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A modified k-nearest-neighbors method is introduced that provides an efficient nonlinear estimate of the intensity of a point process (field) based on locations of events. The method is applied to perform a detailed statistical analysis of the spatial structure of the seismic intensity field (in the context of this paper, the term "intensity" refers to the intensity of the seismic field, and should not be mistaken for the intensity of ground motion caused by an earthquake). The proposed method requires neither a preliminarily delineated area nor a normalization procedure for the estimates. In contrast to many interpolation methods, estimates based on the proposed method are statistically justified. The "uncertainty relation" between the spatial smoothing effect and random errors is established for the method in an explicit form. A procedure for choosing the number of nearest neighbors k* that controls the effective radius of smoothing is suggested. The proposed method is applied to analyze the seismic intensity field in two seismogenic regions surrounding the Kuril Islands and Japan during the period from 1904 to 2011. Spots of increased seismic activity in these regions are localized by the new method, and some quantitative statistical characteristics of these spots are determined and discussed. [ABSTRACT FROM AUTHOR]