Almost Sure L(Gamma)-Norm Convergence for Data-Based Histogram Density Estimates.
Abstract
Let X1,...,Xn be i.i.d. samples drawn from a d-dimensional distribution with density f. Partition the space R subscript d into a union of disjoint intervals I sub l = I(l,X1,...,Xn) with the form I sub l = ( x = (x(1),...,x(d): - infinity < a sub li < or = x(i) <or = b sub li < infinity, i = 1,...,d). Define the database histogram estimate of f(x) based on this partition as fn(x) = The number of X1,...,Xn falling into I sub l + n times the volume of I sub l, for x is an element of I sub l, l = 1,2... For given constant r > or = 1 we obtain the sufficient condition for limit as n approached infinity of the Integral over the R subscript d of the absolute value of (f sub n)(x) - f(x) to the rth power dx = O. The results give substantial improvements upon existing results. Keywords: Data based; Density estimator; Empirical distribution; Histogram.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1987
- Accession Number
- ADA189944
Entities
People
- L. C. Zhao
- Paruchuri R. Krishnaiah
- X. R. Chen
Organizations
- University of Pittsburgh