Electronic Journal of Statistics

Estimation of a delta-contaminated density of a random intensity of Poisson data

Daniela De Canditiis and Marianna Pensky

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In the present paper, we constructed an estimator of a delta contaminated mixing density function $g(\lambda)$ of an intensity $\lambda$ of the Poisson distribution. The estimator is based on an expansion of the continuous portion $g_{0}(\lambda)$ of the unknown pdf over an overcomplete dictionary with the recovery of the coefficients obtained as the solution of an optimization problem with Lasso penalty. In order to apply Lasso technique in the, so called, prediction setting where it requires virtually no assumptions on the dictionary and, moreover, to ensure fast convergence of Lasso estimator, we use a novel formulation of the optimization problem based on the inversion of the dictionary elements.

We formulate conditions on the dictionary and the unknown mixing density that yield a sharp oracle inequality for the norm of the difference between $g_{0}(\lambda)$ and its estimator and, thus, obtain a smaller error than in a minimax setting. Numerical simulations and comparisons with the Laguerre functions based estimator recently constructed by [8] also show advantages of our procedure. At last, we apply the technique developed in the paper to estimation of a delta contaminated mixing density of the Poisson intensity of the Saturn’s rings data.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 683-705.

Received: August 2015
First available in Project Euclid: 16 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62C12: Empirical decision procedures; empirical Bayes procedures
Secondary: 62P35: Applications to physics

Mixing density Poisson distribution empirical Bayes Lasso penalty


De Canditiis, Daniela; Pensky, Marianna. Estimation of a delta-contaminated density of a random intensity of Poisson data. Electron. J. Statist. 10 (2016), no. 1, 683--705. doi:10.1214/16-EJS1118. https://projecteuclid.org/euclid.ejs/1458133058

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  • [1] Antoniadis, A., Sapatinas, T. (2004). Wavelet shringkage for natural exponential families with quadratic variance functions, Biometrika, 88, 805–820.
  • [2] Besbeas, P., De Feis, I., Sapatinas, T. (2004). A comparative simulation study of wavelet shrinkage estimators for Poisson counts, International Statistical Review, 72, 209–237.
  • [3] Brown, L. D., Cai, T. T., Zhou, H. (2010). Nonparametric regression in exponential families, Ann. Statist., 38, 2005–2046.
  • [4] Bruni, V., De Canditiis, D., Vitulano, D. (2012). Time-scale energy based analysis of contours of real-world shapes, Mathematics and Computer in Simulation, 82, 2891–2907.
  • [5] Bühlmann, P., van de Geer, S. (2011)., Statistics for High-Dimensional Data: Methods, Theory and Applications, Springer.
  • [6] Chow, Y.-S., Geman, S., Wu, L.-D. (1983). Consistent cross-validated density estimaton, Ann. Statist., 11, 25–38.
  • [7] Colwell, J. E., Nicholson, P. D., Tiscareno, M. S., Murray, C. D., French, R. G., Marouf, E. A. (2009). The Structure of Saturn’s Rings, in, Saturn from Cassini-Huygens, Dougherty, M., Esposito, L., Krimigis, S. Eds., Springer.
  • [8] Comte, F., Genon-Catalot, V. (2015). Adaptive Laguerre density estimation for mixed Poisson models, Electr. Journ. Statist., 9, 1113–1149.
  • [9] Esposito, L. W., Barth, C. A., Colwell, J. E., Lawrence, G. M., McClintock, W. E., Stewart, A. I. F., Keller, H. U., Korth, A., Lauche, H., Festou, M. C., Lane, A. L., Hansen, C. J., Maki, J. N., West, R. A., Jahn, H., Reulke, R., Warlich, K., Shemansky, D. E., Yung, Y. L. (2004). The Cassini ultraviolet imaging spectrograph investigation, Space Sci. Rev., 115, 294–361.
  • [10] Fryzlewicz, P., Nason G. P. (2004). A Haar-Fisz algorithm for Poisson intensity estimation, Journ. Computat. Graph. Statis., 13, 621–638.
  • [11] Harmany, Z., Marcia, R., Willett, R. (2012). This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms: Theory and Practice, IEEE Trans. Image Processing, 21, 1084–1096.
  • [12] Herngartner, N. W. (1997). Adaptive demixing in Poisson mixture models, Ann. Stat., 25, 917–928.
  • [13] Hirakawa, K., Wolfe, P. J. (2012). Skellam shrinkage: Wavelet-based intensity estimation for inhomogeneous Poisson data, IEEE Trans. Inf. Theory, 58, 1080–1093.
  • [14] Kolaczyk, E. D. (1999). Bayesian multiscale models for Poisson processes, Journ. Amer. Statist. Assoc., 94, 920–933.
  • [15] Lambert, D., Tierney, L. (1984). Asymptotic properties of maximum likelihood estimates in the mixed Poisson model, Ann. Statist., 12, 1388-1399.
  • [16] Lord, D., Washington, S. P., Ivan, J. N. (2005). Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: Balancing statistical fit and theory, Accid. Anal. Prevent., 37, 35–46.
  • [17] Mallat, S. (2009)., A Wavelet Tour of Signal Processing: The Sparse Way, 3rd ed. Elsevier.
  • [18] Pensky, M. (2016). Solution of linear ill-posed problems using overcomplete dictionaries, Ann. Stat., to appear.
  • [19] Timmermann, K. E., Nowak, R. D. (1999). Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging, IEEE Trans. Inf. Theory, 45, 846–862.
  • [20] Walter, G. (1985). Orthogonal polynomials estimators of the prior distribution of a compound Poisson distribution, Sankhya, Ser. A, 47, 222–230.
  • [21] Walter, G., Hamedani, G. (1991). Bayes empirical Bayes estimation for natural exponential families with quadratic variance function, Ann. Statist., 19, 1191–1224.
  • [22] Zhang, C.-H. (1995). On estimating mixing densities in discrete exponential family models, Ann. Statist., 23, 929–947.
  • [23] Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net, JRSS, Ser. B, 67, Part 2, 301–320.