Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T18:00:30.061Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fourier inversion method for the estimation of a density and its derivatives

Published online by Cambridge University Press:  09 April 2009

J. Blum  and
V. Susarla
J. Blum
Affiliation:
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, U.S.A.
V. Susarla
Affiliation:
Mathematics Research Center, Madison, Wisconsin, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

During the last two decades a variety of methods have been developed for the problem of estimation of unknown density f wrt Lebesgue measure and its vth dervative g(= f(v)) using i.i.d random variables X1, …, Xn when X1 ∼ f. For example, see Wegman (1972). In almost all the papers on the estimation of f(x) or g(x), various authors assumed the existence of derivatives of f of order r(>v) at x to obtain rates for the mean-square convergences and other desirable properties for their estimators. Here it is shown that if

then estimators gˇ(x) can be constructed for which E[(gˇ(x)-g(x))2] = O(n−(2a-δ)/(2a+2v+1)) for anygiveng δ > 0. Simiar statements hold for almost sure convergence of gˇ(x). It can also be shown that (gˇ(x1), gˇ(x2)) is asymptotically bivariate normal under certain conditions for x1, ≠ x2. If (Al) is satisfied with a ≦ l, then our estimators have all the desirable properties while other methods are not applicable in this situation since they require differentiability conditions on g. (For example, see Susarla and Kumar (1975) and its references.) Our estimators are defined by using the inversion theorem for some absolutely integrable characteristic functions. The motivation for our estimators is given in O'Bryan and Susarla (1975, 76) and Susarla and O'Bryan (1975).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 2 , March 1977 , pp. 166 - 171
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Dvoretzky, A., Kiefer, J., and Wolfowitz, J. (1956), ‘Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator’, Ann. Math. Statist. 24, 642–669.CrossRefGoogle Scholar
[2]Nadaraya, E. A. (1965), ‘On non-parametric estimates of density functions and regression curves’, Theor. Prob. Appl. 10, 186–190.CrossRefGoogle Scholar
[3]O'Bryan, T. and Susarla, V. (1975), ‘An empirical Bayes estimation problem with nonidentical components involving normal distributions’, Communications in Statistics 4, 1033–1042.CrossRefGoogle Scholar
[4]O'Bryan, T. and Susarla, V. (1976), ‘Rates in the empirical Bayes estimation problem with non-identical components. Case of normal distributions’, Ann. Inst. Statist. Math. 28, Part A, 389–397.CrossRefGoogle Scholar
[5]Parzen, E. (1962), ‘On the estimation of probability density and mode’, Ann. Math. Statist. 33, 1065–1076.CrossRefGoogle Scholar
[6]Singh, R. S. (1974), ‘Estimation of derivatives of the average of μ-densities and sequencecompound estimation in exponential families’, RM-318, Department of Statistics and Probability, Michigan State University.Google Scholar
[7]Susarla, V. and Kumar, S. (1975), ‘Estimation of partial derivatives of the average of densities belonging to a family of densities’, J. Austral. Math. Soc. 20, 230–241.CrossRefGoogle Scholar
[8]Susarla, V. and O'Bryan, T. (1975), ‘An empirical Bayes two action problem with nonidentical components for a translated exponential distribution’, Communications in Statistics 4, 767–775.CrossRefGoogle Scholar
[9]Wahba, G. (1971), ‘A polynomial algorithm for density estimation’, Ann. Math. Statist. 42, 1870–1886.CrossRefGoogle Scholar
[10]Wegman, E. J. (1972), ‘Nonparametric probability density estimation: I. A summary of available methods’, Technometrics 14, 533–546.CrossRefGoogle Scholar