Abstract
We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0, T]. We consider regimes where the sampling rate Δ = Δ T → 0 as T → ∞. We propose an adaptive wavelet threshold density estimator and study its performance for the L p loss, p ≥ 1, over Besov spaces. We achieve minimax rates of convergence for sampling rates Δ T that vanish with T at arbitrary polynomial rate. In the same spirit as Buchmann and Grübel (2003) and Duval (2012), the estimation procedure is based on the inversion of the compounding operator. The inverse has no closed form expression and is approached with a fixed point technique.
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References
E. E. Alvarez, “Estimation in Stationary Markov Renewal Processes, with Application to Earthquake Forecasting in Turkey”, Methodology and Computing in Appl. Probab. 7, 119–130 (2005).
M. Bec and C. Lacour, “Adaptive Kernel Estimation of the Lévy Density”, Hal preprint 0058322 (2011).
B. Buchmann and R. Grübel, “Decompounding: an Estimation Problem for Poisson Random Sums”, Ann. Statist. 31, 1054–1074 (2003).
A. Cohen, Numerical Analysis of Wavelet Methods, in Studies in Mathematics and Its Applications (2003), Vol. 32.
F. Comte and V. Genon-Catalot, “Nonparametric Estimation for Pure Jump Lévy Processes Based on High Frequency Data”, Stochastic Processes and Their Applications 119, 4088–4123 (2009).
F. Comte and V. Genon-Catalot, “Nonparametric Adaptive Estimation for Pure Jump Lévy Processes”, Ann. de l’I.H.P., Probab. and Statist. 46, 595–617 (2010).
F. Comte and V. Genon-Catalot, “Estimation for Lévy Processes from High Frequency Data within a Long Time Interval”, Ann. Statist. 39, 803–837 (2011).
H. M. Cuppen, O. Morata, and E. Herbst, “Monte Carlo Simulations of H 2 Formation on Stochastically Heated Grains”, Arxiv preprint 0601554v1 (2006).
J. Dedecker, P. Doukhan, G. Lang, R. J. León, S. Louhichi, and C. Prieur, Weak Dependence. With Examples and Applications, in Lecture Notes in Statistics (Springer, 2007).
D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Density Estimation by Wavelet Thresholding”, Ann. Statist. 24(2), 508–539 (1996).
C. Duval, “Adaptive Wavelet Estimation of a Compound Poisson Process”, Arxiv preprint 12033135 (2012).
S. Fedotov and A. Iomin, “Probabilistic Approach to a Proliferation and Migration Dichotomy in the Tumor Cell Invasion”, Physical Review E 77, 031911 (2008).
J. E. Figueroa-López and C. Houdré, “Risk Bounds for the Nonparametric Estimation of Lévy Processes”, in IMS Lecture Notes-Monogr. Ser., Vol. 51: High dimensional probability (2006), pp. 96–116.
W. Härdle, G. Kerkyacharian, D. Picard, and S. Tsybakov, Wavelets, Approximation, and Statistical Applications, in Lecture Notes in Statistics (Springer, 1998), Vol. 129.
A. Helmstetter and D. Sornette, “Diffusion of Epicenters of Earthquake Aftershocks, Omori’s Law, and Generalized Continuous-Time Random Walk Models”, Amer. Phys, Soc. 66, 061104 (2002).
J. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, and R. Matzler, “In Vivo Anomalous Diffusion and Weak Ergodicity Breaking of Lipid Granules”, Arxiv preprint 10100347v2 (2010).
G. Kerkyacharian and D. Picard, “Thresholding Algorithms, Maxisets and Well-Concentrated Bases”, Test 9(2), 283–344 (2000).
M. Kessler, “Estimation of an Ergodic Diffusion from Discrete Observations”, Scandinav. J. Statist. 24, 211–229 (1997).
M. Kotulski, “Asymptotic Distributions of the Continuous-Time Random Walks: A Probabilistic Approach”, J. Statist. Phys. 81, 777–792 (1995).
T. Lindvall, Lectures on the Coupling Method, in Dover Publications (1992).
J. Masoliver, M. Montero, and J. Perelló, “The Continuous Time Random Walk Formalism in Financial Markets”, Arxiv preprint 0611138v1 (2006).
M. M. Meerschaert and H. P. Scheffler, “Limit Theorems for Continuous-Time Random Walks with Infinite Mean Waiting Times”, J. Appl. Probab. 41, 623–638 (2004).
M. M. Meerschaert and H. P. Scheffler, “Limit Theorems for Continuous Time Random Walks with Slowly Varying Waiting Times”, Statist. Probab. Letters 71, 15–22 (2005).
M. Neumann and M. Rei, “Nonparametric Estimation for Lévy Processes from Low-Frequency Observations”, Bernoulli 15, 223–248 (2009).
I. Rodriguez-Iturbe, D. R. Cox, and V. Isham, “A Point Process Model for Rainfall: Further Developments”, Proc. Roy. Soc. London. Series A, Math. and Phys. Sci. 417(1853), 283–298 (1988).
E. Scalas, R. Gorenflo, H. Luckock, F. Mainardi, M. Mantelli, and M. Raberto, “Anomalous Waiting Times in High-Frequency Financial Data”, Arxiv preprint 0505210v1 (2005).
E. Scalas, “The Application of Continuous-Time Random Walks in Finance and Economics”, Physica A 362, 225–239 (2006).
B. van Es, S. Gugushvili and P. Spreij, “A Kernel Type Nonparametric Density Estimator for Decompounding”, Bernoulli 13, 672–694 (2007).
Y. Vardi, “Nonparametric Estimation in Renewal Processes”, Ann. Statist. 10(3), 772–785 (1982).
N. W. Watkins and D. Credgington, “A Kinetic Equation for Linear Fractional Stable Motion with Applications to Space Plasma Physics”, Arxiv preprint 0803.2833v1 (2008).
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Duval, C. Nonparametric estimation of a renewal reward process from discrete data. Math. Meth. Stat. 22, 28–56 (2013). https://doi.org/10.3103/S106653071301002X
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DOI: https://doi.org/10.3103/S106653071301002X
Keywords
- renewal reward process
- continuous time random walk
- compound Poisson process
- discretely observed random process
- wavelet density estimation