Stat.Nonparametric.Local­Poly.Random­Design­Rate

Global smoothness implies membership in the local Hölder class. For a window [x₀ − r, x₀ + r] of positive radius, if β > 0, H ≥ 0, m is p = ⌈β⌉ − 1 times continuously differentiable on the whole real line, and its p-th derivative satisfies |m⁽ᵖ⁾(x) − m⁽ᵖ⁾(y)| ≤ H |x − y|^(β − p) for all x, y in the window, then m belongs to the local Hölder class of smoothness β and constant H on the window.

Bias of the local-polynomial intercept on a fixed design. Let the kernel K be nonnegative and zero outside [−1, 1], with evaluation point x₀ and window radius r, and let the regression function m be, on the window [x₀ − r, x₀ + r], p = ⌈β⌉ − 1 times continuously differentiable with p-th derivative Hölder of order β − p and constant H. For a positive bandwidth h not exceeding r and design points x_1, …, x_N whose weighted design matrix M is invertible with √(M₀₀ (M⁻¹)₀₀) ≤ L, the intercept computed from the noise-free responses m(x_i) satisfies |∑_i S_i m(x_i) − m(x₀)| ≤ (H / p!) · L · h^β.

Conditional mean-squared error on a design with bounded leverage, dimension one. Let the kernel K be nonnegative, at most K_max and zero outside [−1, 1], with evaluation point x₀ and window radius r, and let the regression function m be, on the window [x₀ − r, x₀ + r], p = ⌈β⌉ − 1 times continuously differentiable with p-th derivative Hölder of order β − p and constant H. Take a positive bandwidth h not exceeding r, a clipping level M with |m(x₀)| ≤ M, and design points x_1, …, x_N, and let the responses be independent with Y_i drawn from a conditional outcome distribution that at each x_i in the window [x₀ − r, x₀ + r] has a finite second moment, mean m(x_i) and variance at most σ². If the weighted design matrix M_x is invertible, (M_x⁻¹)₀₀ ≤ 2 c_inv/(N h) and √((M_x)₀₀ (M_x⁻¹)₀₀) ≤ √(2 c_inv (c_top + 1)), then the degree-p intercept clipped to [−M, M] satisfies E[(m̂ − m(x₀))²] ≤ C_bias² h^(2β) + C_var/(N h), the expectation being over the responses.

On the moment event the design is good. Let the design density lie between f_min > 0 and f_max on [x₀ − r, x₀ + r], and the kernel K be measurable, nonnegative, bounded, zero outside [−1, 1] and at least K_min > 0 on [−Δ, Δ]. Take a positive sample size N, a positive bandwidth h not exceeding r, and design points x_1, …, x_N for which every empirical moment (N h)⁻¹ ∑_i K(u_i) u_i^(j+k), u_i = (x_i − x₀)/h, 0 ≤ j, k ≤ p, is within the design tolerance t₀ of its expectation. Then the weighted design matrix M with entries ∑_i K(u_i) (x_i − x₀)^(j+k) is invertible, (M⁻¹)₀₀ ≤ 2 c_inv/(N h), and √(M₀₀ (M⁻¹)₀₀) ≤ √(2 c_inv (c_top + 1)).

All kernel-weighted design moments are close at once, dimension one. Let the design points X_1, …, X_N be independent draws from a density that is at most f_max on an interval [x₀ − r, x₀ + r], and let the kernel K be measurable, nonnegative, at most K_max, and zero outside [−1, 1]. For a polynomial degree p, a positive sample size N, a positive bandwidth h not exceeding r and a positive tolerance t, consider the (p+1) × (p+1) matrix of empirical moments (N h)⁻¹ ∑_i K(u_i) u_i^(j+k), u_i = (X_i − x₀)/h. Then the probability that some entry differs from its expectation by more than t is at most 2 (p+1)² exp(−N h t² / (4 (2 f_max K_max² + K_max t))).

Let the design density f lie between f_min > 0 and f_max on [x₀ − r, x₀ + r], and the kernel K be measurable, nonnegative, bounded, zero outside [−1, 1] and at least K_min > 0 on [−Δ, Δ]. For a positive bandwidth h not exceeding r, the population moment matrix with (j,k) entry ∫ K(u) f(x₀ + h u) u^(j+k) du, 0 ≤ j, k ≤ p, is positive definite.

Let the design density f lie between f_min > 0 and f_max on [x₀ − r, x₀ + r], and the kernel K be measurable, nonnegative, bounded, zero outside [−1, 1] and at least K_min > 0 on [−Δ, Δ]. For a positive bandwidth h not exceeding r, let T be the population moment matrix with (j,k) entry ∫ K(u) f(x₀ + h u) u^(j+k) du. Then every row of T⁻¹ has absolute sum at most c_row = (p+1) tr(R⁻¹)/(f_min K_min), where R is the monomial moment matrix on [−Δ, Δ].

Pointwise rate N^(−2β/(2β+1)) for the clipped local-polynomial estimator at an interior point, dimension one. Let the design density lie between positive constants f_min ≤ f_max on an interval [x₀ − r, x₀ + r], and let the kernel K be measurable, nonnegative, at most K_max, zero outside [−1, 1] and at least K_min > 0 on [−Δ, Δ]. Suppose for some β > 0 and p = ⌈β⌉ − 1 the regression function m is, on that interval, p times continuously differentiable with p-th derivative Hölder of order β − p and constant H, the sample size satisfies N ≥ max(1, ⌈r^(−(2β+1))⌉), and the clipping level M satisfies |m(x₀)| ≤ M. Let (X_i, Y_i), i = 1, …, N, be independent pairs with X_i drawn from the design density and Y_i drawn, given X_i, from a conditional outcome distribution that at almost every design point x in that interval has a finite second moment, mean m(x) and variance at most σ². Then the degree-p kernel-weighted least-squares intercept at x₀ with bandwidth h = N^(−1/(2β+1)) (taken to be zero when the weighted design matrix is singular), clipped to [−M, M], satisfies E[(m̂(x₀) − m(x₀))²] ≤ C · N^(−2β/(2β+1)) with C = C_bias² + C_var + 16 M² (p+1)² / c₀, an explicit function of f_min, f_max, K_min, K_max, Δ, the kernel moments, p, H, σ and M that does not depend on N.

Unconditional mean-squared error of the clipped local-polynomial estimator at an interior point, dimension one. Let the design density lie between positive constants f_min ≤ f_max on an interval [x₀ − r, x₀ + r], and let the kernel K be measurable, nonnegative, at most K_max, zero outside [−1, 1] and at least K_min > 0 on [−Δ, Δ]. Suppose for some β > 0 and p = ⌈β⌉ − 1 the regression function m is, on that interval, p times continuously differentiable with p-th derivative Hölder of order β − p and constant H, the sample size N is positive, the bandwidth h is positive and at most r, and the clipping level M satisfies |m(x₀)| ≤ M. Let (X_i, Y_i), i = 1, …, N, be independent pairs with X_i drawn from the design density and Y_i drawn, given X_i, from a conditional outcome distribution that at almost every design point x in that interval has a finite second moment, mean m(x) and variance at most σ². Then the degree-p kernel-weighted least-squares intercept at x₀ (taken to be zero when the weighted design matrix is singular), clipped to [−M, M], satisfies E[(m̂(x₀) − m(x₀))²] ≤ C_bias² h^(2β) + C_var/(N h) + 16 M² (p+1)² exp(−c₀ N h), where C_bias, C_var and c₀ > 0 are explicit functions of f_min, f_max, K_min, K_max, Δ, the kernel moments, p, H and σ that do not depend on N or h.