pith. machine review for the scientific record. sign in

arxiv: 2605.02253 · v1 · submitted 2026-05-04 · 🧮 math.ST · stat.ME· stat.TH

Recognition: unknown

Simultaneous Inference for Nonlinear Time Series, a Sieve M-regression Approach

Tianpai Luo, Zhou Zhou

Pith reviewed 2026-05-08 02:53 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords sieve M-regressionsimultaneous inferencenonlinear time seriesBahadur representationconfidence regionsbootstrapdependent dataempirical processes
0
0 comments X

The pith

A uniform Bahadur representation for sieve M-estimators enables simultaneous confidence regions over the full predictor space in nonlinear time series.

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

This paper develops methods for simultaneous inference on conditional distributions in nonlinear time series models estimated via sieve M-regression. Previous work on these estimators gave only pointwise results, but here a uniform version of the Bahadur representation is proved that works with dependent observations and an increasing number of basis functions. The representation rests on new high-dimensional empirical process bounds for time-dependent data and an M-decomposition technique that keeps the error terms under control. With this in hand the authors obtain a Gaussian approximation for the maximal deviation of the estimator and build simultaneous confidence regions whose validity is guaranteed by explicit error bounds. A self-convolved bootstrap is introduced to make the regions computable in practice.

Core claim

We establish a uniform Bahadur representation for the sieve M-estimator, accommodating dependent data and a growing number of sieve basis functions. A novel high-dimensional empirical process theory is developed for temporally dependent data, and a specifically designed M-decomposition method is utilized to control high-dimensional complexities. Building on this representation, we develop a convex Gaussian approximation to characterize the asymptotic behavior of the estimator and construct valid simultaneous confidence regions (SCRs).

What carries the argument

The uniform Bahadur representation of the sieve M-estimator, which supplies a linear stochastic approximation that holds uniformly over the predictor space and thereby converts the problem of simultaneous inference into a manageable maximal deviation question.

If this is right

  • The constructed simultaneous confidence regions have rigorous coverage guarantees for the entire function.
  • The self-convolved bootstrap provides a practical way to approximate the distribution of the maximal deviation.
  • Error bounds are available that quantify the approximation quality under the stated dependence and sieve conditions.
  • Both simulations and real-data examples support the finite-sample performance of the procedure.

Where Pith is reading between the lines

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

  • This framework could be adapted to construct simultaneous bands for other nonparametric estimators in dependent settings.
  • It opens the door to uniform inference on derived quantities such as derivatives or integrals of the conditional distribution.
  • Similar M-decomposition ideas might simplify high-dimensional arguments in other time-series problems.

Load-bearing premise

The data satisfy mixing or weak dependence conditions and the sieve basis approximates the target function well enough that the remainder terms in the Bahadur expansion remain negligible uniformly.

What would settle it

Generate data from a nonlinear time series satisfying the paper's regularity conditions and check whether the empirical coverage of the constructed simultaneous confidence regions equals the nominal level; systematic undercoverage would refute the claim.

Figures

Figures reproduced from arXiv: 2605.02253 by Tianpai Luo, Zhou Zhou.

Figure 1
Figure 1. Figure 1: The daily log returns of S&P 500 index data. In financial time series analysis, the quantile autoregression (QAR) model is widely used view at source ↗
Figure 2
Figure 2. Figure 2: 95% SCRs based on sieve estimator Qˆ τ (Yi |Yi−1, Yi−2) for quantile τ = 0.02 (left), τ = 0.5 (middle), and τ = 0.98 (right). The transparent surfaces represent the SCRs, and the opaque blue surface is the estimation Qˆ τ (Yi |Yi−1, Yi−2). 7 Definitions and assumptions We now introduce the necessary definitions and regularity assumptions. These assumptions are standard in nonlinear time series models and s… view at source ↗
read the original abstract

This paper studies simultaneous inference of conditional distributions in nonlinear time series from a sieve M-regression perspective. Existing literature on sieve M-regression has primarily focused on pointwise asymptotics, leaving the development of uncertainty quantification over the entire predictor space unexplored. We address this gap by establishing a uniform Bahadur representation for the sieve M-estimator, accommodating dependent data and a growing number of sieve basis functions. A novel high-dimensional empirical process theory is developed for temporally dependent data, and a specifically designed M-decomposition method is utilized to control high-dimensional complexities. Building on this representation, we develop a convex Gaussian approximation to characterize the asymptotic behavior of the estimator and construct valid simultaneous confidence regions (SCRs). To facilitate practical implementation, we introduce a self-convolved bootstrap algorithm that accurately approximates the distribution of the maximal deviation. Our inferential framework is supported by rigorous error bounds and validated through numerical simulations and real data applications.

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

2 major / 2 minor

Summary. The paper develops simultaneous inference for conditional distributions in nonlinear time series via sieve M-regression. It establishes a uniform Bahadur representation for the sieve M-estimator that accommodates dependent data and growing sieve dimension, introduces a novel high-dimensional empirical process theory for temporally dependent data together with an M-decomposition to control complexities, derives a convex Gaussian approximation, and constructs valid simultaneous confidence regions (SCRs) via a self-convolved bootstrap. The framework is supported by rigorous error bounds, numerical simulations, and real-data applications.

Significance. If the uniform Bahadur representation and the supporting high-dimensional process theory hold under the invoked regularity conditions, the work would fill a notable gap by moving sieve M-regression from pointwise asymptotics to uniform inference over the predictor space for dependent data. The new empirical-process tools and practical bootstrap procedure could see broader use in time-series applications requiring simultaneous bands.

major comments (2)
  1. [Development of the high-dimensional empirical process theory (leading to the uniform representation)] The central uniform Bahadur representation and the subsequent convex Gaussian approximation rest on the novel high-dimensional empirical process theory for temporally dependent data and the tailored M-decomposition. These are stated to hold under regularity conditions on mixing/weak dependence and sieve approximation rates, yet the precise growth-rate restrictions (e.g., how fast the sieve dimension may grow relative to the mixing coefficients) are not displayed explicitly enough to verify that the maximal deviation bounds remain valid when dependence decays slowly.
  2. [M-decomposition step in the proof of the uniform Bahadur representation] The M-decomposition is invoked to cancel dependence-induced covariance terms in the remainder. It is unclear whether the decomposition fully controls the cross terms when the mixing coefficients satisfy only the minimal conditions stated; a concrete counter-example or a sharper bound on the remainder under slow mixing would strengthen the claim.
minor comments (2)
  1. [Abstract] The abstract refers to 'rigorous error bounds' without indicating their order (e.g., o_p(1) uniformly or explicit rates). Adding a short statement of the achieved rate would help readers assess the tightness of the approximation.
  2. [Numerical experiments] In the simulation section, the choice of sieve basis (e.g., B-splines versus Fourier) and the precise dependence structure (ARMA order, mixing rate) used in the data-generating process should be stated more explicitly so that the reported coverage probabilities can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment below and have prepared revisions to improve clarity and transparency without altering the core results.

read point-by-point responses

  1. Referee: [Development of the high-dimensional empirical process theory (leading to the uniform representation)] The central uniform Bahadur representation and the subsequent convex Gaussian approximation rest on the novel high-dimensional empirical process theory for temporally dependent data and the tailored M-decomposition. These are stated to hold under regularity conditions on mixing/weak dependence and sieve approximation rates, yet the precise growth-rate restrictions (e.g., how fast the sieve dimension may grow relative to the mixing coefficients) are not displayed explicitly enough to verify that the maximal deviation bounds remain valid when dependence decays slowly.

    Authors: We appreciate the referee's observation. The admissible growth rates for the sieve dimension m_n are implicitly determined by the mixing conditions in Assumption 2.2 and the maximal inequalities derived in Theorems 3.1 and 3.2, which ensure the uniform Bahadur remainder is o_p(1) uniformly over the predictor space. To make these restrictions fully explicit and verifiable, especially for slowly decaying dependence, we will add a dedicated remark (new Remark 3.3) in the revised manuscript that tabulates the required relations between m_n, n, and the mixing coefficients α(k) under both strong and weak mixing regimes. This will directly address the concern without changing the stated assumptions. revision: yes

  2. Referee: [M-decomposition step in the proof of the uniform Bahadur representation] The M-decomposition is invoked to cancel dependence-induced covariance terms in the remainder. It is unclear whether the decomposition fully controls the cross terms when the mixing coefficients satisfy only the minimal conditions stated; a concrete counter-example or a sharper bound on the remainder under slow mixing would strengthen the claim.

    Authors: The M-decomposition (detailed in Appendix B.2) splits the centered empirical process into a martingale-difference sequence plus a remainder whose L2 norm is bounded using covariance inequalities that hold under the minimal α-mixing summability in Assumption 2.2. The cross terms are thereby controlled at the rate required for the uniform representation. We agree that an explicit illustration would improve readability. In the revision we will expand the proof sketch with a short paragraph showing the bound on the remainder under a canonical slow-mixing example (e.g., AR(1) with root close to unity), confirming that no stronger mixing rate is needed. No counter-example is required because the existing maximal inequality already covers the minimal case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation from novel first-principles theory

full rationale

The paper's core chain establishes a uniform Bahadur representation via a newly developed high-dimensional empirical process theory for temporally dependent data and a tailored M-decomposition, then proceeds to convex Gaussian approximation and self-convolved bootstrap for SCRs. No quoted steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the novel theory is presented as independent high-dimensional process control under mixing conditions, making the results self-contained rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger inferred from abstract only; actual free parameters or invented entities cannot be identified without full text.

axioms (2)
  • domain assumption Temporal dependence conditions (mixing or weak dependence) sufficient for high-dimensional empirical process theory
    Invoked to control complexities in the uniform representation and Gaussian approximation for time series.
  • domain assumption Sieve basis functions admit growing dimension while preserving approximation properties
    Required for the uniform Bahadur representation to hold with increasing sieve size.

pith-pipeline@v0.9.0 · 5453 in / 1240 out tokens · 59321 ms · 2026-05-08T02:53:15.899454+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

188 extracted references · 13 canonical work pages

  1. [1]

    Bernoulli , volume=

    Asymptotics of nonparametric L-1 regression models with dependent data , author=. Bernoulli , volume=

  2. [2]

    Journal of Business & Economic Statistics , volume =

    Weichi Wu and Zhou Zhou , title =. Journal of Business & Economic Statistics , volume =. 2017 , publisher =

    2017

  3. [3]

    2023 , eprint=

    Simultaneous inference for monotone and smoothly time-varying functions under complex temporal dynamics , author=. 2023 , eprint=

    2023

  4. [4]

    Hjort, Nils and Pollard, David , year =

  5. [5]

    Advances in neural information processing systems , pages=

    Attention is all you need , author=. Advances in neural information processing systems , pages=

  6. [6]

    Bernoulli, to appear , year=

    Detecting long-range dependence for time-varying linear models , author=. Bernoulli, to appear , year=

  7. [7]

    Bernoulli , number =

    Subhra Sankar Dhar and Weichi Wu , title =. Bernoulli , number =

  8. [8]

    2018 , publisher=

    A gentle introduction to empirical process theory and applications , author=. 2018 , publisher=

    2018

  9. [9]

    Vaart, A. W. van der , year=. Asymptotic Statistics , publisher=

  10. [10]

    Statistical science , pages=

    Asymptotics via empirical processes , author=. Statistical science , pages=. 1989 , publisher=

    1989

  11. [11]

    Journal of the American Statistical Association , volume=

    Restoration of monotonicity respecting in dynamic regression , author=. Journal of the American Statistical Association , volume=. 2017 , publisher=

    2017

  12. [12]

    Econometric Reviews , volume=

    Monotonicity-constrained nonparametric estimation and inference for first-price auctions , author=. Econometric Reviews , volume=. 2021 , publisher=

    2021

  13. [13]

    Biometrika , volume=

    Improving point and interval estimators of monotone functions by rearrangement , author=. Biometrika , volume=. 2009 , publisher=

    2009

  14. [14]

    Journal of statistical planning and inference , volume=

    Simultaneous confidence bands for isotonic functions , author=. Journal of statistical planning and inference , volume=. 2009 , publisher=

    2009

  15. [15]

    Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=

    Confidence bands for isotonic dose-response curves , author=. Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=. 1982 , publisher=

    1982

  16. [16]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

    Simultaneous inference of linear models with time varying coefficients , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=. 2010 , publisher=

    2010

  17. [17]

    Bernoulli , volume=

    A simple nonparametric estimator of a strictly monotone regression function , author=. Bernoulli , volume=. 2006 , publisher=

    2006

  18. [18]

    Computational Statistics & Data Analysis , volume=

    Smooth simultaneous confidence band for the error distribution function in nonparametric regression , author=. Computational Statistics & Data Analysis , volume=. 2021 , publisher=

    2021

  19. [19]

    Annual Review of Economics , volume=

    The econometrics of shape restrictions , author=. Annual Review of Economics , volume=. 2018 , publisher=

    2018

  20. [20]

    The Annals of Statistics , pages=

    Monotone nonparametric regression , author=. The Annals of Statistics , pages=. 1988 , publisher=

    1988

  21. [21]

    Bernoulli , volume=

    Towards a general theory for nonlinear locally stationary processes , author=. Bernoulli , volume=. 2019 , publisher=

    2019

  22. [22]

    Journal of the American Statistical Association , volume=

    A geometrical method for removing edge effects from kernel-type nonparametric regression estimators , author=. Journal of the American Statistical Association , volume=. 1991 , publisher=

    1991

  23. [23]

    Proceedings of the National Academy of Sciences , volume=

    Nonlinear system theory: Another look at dependence , author=. Proceedings of the National Academy of Sciences , volume=. 2005 , publisher=

    2005

  24. [24]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

    Non-crossing non-parametric estimates of quantile curves , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=. 2008 , publisher=

    2008

  25. [25]

    Journal of Econometrics , volume=

    Empirical implementation of nonparametric first-price auction models , author=. Journal of Econometrics , volume=. 2012 , publisher=

    2012

  26. [26]

    Journal of Econometrics , volume=

    Nonparametric option pricing under shape restrictions , author=. Journal of Econometrics , volume=. 2003 , publisher=

    2003

  27. [27]

    Econometrica , volume=

    Assortative matching and search , author=. Econometrica , volume=. 2000 , publisher=

    2000

  28. [28]

    American Economic Journal: Microeconomics , volume=

    Strategic entry deterrence and the behavior of pharmaceutical incumbents prior to patent expiration , author=. American Economic Journal: Microeconomics , volume=

  29. [29]

    Statistica Sinica , volume=

    Change point analysis of correlation in non-stationary time series , author=. Statistica Sinica , volume=. 2019 , publisher=

    2019

  30. [30]

    Journal of Business & Economic Statistics , volume=

    Prediction in locally stationary time series , author=. Journal of Business & Economic Statistics , volume=. 2022 , publisher=

    2022

  31. [31]

    The Annals of Statistics , number =

    Zhou Zhou , title =. The Annals of Statistics , number =

  32. [32]

    The Annals of Statistics , volume=

    Detecting relevant changes in the mean of nonstationary processes—A mass excess approach , author=. The Annals of Statistics , volume=. 2019 , publisher=

    2019

  33. [33]

    Loader , title =

    Jiayang Sun and Clive R. Loader , title =. The Annals of Statistics , number =

  34. [34]

    arXiv preprint arXiv:2107.07257 , year=

    Nonparametric, tuning-free estimation of S-shaped functions , author=. arXiv preprint arXiv:2107.07257 , year=

    work page arXiv

  35. [35]

    arXiv preprint arXiv:2011.00328 , year=

    On the rate of convergence of a deep recurrent neural network estimate in a regression problem with dependent data , author=. arXiv preprint arXiv:2011.00328 , year=

    work page arXiv 2011

  36. [36]

    Annals of Statistics, 43 (5), 2015 , year=

    Nonparametric confidence intervals for monotone functions , author=. Annals of Statistics, 43 (5), 2015 , year=

    2015

  37. [37]

    The Annals of Statistics , volume=

    Confidence intervals for multiple isotonic regression and other monotone models , author=. The Annals of Statistics , volume=. 2021 , publisher=

    2021

  38. [38]

    The Annals of Statistics , volume=

    Coverage of credible intervals in nonparametric monotone regression , author=. The Annals of Statistics , volume=

  39. [39]

    Journal of the American Statistical Association , volume=

    Inference for monotone functions under short-and long-range dependence: Confidence intervals and new universal limits , author=. Journal of the American Statistical Association , volume=. 2016 , publisher=

    2016

  40. [40]

    1969 , publisher=

    Estimation of Isotonic Regression , author=. 1969 , publisher=

    1969

  41. [41]

    Bernoulli , volume=

    On matrix estimation under monotonicity constraints , author=. Bernoulli , volume=. 2018 , publisher=

    2018

  42. [42]

    A general asymptotic scheme for inference under order restrictions , author=. Ann. Statist. , volume=

  43. [43]

    Bernoulli, to appear , year=

    Multiscale jump testing and estimation under complex temporal dynamics , author=. Bernoulli, to appear , year=

  44. [44]

    Statistica Sinica , volume=

    ESTIMATING A MONOTONE TREND , author=. Statistica Sinica , volume=

  45. [45]

    The Annals of Statistics , pages=

    Estimating a smooth monotone regression function , author=. The Annals of Statistics , pages=. 1991 , publisher=

    1991

  46. [46]

    Econometrica: journal of the Econometric Society , pages=

    Identification and estimation of local average treatment effects , author=. Econometrica: journal of the Econometric Society , pages=. 1994 , publisher=

    1994

  47. [47]

    The Annals of Statistics , volume=

    NONPARAMETRIC REGRESSION FOR LOCALLY STATIONARY TIME SERIES , author=. The Annals of Statistics , volume=

  48. [48]

    Bernoulli , volume=

    Nonparametric specification for non-stationary time series regression , author=. Bernoulli , volume=

  49. [49]

    The Annals of Statistics , volume=

    Time-varying nonlinear regression models: nonparametric estimation and model selection , author=. The Annals of Statistics , volume=

  50. [50]

    Journal of the American Statistical Association , volume=

    Estimation and identification of a varying-coefficient additive model for locally stationary processes , author=. Journal of the American Statistical Association , volume=. 2019 , publisher=

    2019

  51. [51]

    2022 , journal =

    Locally Stationary Quantile Regression for Inflation and Interest Rates , author =. 2022 , journal =

    2022

  52. [52]

    Statistica Sinica , number =

    Gaussian Approximations for Non-stationary Multiple Time Series , author =. Statistica Sinica , number =

  53. [53]

    Journal of Econometrics , volume=

    Simultaneous inference for time-varying models , author=. Journal of Econometrics , volume=. 2022 , publisher=

    2022

  54. [54]

    The Annals of Statistics , volume=

    Statistical estimation in varying coefficient models , author=. The Annals of Statistics , volume=. 1999 , publisher=

    1999

  55. [55]

    Biometrika , volume=

    Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data , author=. Biometrika , volume=. 1998 , publisher=

    1998

  56. [56]

    Journal of Mathematical Analysis and Applications , volume=

    Measure preserving transformations and rearrangements , author=. Journal of Mathematical Analysis and Applications , volume=. 1970 , publisher=

    1970

  57. [57]

    1988 , publisher=

    Interpolation of operators , author=. 1988 , publisher=

    1988

  58. [58]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

    Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=. 2000 , publisher=

    2000

  59. [59]

    The annals of Statistics , volume=

    Fitting time series models to nonstationary processes , author=. The annals of Statistics , volume=. 1997 , publisher=

    1997

  60. [60]

    van der Laan and Marco Carone , title =

    Ted Westling and Mark J. van der Laan and Marco Carone , title =. Electronic Journal of Statistics , number =

  61. [61]

    Numerische mathematik , volume=

    Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation , author=. Numerische mathematik , volume=. 1978 , publisher=

    1978

  62. [62]

    1999 , publisher=

    Subsampling , author=. 1999 , publisher=

    1999

  63. [63]

    Bernoulli , number =

    Fabian Mies and Ansgar Steland , title =. Bernoulli , number =. 2023 , doi =

    2023

  64. [64]

    On the Maximal Perimeter of a Convex Set in

    Nazarov, Fedor , booktitle =. On the Maximal Perimeter of a Convex Set in. 2003 , publisher=

    2003

  65. [65]

    The Annals of Probability , pages=

    Central Limit Theorems and Bootstrap in High Dimensions , author=. The Annals of Probability , pages=. 2017 , publisher=

    2017

  66. [66]

    Bulletin of the American Meteorological Society , volume=

    Enlightening global dimming and brightening , author=. Bulletin of the American Meteorological Society , volume=. 2012 , publisher=

    2012

  67. [67]

    Journal of Geophysical Research: Atmospheres , volume=

    Relationship between sunshine duration and temperature trends across Europe since the second half of the twentieth century , author=. Journal of Geophysical Research: Atmospheres , volume=. 2015 , publisher=

    2015

  68. [68]

    Journal of the American Statistical Association , year=2013, volume=

    Zhou Zhou , title=. Journal of the American Statistical Association , year=2013, volume=

    2013

  69. [69]

    international Journal of Climatology , volume=

    Relationship between sunshine duration and air temperature and contemporary global warming , author=. international Journal of Climatology , volume=. 2015 , publisher=

    2015

  70. [70]

    Global warming will happen faster than we think , volume =

    Xu, Yangyang and Ramanathan, Veerabhadran and Victor, David , year =. Global warming will happen faster than we think , volume =. Nature , doi =

  71. [71]

    Global warming of 1.5° C

    Impacts of 1.5 C global warming on natural and human systems , author=. Global warming of 1.5° C. , year=

  72. [72]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

    Estimating smooth monotone functions , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=. 1998 , publisher=

    1998

  73. [73]

    2016 , publisher=

    Mathematical Foundations of Infinite-Dimensional Statistical Models , author=. 2016 , publisher=

    2016

  74. [74]

    Dose Finding Using the Biased Coin Up-and-Down Design and Isotonic Regression , urldate =

    Mario Stylianou and Nancy Flournoy , journal =. Dose Finding Using the Biased Coin Up-and-Down Design and Isotonic Regression , urldate =

  75. [75]

    MATCH Commun

    Monotonic regression for the detection of temporal trends in environmental quality data , author=. MATCH Commun. Math. Comput. Chem , volume=

  76. [76]

    Journal of Climate , volume=

    Global warming and the problem of testing for trend in time series data , author=. Journal of Climate , volume=

  77. [77]

    arXiv preprint arXiv:1412.8724 , year=

    A general framework for robust testing and confidence regions in high-dimensional quantile regression , author=. arXiv preprint arXiv:1412.8724 , year=

    work page arXiv

  78. [78]

    Bernoulli , number =

    Xianyang Zhang and Guang Cheng , title =. Bernoulli , number =. 2018 , doi =

    2018

  79. [79]

    The Annals of Statistics , number =

    Ted Westling and Marco Carone , title =. The Annals of Statistics , number =. 2020 , doi =

    2020

  80. [80]

    Electronic Journal of Statistics , number =

    C. Electronic Journal of Statistics , number =. 2014 , doi =

    2014

Showing first 80 references.