Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 2 , pp. 227--237

ASYMPTOTIC DISTRIBUTION OF THE QUADRATIC NORMS OF THE DEVIATION OF ORTHOGONAL SERIES TYPE DENSITY ESTIMATES

By

KUSSIY K. ALYASS and TZE-CHIEN SUN, Wayne State University

SUMMARY. Estimates of two-dimensional density functions based on an orthogonal series are considered. A method that involves Poissonization of the sample size is used to show that the properly normalized quadratic norm of the deviation of the density estimate from the density function itself is asymptotically normally distributed. The k-dimensional analogue is also given. The case where estimates are based on the trigonometric orthonormal system is discussed. Results obtained can be used in test of goodness-of-fit.

AMS (1980) subject classification. Primary G2G05, 62E20, 62H12.

Key words and phrases. Multidimensional density estimates, quadratic norms of the deviation, asymptotic distribution, orthogonal series.

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