Estimation tail parameter for Geometric Brownian motion
Keywords:
Heavy-tailed; Geometric Brownian Motion; Tail index; Hill estimator; Bootstrap; Double Bootstrap and Direct method*
Abstract
Right-tailed distributions are very important in many applications. There are many studies estimating the tail index. In this paper, we will estimate the tail parameter using the three (the Direct, Bootstrap and Double Bootstrap) methods. Our aim is to illustrate the best way to estimate the -stable with using simulation and real data for the daily Iraqi financial market dataset.
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References
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Hall, P. (1990). Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. Journal of Multivariate Analysis, 32(2), 177-203.
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Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Annals of Statistics, 1163-1174.
Iacus, S. M. (2009). Simulation and Inference for stochastic Differential Equation with R Examples. Springer Science & Business Media.
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Imkeller, P. & Schmalfuss, B. (2001). The conjugacy of stochastic and random differential equations and the existence of global attractors. Journal of Dynamics and Differential Equations, 13(2), 215-249.
Masona, D. M. (1982). Law of Large Number for Sums of Extreme Values. The Annals of Probability, 754-764.
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Nemeth, L. & Zempleni, A. (2020). Regression estimation for the tail index. Journal of Statistical Theory and Practice, 14(3), 1-23.
Peng, L. & Qi, Y. (2017). Inference for Heavy-Tailed Data: Applications in Insurance and Finance. Academic press.
Qi, Y. (2008). Bootstrap and empirical likelihood methods in extremes. Extremes, 11(1), 81-97.
Rolski, T., Schmidli, H., Scmidt, V. & Teugels, J. L. (2009). Stochastic Processes for Insurance and Finance. (Vol. 505). John wiley & Sons.
Bahat, D. , Rabinovich, A. & Frid, V. (2005). Tensile Fracturing in Rocks (p.570). Springer-verlag Berlin Heidelberg.
Caeiro, F., & Gomes, M. I. (2006). A new class of estimators of a “ scale” second order parameter. Extremes, 9(3-4), 193-211.
Ciuperca, G. & Mercadier, C. (2010). Semi-parametric estimation for heavy tailed distributions. Extremes, 13(1), 55-87.
Danielsson, J., de Haan, L., Peng, L. & de Vries, C. G. (2001). Using a bootstrap method to choose the sample fraction in tail index estimation. Journal of Multivariate Analysis, 76(2), 226-248.
Danielsson, J., Ergun, L. M., de Haan, L. & de Vries, C. G. (2016). Tail index estimation: Quantile driven threshold selection. Available at SSRN 2717478.
Drees, H., & Kaufmann, E. (1998). Selecting the optimal sample fraction in univariate extreme value estimation. Stochastic Processes and their applications, 75(2), 149-172.
Dunbar, S. R. (2016). Stochastic Processes and Advanced Mathematical Finance. The Definition of Brownian Motion and the Wiener Process, Department of Mathematics, 203, 68588-0130.
Fisher, B. & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 6, 637- 654.
Foss, S., Korshunov, D. & Zachary, S. (2011). An Introduction to Heavy-Tailed and Sub exponential Distributions. (vol. 6, pp. 0090-6778). New York: springer.
Franke, J. , Hardle, W. K. & Hafner, C. M. (2004). Statistics of Financial Markets ( vol. 2). Berlin: Springer.
Gomes, M. I., De Haan, L. & Peng, L. (2002). Semi-Parametric Estimation of the Second Order Parameter in Statistics of Extremes. Extremes, 59(4), 387-414.
Gomes, M. I., Pestana, D. & Caeiro, F. (2009). A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator. Statistics & Probability Letters, 79(3), 295-303.
Gomes, M.I., & Pestana, D. (2007). A simple second-order reduced bias tail index estimator. Journal of statistical computation and simulation, 77(6), 487-502.
Hall, P. (1990). Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. Journal of Multivariate Analysis, 32(2), 177-203.
Hashemifard, Z., Amindavar, H. & Amini, A. (2016). Parameters Estimation for Continuous-Time Heavy-Tailed Signals Modeled By α- Stable Autoregressive Processes. Digital signal processing, 57, 79-92.
Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Annals of Statistics, 1163-1174.
Iacus, S. M. (2009). Simulation and Inference for stochastic Differential Equation with R Examples. Springer Science & Business Media.
Iacus, S. M. (2011). Option Pricing and Estimation of Financial Model with R. New York: Wiley.
Imkeller, P. & Schmalfuss, B. (2001). The conjugacy of stochastic and random differential equations and the existence of global attractors. Journal of Dynamics and Differential Equations, 13(2), 215-249.
Masona, D. M. (1982). Law of Large Number for Sums of Extreme Values. The Annals of Probability, 754-764.
Mikosch, T. (1999). Regular Variation, Sub exponentiality and Their Applications in Probability Theory (vol.99). Eindhoven, the Netherlands: Eindhoven University of Technology.
Mikosch, T. (2004). Elementary Stochastic Calculus (5 th ed.).World Scientific, Denmark.
Nemeth, L. & Zempleni, A. (2020). Regression estimation for the tail index. Journal of Statistical Theory and Practice, 14(3), 1-23.
Peng, L. & Qi, Y. (2017). Inference for Heavy-Tailed Data: Applications in Insurance and Finance. Academic press.
Qi, Y. (2008). Bootstrap and empirical likelihood methods in extremes. Extremes, 11(1), 81-97.
Rolski, T., Schmidli, H., Scmidt, V. & Teugels, J. L. (2009). Stochastic Processes for Insurance and Finance. (Vol. 505). John wiley & Sons.
Published
2021-09-14
How to Cite
Abd Hassan, N., & Muhannad F. Al-Saadony. (2021). Estimation tail parameter for Geometric Brownian motion. Al-Qadisiyah Journal of Pure Science, 26(5), math 1-15. https://doi.org/10.29350/qjps.2021.26.5.1440
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Section
Mathematics
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