Marginal integration for a nonparametric additive model

For \(E(Y|X = x) = \mu + m_1(x^1) + \dots + m_d(x^d)\) (3.5) with the location normalisation \(E[m_j(X^j)] = 0\) (3.6), which makes \(\mu = E(Y)\) (3.7) and \(m_1(x^1) = \int E(Y|X = x)p_{-1}(x^{-1})dx^{-1} - \mu\) (3.8). Replacing the conditional mean by the kernel estimator (3.9) and the outer integral by a sample average gives \(\hat m_1(x^1) = n^{-1}\sum_i \hat g(x^1, X_i^{(-1)}) - \hat\mu\).

Usage

morie_marginal_integration(x, y, bandwidth = NULL, j = 1L, grid = NULL)

Arguments

x

numeric matrix of continuous covariates.

y

numeric response.

bandwidth

h1, or c(h1, h2); Silverman-based when NULL.

j

which component to estimate (1-based in R).

grid

evaluation points.

Value

list: grid, m_hat, mu_hat, component, h1, h2, normalisation, mean_of_m_hat, smoothness_required, curse_of_dimensionality, n, d, method.

Details

Hold the component of interest fixed, average the fitted surface over the OTHERS at their observed values, subtract the mean. The normalisation is what makes that meaningful: without \(E[m_j] = 0\) each component could absorb a constant and \(\mu\) shed one, so nothing would be identified.

This estimator carries the curse of dimensionality, which is why the chapter goes on to develop others. \(\hat g\) smooths the full covariate with a \((d-1)\)-dimensional kernel, so Theorem 3.1 needs the components \(q\) times differentiable for some \(q > d - 1\) – the requirement GROWS with d. smoothness_required returns that q, so the cost is a number rather than a caveat. Mirrors morie.fn.hrzmir.

References

Horowitz, Sec. 3.1.1, eqs. (3.5)-(3.9), Theorem 3.1; Linton and Nielsen (1995), Linton and Hardle (1996).

Examples

set.seed(1)
x <- matrix(runif(400, -1, 1), ncol = 2)
y <- 2 + sin(pi * x[, 1]) + x[, 2]^2 - 1 / 3 + rnorm(200) * 0.2
morie_marginal_integration(x, y, j = 1)$smoothness_required
#> [1] 2