dependence
AU  - Eduardo Fonseca Mendes
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23220
UR  - https://arxiv.org/abs/2609.23220
AB  - 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.
ER  -