Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T10:27:56.814Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of Partial derivatives of the average of densities belonging to a family of densities

Published online by Cambridge University Press:  09 April 2009

V. Susarla  and
S. Kumar
V. Susarla
Affiliation:
University of Wisconsin-Milwaukee Milwaukee, U. S. A.
S. Kumar
Affiliation:
Bureau of Indian Affairs Ottawa, Canada
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.

Recently, attention has been drawn to the problem of estimation of a k-variate probability density and its partial derivatives of various orders. Specifically, let X1, …, Xn be i.i.d. k-variate random variables with common density f wrt Lebesgue measure μ on the k-dimensional σ-field Bk. Parzen (1962) in the k = 1 case and Cacoullos (1966) in the k ≧ 1 case gave the asymptotic properties of a class of kernel estimates fn(x), x ∈ Rk, of f(x) based on X1, …, Xn. The asymptotic properties given in the above two papers concern consistency, asymp-totic unbiasedness, bounds for the mean squared error and asymptotic normality of fn. Also in the context of an empirical Bayes two-action problem, Johns and Van Ryzin (1972) introduced kernel estimates for f(x) and the derivative f'(x)for x∈R1 when f is a mixture of univariate exponential densities wrt Lebesgue measure on B1. They also investigated the asymptotic unbiasedness and themean squared error convergence properties of these estimates. Lin (1968) statedsome generalizations of the results of Johns and Van Ryzin, with applicationsto empirical Bayes decision problems.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 2 , September 1975 , pp. 230 - 241
Copyright
Copyright © Australian Mathematical Society 1975

References

Cacoullos, T. (1966), ‘Estimation of a multivariate density’, Ann. Inst. Statist. Math. 18, 179–189.CrossRefGoogle Scholar
Feller, W. (1957), An introduction to probability theory and its applications, vol. I. (John Wiley & Sons, N.Y. 1957).Google Scholar
Hoeffding, W. and Robbins, H. (1948), ‘The central limit theorem for dependent random variables’, Duke Math. J. 15, 773–780.CrossRefGoogle Scholar
Johns, M. V. Jr, and Van Ryzin, J. R. (1972), ‘Convergence rates for empirical Bayes two-action problems II. Continuous case’, Ann. Math. Statist. 43, 934–947.CrossRefGoogle Scholar
Pi-Erh, Lin (1968), ‘Estimating partial derivatives of an unconditinal multivariate density (preliminary report’), Ann. Math. Statist. 39, 1371.Google Scholar
Parzen, E. (1962), ‘On the estimation of probability density and mode’, Ann. Math. Statist. 33, 1065–1076.CrossRefGoogle Scholar
Susarla, V. (1974), ‘Rate of convergence in the sequence-compound squared-distance loss estimation problem for a family of m-variate normal distributions’, Ann. Statist. 2, 118–133.CrossRefGoogle Scholar
Susarla, V. (1974), ‘Rates of convergence in sequence-compound squared-distance loss estimation and linear-loss two-action problems for a family of scale parameter exponential distributions’, Ann. Inst. Statist. Math. 26, 53–67.CrossRefGoogle Scholar
Van Ryzin, J. R. (1969), ‘On strong consistency of density estimates’, Ann. Math. Statist. 40, 1765–1772.CrossRefGoogle Scholar