Nonparametric Regression When Estimating the Probability of Success: A Comparison of Four Extant Estimators

Abstract

For the random variables Y, X ₁,..., X _p, where Y is binary, let M(x ₁,..., x _p) = P(Y = 1

(X₁,... X _p) = (x ₁,... x _p)). The article compares four smoothers aimed at estimating M(x ₁,...,x _p), three of which can be used when p > 1. Evidently there are no published comparisons of smoothers when p > 1 and Y is binary. One of the estimators stems from Hosmer and Lemeshow (1989, 85), which is limited to p = 1. A simple modification of this estimator (called method E3 here) is proposed that can be used when p > 1. Roughly, a weighted mean of the Y values is used, where the weights are based on a robust analog of Mahalanobis distance that replaces the usual covariance matrix with the minimum volume estimator. Another estimator stems from Signorini and Jones (1984) and is based in part on an estimate of the probability density function of X ₁,..., X _p. Here, an adaptive kernel density estimator is used. 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, are 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. The estimator studied by Signorini and Jones (1984) is not recommended, particularly when p > 1.