Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion

Article

Abstract

We study the asymptotic property of simple estimator of the parameter of a skew Brownian motion when one observes its positions on a fixed grid—or equivalently of a simple random walk with a bias at 0. This estimator, nothing more than the maximum likelihood estimator, is based only on the number of passages of the random walk at 0. It is very simple to set up, is consistent and is asymptotically mixed normal. We believe that this simplified framework is helpful to understand the asymptotic behavior of the maximum likelihood of the skew Brownian motion observed at discrete times which is studied in a companion paper.

Keywords

Skew random walk Skew Brownian motion Maximum likelihood estimator Local asymptotic mixed normality Local time Null recurrent process 

Notes

Acknowledgements

This work has been developed within the framework of the Inria’s Équipe Associée ANESTOC-TOSCA between France and Chile. It is associated to a joint work with E. Mordecki and S. Torres on the estimation of the parameter of the Skew Brownian motion. The author wishes to thank them for interesting discussion on this topic. The author also wishes the referees for their careful reading and having suggested corrections, improvements and for having pointing out the attention toward Bayesian estimation.

References

  1. Alvarez LHR, Salminen P (2016) Timing in the presence of directional predictability: optimal stopping of Skew Brownian Motion (2016). Preprint arxiv:1608.04537
  2. Appuhamillage T, Bokil V, Thomann E, Waymire E, Wood B (2011) Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann Appl Probab 21(1):183–214. doi: 10.1214/10-AAP691 MathSciNetCrossRefMATHGoogle Scholar
  3. Barahona M, Rifo L, Sepúlveda M, Torres S (2016) A simulation-based study on Bayesian estimators for the skew Brownian motion. Entropy 18.7, Paper No. 241, 14. issn: 1099-4300. doi: 10.3390/e18070241
  4. Bass RF, Khoshnevisan D (1993) Rates of convergence to Brownian local time. Stoch Process Appl 47(2):197–213. doi: 10.1016/0304-4149(93)90014-U MathSciNetCrossRefMATHGoogle Scholar
  5. Berry AC (1941) The accuracy of the Gaussian approximation to the sum of independent variates. Trans Am Math Soc 49:122–136MathSciNetCrossRefMATHGoogle Scholar
  6. Blyth CR (1986) Approximate binomial confidence limits. J Am Stat Assoc 81(395):843–855MathSciNetCrossRefMATHGoogle Scholar
  7. Bossy M, Champagnat N, Maire S, Talay D (2010) Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. M2AN Math Model Numer Anal 44(5):997–1048. doi: 10.1051/m2an/2010050 MathSciNetCrossRefMATHGoogle Scholar
  8. Brown LD, Cai TT, DasGupta A (2002) Confidence intervals for a binomial proportion and asymptotic expansions. Ann Stat 30(1):160–201. doi: 10.1214/aos/1015362189 MathSciNetCrossRefMATHGoogle Scholar
  9. Cantrell RS, Cosner C (1999) Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design. Theor Popul Biol 55(2):189–207. doi: 10.1006/tpbi.1998.1397 CrossRefMATHGoogle Scholar
  10. Chung KL, Durrett R (1976) Downcrossings and local time. Z Wahrscheinlichkeitstheorie und Verw Gebiete 35(2):147–149MathSciNetCrossRefMATHGoogle Scholar
  11. Chung KL, Hunt GA (1949) On the zeros of \(\sum _{1}^{n}\pm 1\). Ann Math 50(2):385–400MathSciNetCrossRefMATHGoogle Scholar
  12. Csáki E, Révész P (1983) A combinatorial proof of a theorem of P. Lévy on the local time. Acta Sci Math (Szeged) 45(1-4):119–129MathSciNetMATHGoogle Scholar
  13. Csörgo M, Horváth L (1989) On best possible approximations of local time. Stat Probab Lett 8(4):301–306. doi: 10.1016/0167-7152(89)90036-9 MathSciNetCrossRefMATHGoogle Scholar
  14. Delattre S, Hoffmann M (2002) Asymptotic equivalence for a null recurrent diffusion. Bernoulli 8(2):139–174MathSciNetMATHGoogle Scholar
  15. Esseen C-G (1942) On the Liapounoff limit of error in the theory of probability. Ark Mat Astr Fys 28A(9):19MathSciNetMATHGoogle Scholar
  16. Étoré P (2006) On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients. Electron J Probab 11(9):249–275. doi: 10.1214/EJP.v11-311 MathSciNetCrossRefMATHGoogle Scholar
  17. Fernholz ER, Ichiba T, Karatzas I (2013) Two Brownian particles with rank-based characteristics and skew-elastic collisions. Stoch Process Appl 123(8):2999–3026. doi: 10.1016/j.spa.2013.03.019 MathSciNetCrossRefMATHGoogle Scholar
  18. Florens D (1998) Estimation of the diffusion coefficient from crossings. Stat Inference Stoch Process 1(2):175–195. doi: 10.1023/A:1009927813898 MathSciNetCrossRefMATHGoogle Scholar
  19. Harrison JM, Shepp LA (1981) On skew Brownian motion. Ann Probab 9(2):309–313MathSciNetCrossRefMATHGoogle Scholar
  20. Ibragimov IA, Has’ minskiĭ RZ (1981) Statistical estimation. Asymptotic theory. Applications of mathematics. Springer, New YorkGoogle Scholar
  21. Itô K, McKean HP Jr (1963) Brownian motions on a half line. Ill J Math 7:181–231MathSciNetMATHGoogle Scholar
  22. Itô K, McKean HP Jr (1974) Diffusion processes and their sample paths, 2nd edn. Springer, BerlinMATHGoogle Scholar
  23. Jacod J (1998) Rates of convergence to the local time of a diffusion. Ann Inst H Poincaré Probab Stat 34(4):505–544. doi: 10.1016/S0246-0203(98)80026-5 MathSciNetCrossRefMATHGoogle Scholar
  24. Jeganathan P (1982) On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal. Sankhyā Ser A 44(2):173–212MathSciNetMATHGoogle Scholar
  25. Jeganathan P (1983) Some asymptotic properties of risk functions when the limit of the experiment is mixed normal. Sankhyā Ser A 45(1):66–87MathSciNetMATHGoogle Scholar
  26. Khoshnevisan D (1994) Exact rates of convergence to Brownian local time. Ann Probab 22(3):1295–1330MathSciNetCrossRefMATHGoogle Scholar
  27. Lamperti J (1958) An occupation time theorem for a class of stochastic processes. Trans Am Math Soc 88:380–387MathSciNetCrossRefMATHGoogle Scholar
  28. Le Gall J-F (1985) One-dimensional stochastic differential equations involving the local times of the unknown process. In: Stochastic analysis and applications. Vol. 1095. Lecture Notes in Mathematics. Springer Verlag, 51–82Google Scholar
  29. LeCam L (1953) On some asymptotic properties of maximum likelihood estimates and related Bayes’ estimates. Univ Calif Publ Stat 1:277–329MathSciNetGoogle Scholar
  30. Lejay A Pigato P (2017) Statistical estimation of the Oscillating Brownian Motion. Preprint arxiv:1701.02129
  31. Lejay A (2006) On the constructions of the skew Brownian motion. Probab Surv 3:413–466. doi: 10.1214/154957807000000013 MathSciNetCrossRefMATHGoogle Scholar
  32. Lejay A, Mordecki E, Torres S (2014) Is a Brownian motion skew? Scand J Stat 41(2):346–364. doi: 10.1111/sjos.12033 MathSciNetCrossRefMATHGoogle Scholar
  33. Lejay A, Mordecki E, Torres S (2017) Two consistent estimators for the Skew Brownian motion. PreprintGoogle Scholar
  34. Lejay A, Pichot G (2012) Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps. J Comput Phys 231(21):7299–7314. doi: 10.1016/j.jcp.2012.07.011 MathSciNetCrossRefMATHGoogle Scholar
  35. Lipton A, Sepp A (2011) Filling the Gap. Risk (Oct. 2011), 86–91Google Scholar
  36. Min A, Reeve JD, Xiao M, Xu D (2012) Identification of diffusion coefficient in nonhomogeneous landscapes. Neural information processing. Springer, Berlin. doi: 10.1007/978-3-642-34481-7_36 Google Scholar
  37. Nagaev SV, Chebotarev VI (2011) On an estimate for the closeness of the binomial distribution to the normal distribution. Dokl Akad Nauk 436(1):26–28. doi: 10.1134/S1064562411010030 MathSciNetGoogle Scholar
  38. Nagaev SV, Chebotarev VI (2012) On the estimation of the closeness of the binomial distribution to the normal distribution. Theory Probab Appl 56(2):213–239MathSciNetCrossRefMATHGoogle Scholar
  39. Révész P (1990) Random walk in random and nonrandom environments. World Scientific Publishing, Teaneck. doi: 10.1142/1107 CrossRefMATHGoogle Scholar
  40. Revuz D, Yor M (1999) Continuous martingales and Brownian motion. 3rd ed. Vol. 293. Grundlehren der Mathematischen Wissenschaften.Springer-Verlag, Berlin, doi: 10.1007/978-3-662-06400-9
  41. Robert CP (2007) The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer texts in statistics, 2nd edn. Springer, New YorkGoogle Scholar
  42. Ross TD (2003) Accurate confidence intervals for binomial proportion and Poisson rate estimation. Comput Biol Med 33(6):509–531CrossRefGoogle Scholar
  43. Walsh J (1978) A diffusion with discontinuous local time. In: Temps locaux. Vol. 52–53. Astérisques. SociétéMathématique de France, 37–45Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Université de Lorraine, IECL, UMR 7502Vandœuvre-lès-NancyFrance
  2. 2.CNRS, IECL, UMR 7502Vandœuvre-lés-NancyFrance
  3. 3.InriaVillers-lès-NancyFrance

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