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arxiv: 1906.10967 · v1 · pith:H3CWU5OYnew · submitted 2019-06-26 · 🧮 math.ST · stat.TH

Preliminary test estimation in ULAN models

Davy Paindaveine , Jos\'ea Rasoafaraniaina , Thomas Verdebout This is my paper

Pith reviewed 2026-05-25 15:21 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords preliminary test estimationULAN modelsBahadur representationsasymptotic theorycovariance estimationlinear regressionmultisample modelslocal asymptotic normality
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The pith

In ULAN models, preliminary test estimators based on estimators with generic Bahadur-type representations admit a general asymptotic theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a general asymptotic theory for preliminary test estimators in uniformly locally asymptotically normal models. These estimators arise when there is prior suspicion that the target parameter lies in a submodel. The theory holds whenever the component estimators admit generic Bahadur-type representations. This setup permits direct asymptotic comparisons between the preliminary test estimators and ordinary estimators. The same theory recovers classical results in linear regression and applies to covariance estimation across multiple samples under suspected equality or proportionality.

Core claim

In ULAN models, a general asymptotic theory can be derived for preliminary test estimators based on estimators admitting generic Bahadur-type representations. This allows for a detailed comparison between classical estimators and preliminary test estimators in ULAN models. The results reduce to some classical results in standard linear regression models and are illustrated in multisample setups where m covariance matrices are estimated under the suspicion that they are equal, proportional, or share a common scale. Simulation results confirm the theoretical findings.

What carries the argument

Uniformly locally asymptotically normal (ULAN) models combined with generic Bahadur-type representations of the underlying estimators.

If this is right

  • The results recover known classical statements in standard linear regression.
  • The same asymptotic expansions apply to multisample covariance estimation when equality, proportionality, or common scale is suspected.
  • Direct asymptotic risk comparisons between preliminary test estimators and ordinary estimators become available in any ULAN model.
  • Simulation evidence supports the accuracy of the derived limiting distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Bahadur-based argument could be applied to other locally asymptotically normal families such as certain time-series or spatial models.
  • The framework supplies a route to study preliminary test procedures in high-dimensional regimes where submodel suspicion is frequent.
  • It opens a path toward adaptive versions that choose the preliminary test threshold from the data.

Load-bearing premise

The estimators used to build the preliminary test must admit generic Bahadur-type representations.

What would settle it

In a concrete ULAN model, compute the actual limiting distribution of a preliminary test estimator whose base estimators have Bahadur representations and check whether it matches the derived asymptotic law.

Figures

Figures reproduced from arXiv: 1906.10967 by Davy Paindaveine, Jos\'ea Rasoafaraniaina, Thomas Verdebout.

Figure 1
Figure 1. Figure 1: Illustration of the various situations where asymptotics are de [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of AMSEs θ,τ ( ˆθU), AMSEs θ,τ ( ˆθC) and AMSEs θ,τ ( ˆθPTE) as func￾tions of kδk 2 , for p = 10, r = 1 and α = .05. 4. Two specific applications In this section, we illustrate the general results obtained above on two par￾ticular cases. First, we consider preliminary test estimation in the simple linear regression model and show that we recover for this model and for the considered estimation proble… view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the empirical performance measures tr[Γ [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
read the original abstract

Preliminary test estimation, which is a natural procedure when it is suspected a priori that the parameter to be estimated might take value in a submodel of the model at hand, is a classical topic in estimation theory. In the present paper, we establish general results on the asymptotic behavior of preliminary test estimators. More precisely, we show that, in uniformly locally asymptotically normal (ULAN) models, a general asymptotic theory can be derived for preliminary test estimators based on estimators admitting generic Bahadur-type representations. This allows for a detailed comparison between classical estimators and preliminary test estimators in ULAN models. Our results, that, in standard linear regression models, are shown to reduce to some classical results, are also illustrated in more modern and involved setups, such as the multisample one where $m$ covariance matrices ${\pmb\Sigma}_1, \ldots, {\pmb\Sigma}_m$ are to be estimated when it is suspected that these matrices might be equal, might be proportional, or might share a common "scale". Simulation results confirm our theoretical findings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a general asymptotic theory for preliminary test estimators in uniformly locally asymptotically normal (ULAN) models. It assumes the underlying estimators admit generic Bahadur-type representations, derives the limiting distributions of the preliminary test estimators, and shows that the results reduce to classical findings in linear regression while providing an illustration in the multisample covariance estimation problem (with m covariance matrices suspected to be equal, proportional, or sharing a common scale), supported by simulation studies.

Significance. If the derivations hold, the work supplies a unified framework for comparing preliminary test estimators against classical ones across the broad ULAN class, extending beyond the regression case. The explicit reduction to known linear-regression results functions as an internal consistency check, and the covariance illustration demonstrates applicability to structured modern problems.

minor comments (3)
  1. [Main results section] The abstract states that results reduce to classical ones in linear regression, but the main text should explicitly identify the section or theorem where this reduction is verified (e.g., by matching the limiting distribution or risk expressions).
  2. [Simulation section] Simulation details (sample sizes, number of replications, specific parameter values for the multisample covariance example) are mentioned but not fully specified, which limits reproducibility of the numerical confirmation.
  3. [Introduction and notation] Notation for the preliminary test statistic and the indicator of the submodel should be introduced with a single, self-contained definition early in the paper to avoid repeated cross-references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation establishes asymptotic results for preliminary test estimators in ULAN models by starting from the standard ULAN property and the assumption that underlying estimators admit generic Bahadur representations; these are external inputs, not defined in terms of the target estimators. The paper explicitly reduces its general results to classical linear regression cases as an internal consistency check and illustrates them in the multisample covariance setting without any self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that close the argument. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim relies on the ULAN property of the model and the Bahadur representation property of the estimators, which are assumptions from the domain of asymptotic statistics.

axioms (2)
  • domain assumption Models are uniformly locally asymptotically normal (ULAN).
    The setting for deriving the general theory.
  • domain assumption Estimators admit generic Bahadur-type representations.
    Necessary condition for the asymptotic theory to hold.

pith-pipeline@v0.9.0 · 5715 in / 1026 out tokens · 24547 ms · 2026-05-25T15:21:32.380975+00:00 · methodology

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    E., and Basu, A

    Ahmed, S. E., and Basu, A. K. (2000). Least squares, preliminary test and Stein-type estimation in general vector AR( p) models. Statistica Neerlandica 54, 47–66

    work page 2000

  2. [2]

    G., Norouzirad, M., and Nadarajah, S

    Arashi, M., Kibria, B. G., Norouzirad, M., and Nadarajah, S. (2014). Improved preliminary test and Stein-rule Liu estimators for the ill-conditioned elliptical linear regression model. Journal of Multivariate Analysis 126, 53–74. REFERENCES24

    work page 2014

  3. [3]

    Bancroft, T. A. (1944). On biases in estimation due to the use of preliminary tests of significance. Annals of Mathematical Statistics 15, 190–204

    work page 1944

  4. [4]

    J., and Ritov, Y

    Bickel, P. J., and Ritov, Y. A. (1996). Inference in hidden Markov models I: Local asymptotic normality in the stationary case. Bernoulli 2, 199–228

    work page 1996

  5. [5]

    C., Klaassen, C

    Drost, F. C., Klaassen, C. A., and Werker, B. J. (1997). Adaptive estimation in time-series models. Annals of Statistics 25, 786–817

    work page 1997

  6. [6]

    Francq, C., and Zakoian, J. M. (2013). Inference in nonstationary asymmetric GARCH models. Annals of Statistics 41, 1970–1998

    work page 2013

  7. [7]

    and Verdebout, Th

    Garcia-Portugues, E., Paindaveine, D. and Verdebout, Th. (2019). On optimal tests for rota- tional symmetry against new classes of hyperspherical distributions. Submitted

    work page 2019

  8. [8]

    E., Lieberman, O., and Giles, J

    Giles, D. E., Lieberman, O., and Giles, J. A. (1992). The optimal size of a preliminary test of linear restrictions in a misspecified regression model. Journal of the American Statistical Association 87, 1153–1157

    work page 1992

  9. [9]

    A., and Giles, D

    Giles, J. A., and Giles, D. E. (1993). Pre-test estimation and testing in econometrics: recent developments. Journal of Economic Surveys 7, 145–197

    work page 1993

  10. [10]

    Guta, M., and Kiukas, J. (2015). Equivalence classes and local asymptotic normality in system identification for quantum Markov chains. Communications in Mathematical Physics 335, 1397–1428

    work page 2015

  11. [11]

    and Paindaveine, D

    Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Annals of Statistics 34, 2707–2756

    work page 2006

  12. [12]

    and Paindaveine, D

    Hallin, M. and Paindaveine, D. (2008). Optimal rank-based tests for homogeneity of scatter. Annals of Statistics 34, 1261–1298

    work page 2008

  13. [13]

    and Paindaveine, D

    Hallin, M. and Paindaveine, D. (2009). Optimal tests for homogeneity of covariance, scale, and shape. Journal of Multivariate Analysis 29, 422–444

    work page 2009

  14. [14]

    and Verdebout, Th

    Hallin, M., Paindaveine, D. and Verdebout, Th. (2010). Optimal rank-based testing for principal components. Annals of Statistics 38, 3245–3299

    work page 2010

  15. [15]

    and Verdebout, Th

    Hallin, M., Paindaveine, D. and Verdebout, Th. (2013). Optimal rank-based tests for common principal components. Bernoulli 19, 2524–2556

    work page 2013

  16. [16]

    and Verdebout, Th

    Hallin, M., Paindaveine, D. and Verdebout, Th. (2014). Efficient R-estimation of principal REFERENCES25 and common principal components. Journal of the American Statistical Association 109, 1071–1083

    work page 2014

  17. [17]

    Hallin, M., Taniguchi, M., Serroukh, A., and Choy, K. (1999). Local asymptotic normality for regression models with long-memory disturbance. Annals of Statistics 27, 2054–2080

    work page 1999

  18. [18]

    Kahn, J., and Guta, M. (2009). Local asymptotic normality for finite dimensional quantum systems. Communications in Mathematical Physics 289, 597–652

    work page 2009

  19. [19]

    G., and Saleh, A

    Kibria, B. G., and Saleh, A. M. E. (2004). Preliminary test ridge regression estimators with student’s t errors and conflicting test-statistics. Metrika 59, 105–124

    work page 2004

  20. [20]

    and Verdebout, Th

    Ley, Chr., Swan, Y, Thiam, B. and Verdebout, Th. (2013) Optimal R-estimation of a spherical location. Statistica Sinica 23, 305–333

    work page 2013

  21. [21]

    and Verdebout, Th

    Ley, Chr. and Verdebout, Th. (2017). Modern Directional Statistics . Chapman and Hall, CRC press

    work page 2017

  22. [22]

    Maeyama, Y., Tamaki, K., and Taniguchi, M. (2011). Preliminary test estimation for spectra. Statistics and Probability Letters 81, 1580–1587

    work page 2011

  23. [23]

    Ohtani, K., and Toyoda, T. (1980). Estimation of regression coefficients after a preliminary test for homoscedasticity. Journal of Econometrics 12, 151–159

    work page 1980

  24. [24]

    and Verdebout, Th

    Paindaveine, D., Rasoafaraniaina, J. and Verdebout, Th. (2017). Preliminary test estimation for multi-sample principal components. Econometrics and Statistics 2, 106–116

    work page 2017

  25. [25]

    Saleh, A. M. E. (2006). Theory of Preliminary Test and Stein-type Estimation with Applications (Vol. 517). John Wiley and Sons

    work page 2006

  26. [26]

    K., and Saleh, A

    Sen, P. K., and Saleh, A. M. E. (1979). Nonparametric estimation of location parameter after a preliminary test on regression in the multivariate case. Journal of Multivariate Analysis 9, 322–331

    work page 1979

  27. [27]

    K., and Saleh, A

    Sen, P. K., and Saleh, A. E. (1987). On preliminary test and shrinkage M-estimation in linear models. Annals of Statistics 15, 1580–1592

    work page 1987

  28. [28]

    and Kakizawa, Y

    Tanigushi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Springer

    work page 2000

  29. [29]

    T., Zou, G., and Ohtani, K

    Wan, A. T., Zou, G., and Ohtani, K. (2006). Further results on optimal critical values of pre-test when estimating the regression error variance. The Econometrics Journal 9, 159–176. REFERENCES26 ECARES and D´ epartement de Math´ ematique, Universit´ e libre de Bruxelles (ULB) E-mails: dpaindav@ulb.ac.be, rrasoafa@ulb.ac.be, tverdebo@ulb.ac.be

    work page 2006