The use of variable kernel mass in density estimation

• Autorzy: Sturgeon M. L.
• Języki publikacji: EN

Abstrakty

EN

A new theoretical approach is developed for estimating a univariate density f(x), for large n, using a nonnegative symmetric kernel with variable mass. Compared to kernels of order 4, kernels of variable mass asymptotically achieve (i) smaller variance, (ii) essentially the same bias, and so (iii) a reduced MISE of order O(n^−8/9). The analysis uses a common MISE-optimal bandwidth h and locally adapted kernel mass M(X) = 1 − (f″(X)h²)/(24 f(X)), to be estimated at the kernel center, where f(X) is the average f value over an interval of length h centered on X. Mass adaptation derives from considering the expected effect of negative mass, in kernels of order 4, upon the positive part of such kernels. Unlike the Abramson procedure for varying local bandwidth, this procedure does not require any special accommodation for small values of f(X), for f in C⁴.

Słowa kluczowe

EN

kernel mass adaptation; higher order kernels

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2003
• Tom: Vol. 23, Fasc. 1
• Strony: 189--207
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Sturgeon M. L.

• Department: of Mathematics, Salem State College, 13 Castle Circle, Peabody, MA 01960, U.S.A.

Bibliografia

• [1] A. Berlinet, Hierarchies of higher order kernels, Probab. Theory Related Fields 94 (1993), pp. 489-504.
• [2] J. Fan, P. Hall, M. Martin and P. Patil, On local smoothing of nonparametric curve estimators, JASA 91 (1996), pp. 258-266.
• [3] L. Gajek, On improving density estimators which are not bona fide functions, Ann. Statist. 14 (1986), pp. 1612-1618.
• [4] M. C. Jones, Variable kernel density estimators and variable kernel density estimators, Austral. J. Statist. 32 (1990), pp. 361-371.
• [5] I. McKay, A note on bias reduction in variable-kernel estimates, Canad. J. Statist. 21 (1993), pp. 367-375.
• [6] E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist. 33 (1962), pp. 1065-1076.
• [7] B. W. Silverman, Density estimation for statistics and data analysis, Chapman and Hall, New York 1990.
• [8] M. Sturgeon, Mass shifting roles of negative kernel mass in density estimation, Ann. Inst, Statist. Math. 48 (1996), pp. 675-686.
• [9] G. R Terrell and D. W. Scott, On improving convergence rates for nonnegative kernel density estimators, Ann. Statist. 8 (1980), pp. 1160-1163.
• [10] G. R. Terrell and D. W. Scott, Variable kernel density estimation, Ann. Statist. 20 (1992), pp. 1236-1265.