摘要：
We consider maximum likelihood estimation (MLE) of a regression function using sieves defined by Bernstein polynomials, in terms of their order and coefficients. In case we know that the regression function satisfies certain shape- restriction like monotonicity or convexity, we can impose corresponding restriction through the coefficients of the Bernstein polynomials in the sieves so that the estimate also satisfies the shape-restriction. For sieve MLE of this type, we establish its consistency when the regression function is continuous and its rate of convergence when its derivative satisfies Lipschitz condition. Under the same condition, we also show that the integral of the estimate converges weakly to that of the regression function. Simulation studies are presented to evaluate its numerical performance. In addition to excellent confidence interval estimates of area under the regression function, sieve MLE performs better than the Bayesian method based on Bernstein polynomials and density-regression method, reported in Chang et al. (2007).