Abstract
In this paper, a little known computational approach to density estimation based on filtered polynomial approximation is investigated. It is accompanied by the first online available density estimation computer program based on a filtered polynomial approach. The approximation yields the unknown distribution and density as the product of a monotonic increasing polynomial and a filter. The filter may be considered as a target distribution which gets fixed prior to the estimation. The filtered polynomial approach then provides coefficient estimates for (close) algebraic approximations to (a) the unknown density function and (b) the unknown cumulative distribution function as well as (c) a transformation (e.g., normalization) from the unknown data distribution to the filter. This approach provides a high degree of smoothness in its estimates for univariate as well as for multivariate settings. The nice properties as the high degree of smoothness and the ability to select from different target distributions are suited especially in MCMC simulations. Two applications in Sects. 1 and 7 will show the advantages of the filtered polynomial approach over the commonly used kernel estimation method.
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References
Bailey DH (1993) Multiprecision translation and execution of fortran programs. ACM Trans Math Softw 19(3):288–319
Bozdogan H (1987) Model selection and akaike’s information criterion (aic): the general theory and its analytical extensions. Psychometrika 52(3):345–370
Cooley CA, MacEachern SN (1998) Classification via kernel product estimators. Biometrika 85(4):823–833
Cornish E, Fisher R (1937) Moments and cumulates in the specification of distributions. Extrait de la Revue de l’Institute International de Statistique 4:1–14
Elfenbein L (1978) On minimum Von Mises statistic estimators. Ph.D. thesis, George Washington University
Elphinstone CD (1983) A target distribution model for nonparametric density estimation. Commun Stat Theory Methods 12(2):161–198
Elphinstone CD (1985) A method of distribution and density estimation. Ph.D. thesis, University of South Africa, Pretoria
Golub GH, Van Loan CF (1996) Matrix computations. In: The singular value decomposition and unitary matrices, 3rd edn. Johns Hopkins University Press, Baltimore, pp 70–73
Heinzmann D (2005) Computational aspects of filtered polynomial density estimation. Master’s thesis, Swiss Federal Institute of Technology at Lausanne, Switzerland. http://www.math.unizh.ch/user/heinzmann/software
Jaschke SR (2002) The cornish fisher expansion in the context of delta gamma normal approximations. J Risk 4(4):33–52
Jeffreys B, Jeffreys H (1988) Method of mathematical physics. In: Mean-value theorems, 3rd edn. Cambridge University Press, Cambridge, pp 446–448
Jonathon DV (1992) Nonlinear vision. In: Nonlinear systems analysis in vision: overview of kernel methods, 1 edn. CRC Press, London, pp 1–37
Kendall M, Stuart A (1977) The advanced theory of statistics: distribution theory, 4 edn, vol 1. Griffin, London. Provided by the Smithsonian/NASA Astrophysics Data System
Krantz SG (1999) Handbook of complex variables.In: The fundamental theorem of calculus along curves. Birkhaeuser, Boston, pp 22–25
Lancaster P (1966) Error analysis for the Newton–Raphson method. Numerische Math 9(1):55–68
Levenberg K (1944) A method for the solution of certain problems in least squares. Quart Appl Math 2:164–168
Linton OB, Nielsen JP (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. biometrika 82(1):93–100
Mardia KV, Kent JT, Bibby JM (1979) Multivariate analysis. In: Principal component analysis. Academic, London, pp 213–229
R Development Core Team (2006) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. http://www.R-project.org
Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, London
Stone M (1977) An asymptotic equivalence of choice of model by cross validation and akaike’s criterion. J R Stat Soc B39:44–47
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Heinzmann, D. A filtered polynomial approach to density estimation. Comput Stat 23, 343–360 (2008). https://doi.org/10.1007/s00180-007-0070-z
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DOI: https://doi.org/10.1007/s00180-007-0070-z