## Electronic Journal of Statistics

### When is it no longer possible to estimate a compound Poisson process?

Céline Duval

#### Abstract

We consider centered compound Poisson processes with finite variance, discretely observed over $[0,T]$ and let the sampling rate $\Delta=\Delta_{T}\rightarrow\infty$ as $T\rightarrow\infty$. From the central limit theorem, the law of each increment converges to a Gaussian variable. Then, it should not be possible to estimate more than one parameter at the limit. First, from the study of a parametric example we identify two regimes for $\Delta_{T}$ and we observe how the Fisher information degenerates. Then, we generalize these results to the class of compound Poisson processes. We establish a lower bound showing that consistent estimation is impossible when $\Delta_{T}$ grows faster than $\sqrt{T}$. We also prove an asymptotic equivalence result, from which we identify, for instance, regimes where the increments cannot be distinguished from Gaussian variables.

#### Article information

Source
Electron. J. Statist. Volume 8, Number 1 (2014), 274-301.

Dates
First available in Project Euclid: 31 March 2014

https://projecteuclid.org/euclid.ejs/1396271048

Digital Object Identifier
doi:10.1214/14-EJS885

Mathematical Reviews number (MathSciNet)
MR3189556

Zentralblatt MATH identifier
1293.62010

#### Citation

Duval, Céline. When is it no longer possible to estimate a compound Poisson process?. Electron. J. Statist. 8 (2014), no. 1, 274--301. doi:10.1214/14-EJS885. https://projecteuclid.org/euclid.ejs/1396271048

#### References

• [1] Alexandersson, H. (1985). A simple stochastic mode of a precipitation process. Journal of Climate and Applied Meteorology, 24, 1285–1295.
• [2] Athanassoulis, G.A. and Gavriliadis, P.N. (2002). The truncated Hausdorff moment problem solved by using kernel density functions. Probabilistic Engineering Mechanics, 17(3), 273–291.
• [3] Brown, L.D., Carter, A.V., Low, M.G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. The Annals of Statistics, 32, 2074–2097.
• [4] Cont, R. and De Larrard, A. (2013). Price dynamics in a Markovian limit order market. SIAM Journal on Financial Mathematics, 4(1), 1–25.
• [5] Duval, C. and Hoffmann, M. (2011). Statistical inference across time scales. Electronic Journal of Statistics, 5, 2004–2030.
• [6] Embrechts, P., Klüppelberg, C. and Mikosch, M. (1997). Modeling Extremal Events. Springer.
• [7] Gerber, H.U. and Shiu, E. (1998). Pricing perpetual options for jump processes. The North American Actuarial Journal, 2, 101–112.
• [8] Guilbaud, F. and Pham, H. (2012). Optimal high-frequency trading in a pro-rata microstructure with predictive information. Arxiv preprint 1205.3051v1.
• [9] Hong, K.J. and Satchell, S. (2012). Defining single asset price momentum in terms of a stochastic process. Theoretical Economics Letters, 2, 274–277.
• [10] Kotulski, M. (1995). Asymptotic distributions of the continuous-time random walks: A probabilistic approach. Journal of Statistical Physics, 81, 777–792.
• [11] Le Cam, L. and Yang, L.G. (2000). Asymptotics in Statistics: Some Basic Concepts, 2nd edition. Springer-Verlag, New York.
• [12] Levy, J.B. and Taqqu, M.S. (2000). Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards. Bernoulli, 6, 23–44.
• [13] Masoliver, J., Montero, M., Perelló, J. and Weiss, G.H. (2008). Direct and inverse problems with some generalizations and extensions. Arxiv preprint 0308017v2.
• [14] Metzler, R. and Klafter, J. (2004). The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A, 37, 161–208.
• [15] Mikosch, T. (2009). Non-life Insurance Mathematics: An Introduction with the Poisson Process. Springer.
• [16] Önalan, Ö. (2010). Fractional Ornstein-Uhlenbeck processes driven by stable Lévy motion in finance. International Research Journal of Finance and Economics, 42, 129–139.
• [17] Rodriguez, G. and Seatzu, S. (1992). On the solution of the finite moment problem. Journal of Mathematical Analysis and Applications, 171(2), 321–333.
• [18] Russell, J.R. and Engle, R.F. (2005). A discrete-state continuous-time model of financial transactions prices and times. Journal of Business & Economic Statistics, 23(2).
• [19] Shevtsova, I. (2007). On the accuracy of the normal approximation to the distributions of Poisson random sums. PAMM, 7(1), 2080025–2080026.
• [20] Sobezyk, K. and Trȩbicki, J. (1990). Maximum entropy principle in stochastic dynamics. Probabilistic Engineering Mechanics, 5(3), 102–110.
• [21] Tagliani, A. (1999). Hausdorff moment problem and maximum entropy: A unified approach. Applied Mathematics and Computation, 105(2), 291–305.
• [22] Tsybakov, A.B. (2008). Introduction to Nonparametric Estimation. Springer.
• [23] Uchaikin, V.V. and Zolotarev, V.M. (1999). Chance and Stability. Stable Distributions and Their Applications. VSP, Utrecht.
• [24] Watson, G.N. (1922). A Treatise on the Theory of Bessel Functions. Cambridge University Press.