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Statistical Inference about the Drift Parameter in Stochastic Processes
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 2, pp. 107-120 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 60G15, 62F03, 62L10
Keywords: Wiener process; Brownian bridge; symmetric process; sequential methods 
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Stibůrek, David. Statistical Inference about the Drift Parameter in Stochastic Processes. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 2, pp. 107-120. http://geodesic.mathdoc.fr/item/AUPO_2013_52_2_a9/

[1] Billingsley, P.: Convergence of Probability Measures. Second Edition, Wiley, New York, 1999. | MR | Zbl

[2] Csörgő, M., Révész, P.: Strong approximations in probability and statistics. Academic Press, New York, 1981. | MR | Zbl

[3] Horrocks, J., Thompson, M. E.: Modeling Event Times with Multiple Outcomes Using the Wiener Process with Drift. Lifetime Data Analysis 10 (2004), 29–49. | DOI | MR | Zbl

[4] Liptser, R. S., Shiryaev, A. N.: Statistics of Random Processes II. Applications. Springer, New York, 2000. | MR

[5] Mörter, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge, 2010. | MR

[6] Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin, 2003. | MR | Zbl

[7] Redekop, J.: Extreme-value distributions for generalizations of Brownian motion. Ph.D. thesis, University of Waterloo, Waterloo, 1995. | MR

[8] Seshadri, V.: The Inverse Gaussian Distribution: Statistical Theory and Applications. Springer, New York, 1999. | MR | Zbl

[9] Steele, J. M.: Stochastic Calculus and Financial Applications. Springer, New York, 2001. | MR | Zbl