A Bernstein polynomial estimator for f. N(x, y) for an unknown probability density function f(x, y) concentrated on the triangle ,d = {(x, y): 0 _< x, y < l, x + y < 1} or on the square [] = {(x, y): 0 < x, y < 1} is developed. As a measure ofquality the exact order of magnitude for the pointwise mean squared error is established. It is seen that the quality of these Bernstein polynomial estimators is comparable with the quality of the so-called kernel estimators. Further for such estimators uniform weak consistency results and central limit theorems are developed.

Nonparametric regression estimation, smoothed regressograms, Bernstein polynomials,