The Adjoint Projection Pursuit Regression

原文摘要

Consider a projection pursuit regression model y = g(α + β x) + ε, with an arbitrary monotonic link function g. We assume the link function is unknown; thus we can only estimate the direction of β, that is, the ratios among the components of β, say, β 1/β 2. The direction of β is a useful estimand that measures, say, how many units of x 2 is equivalent to one unit of x 1 in terms of potency in affecting the outcome y. Projection pursuit regression usually solves the equation cov(x, y − E(y | β x)) = 0. As an alternative, we propose to solve the adjoint equation cov(x − E(x | β x), y) = 0. The adjoint equation has the advantage that E(x | β x) might be easier to estimate than E(y | β x). We establish two main results for the population case. First, β is the unique solution (up to a multiplicative scalar) to the adjoint equation. Second, β satisfies a fixed-point property based on a modified least squares regression derived from the adjoint equation. We apply the population results to two important empirical problems: diagnosis and estimation. For diagnosis, we check whether the linear least squares estimate for β is valid for the direction of β. In particular, we test whether the estimate agrees with either the adjoint equation or the fixed-point property. The tests are analogous to Tukey's 1 df test. For estimation, we propose the adjoint projection pursuit regression estimate, which solves the empirical adjoint equation. The estimate is √n-consistent for β up to a multiplicative scalar. The rate of convergence is insensitive to the choice of the smoothing parameter in estimating E(x | β x); any window size of order O(1/√n) or smaller achieves the optimal convergence rate. The estimate can be implemented numerically by iterating the modified least squares regression; a simulation study is given to illustrate this procedure.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）