Summary
In this paper we describe the theoretical properties of wavelet based random densities and present algorithms for their generation. We exhibit random densities subject to some standard constraints: smoothness, symmetry, unimodality, and skewness. We also provide three relevant applications of wavelet based-random densities.







Similar content being viewed by others
References
Azzalini, A. (1993), ‘A class of distributions which includes the normal ones’, Scan. J. Statist. 12, 171–178.
Berger, J. O. (1994), ‘An overview of robust Bayesian analysis (with discussion)’, Test 3, 5–124.
Čencov, N. N. (1962), ‘Evaluation of an unknown distribution density from observations’, Doklady (3), 1559–1562.
Chen, J. & Rubin, H. (1986), ‘Drawing a random sample at random’, Comput. Statist. Data Anal. 4, 219–227.
Cohen, A., Daubechies, I. & Vial, P. (1993), ‘Wavelets on the interval and fast wavelet transforms’, Appl. Comput. Harmon. Anal. 1(1), 54–81.
Daubechies, I. (1992), Ten Lectures on Wavelets, number 61 in ‘CBMS-NSF Series in Applied Mathematics’, SIAM, Philadelphia.
Dharmadhikari, S. & Joag-Dev, K. (1988), Unimodality, Convexity, and Application, Academic Press, San Diego.
Ferguson, T. (1974), ‘Trior distributions on spaces of probability measures.’, Annals of Statistics 2, 615–629.
Freedman, D. (1963), ‘On the asymptotic behavior of Bayes’ estimates in the discrete case.’, Annals of Mathematical Statistics 34, 1386–1403.
Good, I. J. & Gaskins, R. A. (1971), ‘Nonparametric roughness penalties for probability densities’, Biometrika 58, 255–277.
Klonias, V. K. (1982), ‘Consistency of two nonparametric maximum penalized likelihood estimators of the probability density function’, Annals of Statistics 10, 811–824.
Lavine, M. (1992), ‘Some aspects of polya tree distributions for statistical modeling’, Annals of Statistics 20, 1222–1235.
Meyer, Y. (1992), Wavelets and Operators, Cambridge Studies in Advanced Mathematics 37, Cambridge University Press, New York.
Muliere, P. & Tardella, L. (1995), Approximating distributions of random functionals of Ferguson-Dirichlet priors, Technical report, Duke University, Discussion Papers ISDS 95-40.
O’Hagan, A. & Leonard, T. (1976), ‘Bayes estimation subject to uncertainty about parameter constraints’, Biometrika 63, 201–202.
Penev, S. & Dechevsky, L. (1997), ‘On non-negative wavelet-based density estimators’, Journal of Nonparametric Statistics 7, 365–394.
Pinheiro, A. & Vidakovic, B. (1997), ‘Estimating the square root of a density via compactly supported wavelets’, Comput. Statist Data Anal. 25, 399–415.
Robert, C. (1994), The Bayesian Choice, Springer-Verlag, New York.
Rubin, H. & Chen, J. (1988), ‘Some stochastic processes related to random density function’, J. Theoret. Probab. 2, 227–237.
Schervish, M. (1995), Theory of Statistics, Springer, New York.
Sethuraman, J. (1994), ‘A constructive definition of Dirichlet priors’, Statistica Sinica 4, 639–650.
Tapia, R. A. & Thompson, J. R. (1978), Nonparametric Probability Density Estimation, The Johns Hopkins University Press, Baltimore.
Vidakovic, B. (1996), ‘A note on random densities via wavelets’, Statist. Probab. Lett. 26, 315–321.
Vidakovic, B. & DasGupta, A. (1996), ‘Efficiency of linear rules for estimating a bounded normal mean’, Sankhyā A 58, 81–100.
Acknowledgment
Research supported by NSF Grant DMS-9626159 at Duke University, and CICYT Grant TIC-95000 at UPM. It started while the second author was visiting UPM under a MEC Grant. We are grateful to an associate editor and two anonymous referees for their comments that improved the paper.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Insua, D.R., Vidakovic, B. Wavelet-based random densities. Computational Statistics 15, 183–203 (2000). https://doi.org/10.1007/s001800000027
Published:
Issue Date:
DOI: https://doi.org/10.1007/s001800000027