We consider multivariate density estimation when the assumptions of identically distributed data or stationary data are relaxed to the assumptions of locally identically distributed data or locally stationary data. We assume that the distribution of the data is changing continuously as function of time. To estimate densities non-parametrically with these local regularity conditions, we need time localization in addition to the usual space localization. We define a time-localized kernel estimator that estimates the density non-parametrically at any given point of time. The consistency of the time-localized kernel estimator is proved and the rates of convergence of the estimator are derived under conditions on the &bgr;-and alpha-mixing coefficients. Both the time-series setting and spatial setting are covered. Copyright 2007 The Author Journal compilation 2007 Blackwell Publishing Ltd.
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