Let f: [a,b] → ℝ be an unknown 2 times differentiable function and consider M to be an α- homogeneous Poisson process on Graf(f). The goal is to estimate f having a sample of the inhomogeneous Poisson process N constructed by dislocating each point of M perpendicularly to Graf(f) by a normal random variable with zero mean and constant variance σ2. The exact formulas for the mean measure and the intensity function of N are obtained. Then, the function f is estimated directly using a hybrid spline approach to penalized maximum likelihood. Simulation results indicate the procedure to be consistent as α → ∞ and σ2 → 0. © 2006 Springer Science+Business Media, LLC.1111134Daley, D.J., Vere-Jones, D., (1988) An Introduction to the Theory of Point Processes, , Springer-Verlag New YorkDias, R., Density estimation via hybrid splines (1998) J Stat Comput Simul, 60, pp. 277-294Dias, R., Sequential adaptive non parametric regression via H-splines (1999) Commun Stat: Comput Simul, 28, pp. 501-515Dudewicz, E.J., Mishra, S.N., (1988) Modern Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, , Wiley Inc. New YorkGarcia, N.L., Maximum likelihood estimation and suboptimal stopping time for a Poisson point process (1995) Rebrape, 9, pp. 67-82. , 1Kutoyants, Y.A., Intensity parameter estimation of an inhomogeneous Poisson process (1979) Probl Control Inform Theory/Probl Upravlen Teor Inform, 8, pp. 137-149. , 2Kutoyants, Y.A., (1998) Statistical Inference for Spatial Poisson Processes, , Springer-Verlag New York Lecture Notes in Statistics, vol. 134Luo, Z., Wahba, G., Hybrid adaptive splines (1997) J Am Stat Assoc, 92, pp. 107-116Ramlau-Hansen, H., Smoothing counting process intensities by means of kernel functions (1983) Ann Stat, 11, pp. 453-466. , 2Resnick, S., (1992) Adventures in Stochastic Processes, , Birkhäuser Boston Inc. Boston, MAWahba, G., (1990) Spline Models for Observational Data, , SIAM Philadelphia, PAWegman, E.J., Wright, I.W., Splines in statistics (1983) J Am Stat Assoc, 78, pp. 351-365