pith. sign in

arxiv: 2606.22730 · v1 · pith:BNJZHGR3new · submitted 2026-06-22 · 🧮 math.ST · stat.TH

Optimal Estimating Equations for Compact-Memory Hawkes Processes

Louis Davis , Conor Kresin This is my paper

Pith reviewed 2026-06-26 06:45 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords Hawkes processesestimating equationscompensatormultivariate point processesasymptotic normalityGodambe informationcompact memorysigned kernels
0
0 comments X

The pith

Integrating predictable functionals of a fixed lag window against dN minus lambda dt produces unbiased estimating equations for multivariate Hawkes processes with compact memory and signed kernels.

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

The paper shows that least squares, Takacs-Fiksel, and other moment estimators belong to one family of compensator-based equations whose unbiasedness follows directly from the definition of the intensity. Within this family the likelihood score is the efficient member, and any finite library of regular functionals delivers the same root-T convergence rate, asymptotic normality, and feasible optimal weighting once identification and rank conditions hold. A projection identity then measures the exact efficiency gap between any chosen library and the full score. If the unification holds, users can trade computational cost for statistical precision while retaining explicit guarantees on rates and covariance.

Core claim

For fixed-dimensional multivariate Hawkes processes with compact memory, nonlinear positive links, and signed kernels allowing inhibition, every suitably regular predictable functional of a fixed lag window yields an unbiased estimating equation when integrated against dN minus lambda dt. Under common regularity, identification, and rank conditions, estimators based on every admissible finite library achieve uniform high-probability and pointwise almost-sure O of square root of log T over T rates, asymptotic normality with Godambe covariance, and admit feasible two-step optimal weighting. A projection identity quantifies each library's exact efficiency loss as the score information outside i

What carries the argument

The compensator integral of a predictable functional against dN minus lambda dt, which enforces unbiasedness and supplies the Godambe information matrix for every admissible library.

If this is right

  • Every admissible finite library of functionals yields estimators with uniform high-probability and almost-sure O of square root of log T over T rates.
  • Asymptotic normality holds with covariance given by the Godambe information matrix.
  • Feasible two-step weighting produces the optimal member within each library.
  • The projection identity gives the precise efficiency loss relative to the likelihood score.
  • The root-T rate cannot be improved uniformly over the class, by the two-point bound.

Where Pith is reading between the lines

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

  • The same compensator construction may apply directly to other marked point processes whose intensity admits a compensator representation.
  • Library design can be guided by the projection identity to target specific features such as inhibition without losing the rate guarantees.
  • The logarithmic burn-in for non-stationary starts suggests the estimators remain usable on finite observation windows that begin far from stationarity.
  • The exhaustive character of the compensator class within finite libraries implies that any new moment estimator in this setting can be compared for efficiency without leaving the framework.

Load-bearing premise

The chosen library of functionals and the link function must satisfy identification and rank conditions so that the Godambe matrix remains invertible.

What would settle it

A simulation of a signed-kernel Hawkes process in which the estimator from a finite admissible library exhibits persistent bias or fails to converge at the claimed rate after burn-in.

read the original abstract

Likelihood is standard for Hawkes-process inference, while less computationally demanding methods have largely developed separately. We show that least squares, Tak\'acs--Fiksel, and related moment-based estimators form a single class of compensator-based estimating equations, with the likelihood score as the efficient benchmark. For fixed-dimensional multivariate Hawkes processes with compact memory, nonlinear positive links, and signed kernels allowing inhibition, every suitably regular predictable functional of a fixed lag window yields an unbiased estimating equation when integrated against $\text{d} N-\lambda\,\text{d} t$. Under common regularity, identification, and rank conditions, estimators based on every admissible finite library achieve uniform high-probability and pointwise almost-sure $\mathcal O(\sqrt{\log(T)/T})$ rates, asymptotic normality with Godambe covariance, and admit feasible two-step optimal weighting. A projection identity quantifies each library's exact efficiency loss as the score information outside its predictable span; a two-point bound shows the root-$T$ scale cannot be improved uniformly. Although compact memory localizes the intensity rather than the stationary law, exponential forgetting yields Bernstein-type concentration and transfers the theory to nonstationary starts after a logarithmic burn-in. Within this scope, the compensator class is exhaustive for finite-library comparisons: it contains the score, gives admissible libraries common guarantees, and quantifies their efficiency gaps exactly.

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 unifies least-squares, Takács-Fiksel, and related moment-based estimators for fixed-dimensional multivariate Hawkes processes with compact memory as members of a single class of compensator-based estimating equations. It shows that every suitably regular predictable functional of a fixed lag window yields an unbiased estimating equation when integrated against dN − λ dt. Under common regularity, identification, and rank conditions, every admissible finite library achieves uniform high-probability and pointwise almost-sure O(√(log T / T)) rates, asymptotic normality with Godambe covariance, and admits feasible two-step optimal weighting. The likelihood score is the efficient benchmark within the class; a projection identity quantifies each library’s exact efficiency loss, while a two-point bound shows that the root-T scale cannot be improved uniformly. Exponential forgetting transfers the results to nonstationary initial conditions after a logarithmic burn-in. The compensator class is exhaustive for finite-library comparisons.

Significance. If the derivations hold, the paper supplies a unified, exhaustive theory for a broad family of computationally lighter alternatives to likelihood inference in Hawkes processes. The exact efficiency-gap quantification via projection, the optimality lower bound, and the extension to signed kernels (allowing inhibition) and nonstationary starts are concrete contributions to point-process statistics. The feasible two-step weighting procedure is of immediate practical value.

minor comments (3)
  1. The abstract is information-dense; a short enumerated list of the main theorems would improve readability for readers scanning the claims.
  2. Notation for the Godambe information matrix and the predictable span should be introduced with a brief reminder of their definitions at the first appearance in the main text.
  3. The statement that the compensator class is 'exhaustive for finite-library comparisons' would benefit from an explicit sentence clarifying the precise sense in which no other finite-library estimator lies outside the class.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, recognition of its contributions, and recommendation for minor revision. We are pleased that the unification of compensator-based estimating equations, the efficiency quantification, and the extensions to signed kernels and nonstationary initials are viewed as concrete advances.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the standard compensator property that any integrable predictable integrand against dN − λ dt yields a martingale (hence unbiased estimating equation), which is an external fact about point processes and does not depend on the paper's library choice, link function, or target rates. All subsequent guarantees (O(√(log T / T)) rates, asymptotic normality with Godambe covariance, two-step weighting, projection identity for efficiency loss) are explicitly conditioned on 'common regularity, identification, and rank conditions' rather than being derived from the results themselves or from any fitted parameter. No self-citation, uniqueness theorem, or ansatz is invoked to close the argument; the compensator class is shown to contain the score as benchmark without circular redefinition. The framework therefore remains self-contained against external martingale theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from explicit statements therein. The central claims rest on 'common regularity, identification, and rank conditions' that are not further specified.

axioms (1)
  • domain assumption Common regularity, identification, and rank conditions hold for the chosen library and link functions.
    Abstract states that the rates, normality, and optimal weighting hold 'under common regularity, identification, and rank conditions.'

pith-pipeline@v0.9.1-grok · 5763 in / 1479 out tokens · 18964 ms · 2026-06-26T06:45:58.814361+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

105 extracted references · 40 canonical work pages

  1. [1]

    Hawkes Processes in Finance , journal =

    Bacry, Emmanuel and Mastromatteo, Iacopo and Muzy, Jean-Fran. Hawkes Processes in Finance , journal =. 2015 , doi =. https://doi.org/10.1142/S2382626615500057 , abstract =

    work page doi:10.1142/s2382626615500057 2015

  2. [2]

    An Introduction to Point Processes from a Martingale Point of View , year =

    Bj. An Introduction to Point Processes from a Martingale Point of View , year =

  3. [4]

    , title =

    van der Vaart, Aad W. , title =. n.d. , note =

  4. [5]

    Journal of the American statistical Association , volume=

    Strictly proper scoring rules, prediction, and estimation , author=. Journal of the American statistical Association , volume=. 2007 , publisher=

    2007

  5. [6]

    USSR computational mathematics and mathematical physics , volume=

    The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , author=. USSR computational mathematics and mathematical physics , volume=. 1967 , publisher=

    1967

  6. [8]

    Canonical and grand canonical

    Georgii, Hans-Otto , journal=. Canonical and grand canonical. 1976 , publisher=

    1976

  7. [9]

    doi:10.1080/14697688.2016.1201589

    Matthias Kirchner , title =. Quantitative Finance , volume =. 2017 , publisher =. doi:10.1080/14697688.2016.1211312 , URL =

    work page doi:10.1080/14697688.2016.1211312 2017

  8. [10]

    Estimation of parameterized pair potentials of marked and non-marked

    Fiksel, Thomas , journal=. Estimation of parameterized pair potentials of marked and non-marked. 1984 , URL=

    1984

  9. [12]

    Rate of convergence to equilibrium of marked

    Br. Rate of convergence to equilibrium of marked. Journal of Applied Probability , volume=. 2002 , publisher=

    2002

  10. [14]

    The Annals of Statistics , number =

    Sophie Donnet and Vincent Rivoirard and Judith Rousseau , title =. The Annals of Statistics , number =. 2020 , doi =

    2020

  11. [15]

    Sparse and low-rank multivariate

    Emmanuel Bacry and Martin Bompaire and St. Sparse and low-rank multivariate. Journal of Machine Learning Research , year =

  12. [17]

    The Annals of Statistics , number =

    Felix Cheysson and Gabriel Lang , title =. The Annals of Statistics , number =. 2022 , doi =

    2022

  13. [18]

    The Annals of Statistics , number =

    Patricia Reynaud-Bouret and Sophie Schbath , title =. The Annals of Statistics , number =. 2010 , doi =

    2010

  14. [19]

    Annals of the Institute of Statistical Mathematics , volume =

    Schoenberg, Frederic Paik , title =. Annals of the Institute of Statistical Mathematics , volume =. 2006 , doi =

    2006

  15. [20]

    On the compensator of step processes in progressively enlarged filtrations and related control problems , volume =

    Bandini, Elena and Confortola, Fulvia and Di Tella, Paolo , year =. On the compensator of step processes in progressively enlarged filtrations and related control problems , volume =. Latin American Journal of Probability and Mathematical Statistics , doi =

  16. [22]

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

    Baddeley, A. and Turner, R. and Møller, J. and Hazelton, M. , title =. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume =. 2005 , month =. doi:10.1111/j.1467-9868.2005.00519.x , url =

    work page doi:10.1111/j.1467-9868.2005.00519.x 2005

  17. [23]

    Point Processes and Queues: Martingale Dynamics , series =

    Br. Point Processes and Queues: Martingale Dynamics , series =

  18. [24]

    Information Divergences and Likelihood Ratios of

    Lasse Leskel. Information Divergences and Likelihood Ratios of. IEEE Transactions on Information Theory , year=

  19. [25]

    Stability of nonlinear

    Pierre Br. Stability of nonlinear. The Annals of Probability , number =. 1996 , doi =

    1996

  20. [31]

    2003 , publisher=

    An introduction to the theory of point processes: volume I: elementary theory and methods , author=. 2003 , publisher=

    2003

  21. [32]

    2008 , publisher=

    An introduction to the theory of point processes: volume II: general theory and structure , author=. 2008 , publisher=

    2008

  22. [34]

    Freedman , title =

    David A. Freedman , title =. The Annals of Probability , number =. 1975 , doi =

    1975

  23. [37]

    V. P. Godambe , title =. The Annals of Mathematical Statistics , number =. 1960 , doi =

    1960

  24. [39]

    Journal of applied econometrics , volume=

    Indirect inference , author=. Journal of applied econometrics , volume=. 1993 , publisher=

    1993

  25. [40]

    V. P. Godambe and C. C. Heyde , journal =. Quasi-Likelihood and Optimal Estimation, Correspondent Paper , urldate =

  26. [41]

    Bernoulli , number =

    Niels Richard Hansen and Patricia Reynaud-Bouret and Vincent Rivoirard , title =. Bernoulli , number =. 2015 , doi =

    2015

  27. [43]

    Large Sample Properties of Generalized Method of Moments Estimators , urldate =

    Lars Peter Hansen , journal =. Large Sample Properties of Generalized Method of Moments Estimators , urldate =

  28. [46]

    Newey , journal =

    Whitney K. Newey , journal =. Efficient Instrumental Variables Estimation of Nonlinear Models , urldate =

  29. [49]

    Bernoulli , number =

    D. Bernoulli , number =. 2024 , doi =

    2024

  30. [50]

    Statistics and Computing , volume=

    Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity , author=. Statistics and Computing , volume=. 2023 , publisher=

    2023

  31. [51]

    Hawkes and David Oakes , journal =

    Alan G. Hawkes and David Oakes , journal =. A Cluster Process Representation of a Self-Exciting Process , urldate =

  32. [52]

    Zeitschrift f

    Jacod, Jean , title =. Zeitschrift f. 1975 , volume =. doi:10.1007/BF00536010 , url =

    work page doi:10.1007/bf00536010 1975

  33. [53]

    2003 , publisher=

    Limit theorems for stochastic processes , author=. 2003 , publisher=

    2003

  34. [54]

    2017 , publisher=

    Random measures, theory and applications , author=. 2017 , publisher=

    2017

  35. [55]

    Annals of the Institute of Statistical Mathematics , volume=

    Parametric estimation of spatial--temporal point processes using the Stoyan--Grabarnik statistic , author=. Annals of the Institute of Statistical Mathematics , volume=. 2023 , publisher=

    2023

  36. [57]

    Lectures on the Poisson Process , publisher=

    Last, Günter and Penrose, Mathew , year=. Lectures on the Poisson Process , publisher=

  37. [58]

    Electronic Communications in Probability , number =

    Th. Electronic Communications in Probability , number =. 2024 , doi =

    2024

  38. [59]

    A Method of Simulated Moments for Estimation of Discrete Response Models Without Numerical Integration , urldate =

    Daniel McFadden , journal =. A Method of Simulated Moments for Estimation of Discrete Response Models Without Numerical Integration , urldate =

  39. [60]

    The asymptotic behaviour of maximum likelihood estimators for stationary point processes , author=. Ann. Inst. Statist. Math , volume=

  40. [61]

    Simulation and the Asymptotics of Optimization Estimators , urldate =

    Ariel Pakes and David Pollard , journal =. Simulation and the Asymptotics of Optimization Estimators , urldate =

  41. [62]

    Zeitschrift f\"

    Central limit theorems for local martingales , author=. Zeitschrift f\"

  42. [63]

    Bulletin of the Belgian Mathematical Society - Simon Stevin , number =

    Patricia Reynaud-Bouret and Emmanuel Roy , title =. Bulletin of the Belgian Mathematical Society - Simon Stevin , number =. 2007 , doi =

    2007

  43. [66]

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

  44. [67]

    Wainwright, Martin J. , year=. High-Dimensional Statistics: A Non-Asymptotic Viewpoint , publisher=

  45. [68]

    Bacry, M

    E. Bacry, M. Bompaire, S. Ga \"i ffas, and J.-F. Muzy. Sparse and low-rank multivariate H awkes processes. Journal of Machine Learning Research, 21 0 (50): 0 1--32, 2020. URL http://jmlr.org/papers/v21/15-114.html

    2020

  46. [69]

    Bandini, F

    E. Bandini, F. Confortola, and P. Di Tella. On the compensator of step processes in progressively enlarged filtrations and related control problems. Latin American Journal of Probability and Mathematical Statistics, 21: 0 95, 01 2024. doi:10.30757/ALEA.v21-05

    work page doi:10.30757/alea.v21-05 2024

  47. [70]

    Bj \"o rk

    T. Bj \"o rk. An introduction to point processes from a martingale point of view, 2011. URL https://www.math.kth.se/matstat/fofu/PointProc3.pdf. Lecture notes, KTH Royal Institute of Technology

    2011

  48. [71]

    Bonnet, M

    A. Bonnet, M. Martinez Herrera, and M. Sangnier. Inference of multivariate exponential hawkes processes with inhibition and application to neuronal activity. Statistics and Computing, 33 0 (4): 0 91, 2023. doi:10.1007/s11222-023-10264-w

    work page doi:10.1007/s11222-023-10264-w 2023

  49. [72]

    L. M. Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR computational mathematics and mathematical physics, 7 0 (3): 0 200--217, 1967

    1967

  50. [73]

    Br \'e maud and L

    P. Br \'e maud and L. Massouli \'e . Stability of nonlinear H awkes processes. The Annals of Probability, 24 0 (3): 0 1563 -- 1588, 1996. doi:10.1214/aop/1065725193. URL https://doi.org/10.1214/aop/1065725193

    work page doi:10.1214/aop/1065725193 1996

  51. [74]

    Br \'e maud, G

    P. Br \'e maud, G. Nappo, and G. L. Torrisi. Rate of convergence to equilibrium of marked H awkes processes. Journal of Applied Probability, 39 0 (1): 0 123--136, 2002. doi:10.1239/jap/1019737993

    work page doi:10.1239/jap/1019737993 2002

  52. [75]

    Carrasco and J.-P

    M. Carrasco and J.-P. Florens. On the asymptotic efficiency of GMM . Econometric Theory, 30 0 (2): 0 372–406, 2014. doi:10.1017/S0266466613000340

    work page doi:10.1017/s0266466613000340 2014

  53. [76]

    Chamberlain

    G. Chamberlain. Asymptotic efficiency in estimation with conditional moment restrictions. Journal of Econometrics, 34 0 (3): 0 305--334, 1987. ISSN 0304-4076. doi:https://doi.org/10.1016/0304-4076(87)90015-7. URL https://www.sciencedirect.com/science/article/pii/0304407687900157

    work page doi:10.1016/0304-4076(87)90015-7 1987

  54. [77]

    Cheysson and G

    F. Cheysson and G. Lang. Spectral estimation of Hawkes processes from count data . The Annals of Statistics, 50 0 (3): 0 1722 -- 1746, 2022. doi:10.1214/22-AOS2173. URL https://doi.org/10.1214/22-AOS2173

    work page doi:10.1214/22-aos2173 2022

  55. [78]

    Clinet and N

    S. Clinet and N. Yoshida. Statistical inference for ergodic point processes and application to limit order book. Stochastic Processes and their Applications, 127 0 (6): 0 1800--1839, 2017. ISSN 0304-4149. doi:https://doi.org/10.1016/j.spa.2016.09.014. URL https://www.sciencedirect.com/science/article/pii/S0304414916301740

    work page doi:10.1016/j.spa.2016.09.014 2017

  56. [79]

    Coeurjolly, Y

    J.-F. Coeurjolly, Y. Guan, M. Khanmohammadi, and R. Waagepetersen. Towards optimal T akacs– F iksel estimation. Spatial Statistics, 18: 0 396--411, 2016. ISSN 2211-6753. doi:https://doi.org/10.1016/j.spasta.2016.08.002. URL https://www.sciencedirect.com/science/article/pii/S2211675316300537

    work page doi:10.1016/j.spasta.2016.08.002 2016

  57. [80]

    Proceedings of the National Academy of Sciences , volume =

    K. Cranmer, J. Brehmer, and G. Louppe. The frontier of simulation-based inference. Proceedings of the National Academy of Sciences, 117 0 (48): 0 30055--30062, 2020. doi:10.1073/pnas.1912789117. URL https://www.pnas.org/doi/abs/10.1073/pnas.1912789117

    work page doi:10.1073/pnas.1912789117 2020

  58. [81]

    Cronie and M

    O. Cronie and M. N. M. Van Lieshout. A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika, 105 0 (2): 0 455--462, 06 2018. ISSN 0006-3444. doi:10.1093/biomet/asy001. URL https://doi.org/10.1093/biomet/asy001

    work page doi:10.1093/biomet/asy001 2018

  59. [82]

    D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes: volume I: elementary theory and methods. Springer, 2003

    2003

  60. [83]

    D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes: volume II: general theory and structure. Springer, 2008

    2008

  61. [84]

    S. G. Donald, G. W. Imbens, and W. K. Newey. Choosing instrumental variables in conditional moment restriction models. Journal of Econometrics, 152 0 (1): 0 28--36, 2009. ISSN 0304-4076. doi:https://doi.org/10.1016/j.jeconom.2008.10.013. URL https://www.sciencedirect.com/science/article/pii/S0304407609000566. Recent Adavances in Nonparametric and Semipara...

    work page doi:10.1016/j.jeconom.2008.10.013 2009

  62. [85]

    Donnet, V

    S. Donnet, V. Rivoirard, and J. Rousseau. Nonparametric B ayesian estimation for multivariate H awkes processes . The Annals of Statistics, 48 0 (5): 0 2698 -- 2727, 2020. doi:10.1214/19-AOS1903. URL https://doi.org/10.1214/19-AOS1903

    work page doi:10.1214/19-aos1903 2020

  63. [86]

    Dzhaparidze and J

    K. Dzhaparidze and J. van Zanten . On B ernstein-type inequalities for martingales. Stochastic Processes and their Applications, 93 0 (1): 0 109--117, 2001. ISSN 0304-4149. doi:https://doi.org/10.1016/S0304-4149(00)00086-7. URL https://www.sciencedirect.com/science/article/pii/S0304414900000867

    work page doi:10.1016/s0304-4149(00)00086-7 2001

  64. [87]

    T. Fiksel. Estimation of parameterized pair potentials of marked and non-marked G ibbsian point processes. Elektron. Inform. Kybernet., 20: 0 270--278, 1984. URL https://cir.nii.ac.jp/crid/1570854176676790528

    arXiv 1984

  65. [88]

    A. R. Gallant and G. Tauchen. Which moments to match? Econometric Theory, 12 0 (4): 0 657–681, 1996. doi:10.1017/S0266466600006976

    work page doi:10.1017/s0266466600006976 1996

  66. [89]

    H.-O. Georgii. Canonical and grand canonical G ibbs states for continuum systems. Communications in Mathematical Physics, 48 0 (1): 0 31--51, 1976. doi:10.1007/BF01609410

    work page doi:10.1007/bf01609410 1976

  67. [90]

    Gneiting and A

    T. Gneiting and A. E. Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association, 102 0 (477): 0 359--378, 2007

    2007

  68. [91]

    V. P. Godambe. An Optimum Property of Regular Maximum Likelihood Estimation . The Annals of Mathematical Statistics, 31 0 (4): 0 1208 -- 1211, 1960. doi:10.1214/aoms/1177705693. URL https://doi.org/10.1214/aoms/1177705693

    work page doi:10.1214/aoms/1177705693 1960

  69. [92]

    V. P. Godambe, editor. Estimating Functions. Oxford University Press, 08 1991. ISBN 9780198522287. doi:10.1093/oso/9780198522287.001.0001. URL https://doi.org/10.1093/oso/9780198522287.001.0001

    work page doi:10.1093/oso/9780198522287.001.0001 1991

  70. [93]

    V. P. Godambe and C. C. Heyde. Quasi-likelihood and optimal estimation, correspondent paper. International Statistical Review / Revue Internationale de Statistique, 55 0 (3): 0 231--244, 1987. ISSN 03067734, 17515823. URL http://www.jstor.org/stable/1403403

    arXiv 1987

  71. [94]

    Gourieroux, A

    C. Gourieroux, A. Monfort, and E. Renault. Indirect inference. Journal of applied econometrics, 8 0 (S1): 0 S85--S118, 1993

    1993

  72. [95]

    Y. Guan, A. Jalilian, and R. Waagepetersen. Quasi-likelihood for spatial point processes. Journal of the Royal Statistical Society Series B: Statistical Methodology, 77 0 (3): 0 677--697, 06 2015. ISSN 1369-7412. doi:10.1111/rssb.12083. URL https://doi.org/10.1111/rssb.12083

    work page doi:10.1111/rssb.12083 2015

  73. [96]

    L. P. Hansen. Large sample properties of generalized method of moments estimators. Econometrica, 50 0 (4): 0 1029--1054, 1982. ISSN 00129682, 14680262. URL http://www.jstor.org/stable/1912775

    arXiv 1982

  74. [97]

    N. R. Hansen, P. Reynaud-Bouret, and V. Rivoirard. Lasso and probabilistic inequalities for multivariate point processes . Bernoulli, 21 0 (1): 0 83 -- 143, 2015. doi:10.3150/13-BEJ562. URL https://doi.org/10.3150/13-BEJ562

    work page doi:10.3150/13-bej562 2015

  75. [98]

    M. Hardy. Combinatorics of partial derivatives. The Electronic Journal of Combinatorics, 13 0 (1): 0 R1, Jan. 2006. doi:10.37236/1027. URL https://www.combinatorics.org/ojs/index.php/eljc/article/view/v13i1r1

    work page doi:10.37236/1027 2006

  76. [99]

    A. G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58 0 (1): 0 83--90, 04 1971. ISSN 0006-3444. doi:10.1093/biomet/58.1.83. URL https://doi.org/10.1093/biomet/58.1.83

    work page doi:10.1093/biomet/58.1.83 1971

  77. [100]

    A. G. Hawkes and D. Oakes. A cluster process representation of a self-exciting process. Journal of Applied Probability, 11 0 (3): 0 493--503, 1974. ISSN 00219002. URL http://www.jstor.org/stable/3212693

    arXiv 1974

  78. [101]

    L. Horváth. Gronwall– B ellman type integral inequalities in measure spaces. Journal of Mathematical Analysis and Applications, 202 0 (1): 0 183--193, 1996. ISSN 0022-247X. doi:https://doi.org/10.1006/jmaa.1996.0311. URL https://www.sciencedirect.com/science/article/pii/S0022247X9690311X

    work page doi:10.1006/jmaa.1996.0311 1996

  79. [102]

    Jacod and A

    J. Jacod and A. Shiryaev. Limit theorems for stochastic processes, volume 288. Springer Science & Business Media, 2003

    2003

  80. [103]

    Kallenberg

    O. Kallenberg. Random measures, theory and applications, volume 1. Springer, 2017

    2017

Showing first 80 references.