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International Theoretical Science Journal
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DOI 10.34229/KCA2522-9664.24.2.9
UDC 519.21
Henghsiu Tsai1, A.V. Nikitin2


1 Academia Sinica, Taipei, Taiwan

htsai@stat.sinica.edu.tw

2 National University "Ostroh Academy," Ostrog, Ukraine; Jan Kochanowski University, Kielce, Poland

anatolii.nikitin@oa.edu.ua; anatolii.nikitin@ujk.edu.pl

THRESHOLD MODELS FOR LEVY PROCESSES AND APPROXIMATE
MAXIMUM LIKELIHOOD ESTIMATION

Abstract. Using the Levy process (the solution to the Ito–Skorokhod stochastic differential equation) we propose the construction of the model of the threshold process and the approximate maximum likelihood method based on approximation of the logarithmic function of the likelihood of observations. The estimates for the parameters of the two-mode threshold jump process with discretely sampled data are found. We show that by checking the likelihood ratio, determining the presence of threshold effects is possible.

Keywords: threshold jump process, approximate maximum likelihood method, stochastic differential equation.


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REFERENCES

  1. Skorohod A.V. Studies in the theory of random processes. Dover Publication, Reprint, 1962.

  2. Yu T.-H., Tsai H., Rachinger H. Approximate maximum likelihood estimation of a threshold diffusion process. Computational Statistics & Data Analysis. 2020. Vol. 142. 106823.

  3. Rachinger H., Lin E.M.H., Tsai H. A bootstrap test for threshold effects in a diffusion process. Computational Statistics. 2023. https://doi.org/10.1007/s00180-023-01375-z .

  4. Aпt-Sahalia Y. Maximum likelihood estimation of discretely sampled diffusions: А closed form approximation approach. Econometrica. 2002. Vol. 70, N 1. Р. 223–262.

  5. Chan K.-S. Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Ann. Statist. 1993. Vol. 21, N 1. Р. 520–533.

  6. Su F., Chan K.S. Quasi-likelihood estimation of a threshold diffusion process. J. Econometrics. 2015. Vol. 189, N 2. Р. 473–484.

  7. Iacus S. Simulation and inference for stochastic processes with R examples. Springer, 2008. 300 p.

  8. Chabanyuk Y., Nikitin A., Khimka U. Asymptotic analyzes for complex evolutionary systems with Markov and Semi-Markov switching using approximation schemes. Wiley-ISTE, 2020. 240 p.

  9. Knopova V. On recurrence and transience of some Levy-type processes in R. Theory of Probability and Mathematical Statistics. 2023. Vol. 108. P. 59–75.

  10. Gihman I.I., Skorohod A.V. Stochastic differential equations and their applications. Kyiv, Naukova dumka, 1982. 612 p.

  11. Uhlenbeck G.E., Ornstein L.S. On the theory of Brownian motion. Phys. Rev. 1930. Vol. 36. Р. 823–841.

  12. Milstein G.N. Numerical integration of stochastic differential equations. Boston: Kluwer Academic Publishers, 1995. 178 p.




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