Statistical Inference about the Drift Parameter in Stochastic Processes

David Stibůrek

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2013)

  • Volume: 52, Issue: 2, page 107-120
  • ISSN: 0231-9721

Abstract

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In statistical inference on the drift parameter a in the Wiener process with a constant drift Y t = a t + W t there is a large number of options how to do it. We may, for example, base this inference on the properties of the standard normal distribution applied to the differences between the observed values of the process at discrete times. Although such methods are very simple, it turns out that more appropriate is to use the sequential methods. For the hypotheses testing about the drift parameter it is more proper to standardize the observed process, and to use the sequential methods based on the first time when the process reaches either B or - B , where B > 0 , until some given time. These methods can be generalized to other processes, for instance, to the Brownian bridges.

How to cite

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Stibůrek, David. "Statistical Inference about the Drift Parameter in Stochastic Processes." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 52.2 (2013): 107-120. <http://eudml.org/doc/260770>.

@article{Stibůrek2013,
abstract = {In statistical inference on the drift parameter $a$ in the Wiener process with a constant drift $Y_\{t\} = at+W_\{t\}$ there is a large number of options how to do it. We may, for example, base this inference on the properties of the standard normal distribution applied to the differences between the observed values of the process at discrete times. Although such methods are very simple, it turns out that more appropriate is to use the sequential methods. For the hypotheses testing about the drift parameter it is more proper to standardize the observed process, and to use the sequential methods based on the first time when the process reaches either $B$ or $-B$, where $B>0$, until some given time. These methods can be generalized to other processes, for instance, to the Brownian bridges.},
author = {Stibůrek, David},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Wiener process; Brownian bridge; symmetric process; sequential methods; Wiener process; Brownian bridge; symmetric process; sequential methods},
language = {eng},
number = {2},
pages = {107-120},
publisher = {Palacký University Olomouc},
title = {Statistical Inference about the Drift Parameter in Stochastic Processes},
url = {http://eudml.org/doc/260770},
volume = {52},
year = {2013},
}

TY - JOUR
AU - Stibůrek, David
TI - Statistical Inference about the Drift Parameter in Stochastic Processes
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2013
PB - Palacký University Olomouc
VL - 52
IS - 2
SP - 107
EP - 120
AB - In statistical inference on the drift parameter $a$ in the Wiener process with a constant drift $Y_{t} = at+W_{t}$ there is a large number of options how to do it. We may, for example, base this inference on the properties of the standard normal distribution applied to the differences between the observed values of the process at discrete times. Although such methods are very simple, it turns out that more appropriate is to use the sequential methods. For the hypotheses testing about the drift parameter it is more proper to standardize the observed process, and to use the sequential methods based on the first time when the process reaches either $B$ or $-B$, where $B>0$, until some given time. These methods can be generalized to other processes, for instance, to the Brownian bridges.
LA - eng
KW - Wiener process; Brownian bridge; symmetric process; sequential methods; Wiener process; Brownian bridge; symmetric process; sequential methods
UR - http://eudml.org/doc/260770
ER -

References

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  4. Liptser, R. S., Shiryaev, A. N., Statistics of Random Processes II. Applications, Springer, New York, 2000. (2000) MR1800858
  5. Mörter, P., Peres, Y., Brownian Motion, Cambridge University Press, Cambridge, 2010. (2010) MR2604525
  6. Øksendal, B., Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 2003. (2003) Zbl1025.60026MR2001996
  7. Redekop, J., Extreme-value distributions for generalizations of Brownian motion, Ph.D. thesis, University of Waterloo, Waterloo, 1995. (1995) MR2693357
  8. Seshadri, V., The Inverse Gaussian Distribution: Statistical Theory and Applications, Springer, New York, 1999. (1999) Zbl0942.62011MR1622488
  9. Steele, J. M., Stochastic Calculus and Financial Applications, Springer, New York, 2001. (2001) Zbl0962.60001MR1783083

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