Nonparametric Regression When Estimating the Probability of Success

For the random variables Y,X1, . . . , Xp, where Y is binary, let M(x1, . . . , xp) = P (Y = 1|(X1, . . . Xp) = (x1, . . . xp)). The paper compares four smoothers aimed at estimating M(x1, . . . , xp), three of which can be used when p > 1. Evidently there are no published comparisons of smoothers when p > 1 and Y is binary. And there are no published results on how the four estimators, considered here, compare. One of the estimators is based on an approach described in Hosmer and Lemeshow (1989, p. 85), which is limited to p = 1. A simple modification of this estimator (called method E3 in the paper) is proposed that can be used when p > 1. No estimator dominated in terms of mean squared error and bias. And for p = 1, the differences among three of the estimators, in terms of mean squared error and bias, is not particularly striking. But for p > 1, differences among the estimators are magnified, with method E3 performing relatively well. An estimator based on the running interval smoother performs about as well as E3, but for general use, E3 is found to be preferable. An estimator studied by Signorini and Jones (1984) is not recommended, particularly when p > 1. keywords: logistic regression, kernel estimators, smoothers.