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Local-linear moment matrix converges to a positive-definite limit

Statement

Let $X_{i}$ be iid real random variables with density $f$ continuous and positive at a fixed point $x$ . Let $K$ be finite-valued Borel, nonnegative, symmetric, with $\int K = 1$ and $0 < μ_{2} = \int u^{2} K (u) d u < \infty$ . Assume compact kernel support, a globally bounded density $f$ , or a bounded rapid-tail kernel satisfying $\sup_{∣ u ∣ \geq A} ∣ u ∣^{3} K (u) \to 0$ . For $h = h_{n} \to 0$ with $n h_{n} \to \infty$ , set $U_{i} = (x - X_{i}) / h_{n}$ and $M_{n} = \frac{1}{n h_{n}} \sum_{i = 1}^{n} K (U_{i}) (\begin{array}{c} 1 \\ U_{i} \end{array}) (1, U_{i}) .$ Then $M_{n} ⟶ f (x) (\begin{array}{cc} 1 & 0 \\ 0 & μ_{2} \end{array}) in probability,$ and $P (M_{n} is nonsingular) \to 1$ . More generally, the proof establishes $(n h_{n})^{- 1} \sum_{i} L (U_{i}) \to f (x) \int L$ for any integrable Borel function $L$ , if $f$ is bounded globally or $L$ has compact support. Boundedness or square integrability of $L$ is unnecessary.

Facts & Assumptions

Given: The sampling, density, kernel and bandwidth hypotheses in the statement.

[F1]

IID means a common law and mutual independence. Identical distribution and IID families.

[F2]

Density integration uses $f d λ$ . The measure with density $f$ relative to $μ$ .

[F3]

A square-integrable variable has variance $E [(V - E V)^{2}]$ . Moments, variance, and covariance on a probability space.

[F4]

Convergence in probability is convergence of each fixed-error event probability to zero. Convergence in probability.

Proof

1.1givenF1F2F3F4algebra

For the general assertion truncate $L$ to $L_{A} = L 1_{{∣ u ∣ \leq A, ∣ L (u) ∣ \leq A}}$ . The expectation of $(n h)^{- 1} \sum_{i} L_{A} (U_{i})$ is $\int L_{A} (u) f (x - h u) d u \to f (x) \int L_{A}$ by uniform density convergence on its fixed compact support. Its variance is at most $(n h)^{- 1} \int L_{A} (u)^{2} f (x - h u) d u = O_{A} ((n h)^{- 1}) \to 0$ . The elementary bound $P (∣ V - E V ∣ > ϵ) \leq Var (V) / ϵ^{2}$ follows by integrating $ϵ^{2} 1_{{∣ V - E V ∣ > ϵ}} \leq (V - E V)^{2}$ . Thus the truncated average converges in probability.

2.1givenF2F4step 1.1algebra

The expectation of the absolute difference between the full and truncated averages is at most $C \int ∣ L - L_{A} ∣$ , using the global density bound or its bound on the shrinking image of the compact support of $L$ . This tends to zero as $A \to \infty$ . Markov's bound, obtained from $ϵ 1_{{Z > ϵ}} \leq Z$ for $Z \geq 0$ , makes this difference uniformly small in probability. Also $\int L_{A} \to \int L$ . First choosing $A$ large and then $n$ large proves the general assertion.

3.1givenF2F4step 1.1step 2.1algebra

For the rapid-tail case and $L (u) = u^{j} K (u)$ , $j = 0, 1, 2$ , restrict first to $∣ x - X_{i} ∣ \leq δ$ , where the density is bounded. The same truncation proof works there. The expected absolute contribution of the remaining observations is at most $ϵ_{h} h^{2 - j} \int_{∣ z - x ∣ > δ} ∣ z - x ∣^{j - 3} f (z) d z \to 0$ , where $ϵ_{h} = \sup_{∣ u ∣ \geq δ / h} ∣ u ∣^{3} K (u) \to 0$ . Thus the same scalar limits hold for these three moments without a global density bound.

4.1givenF4step 2.1algebrastep 3.1∎

Apply steps 1.1–3.1 to $L (u) = K (u) u^{j}$ for $j = 0, 1, 2$ , all integrable by the finite second moment. Symmetry gives $\int u K = 0$ . The entries of $M_{n}$ therefore have the displayed limits. Its determinant tends in probability to $f (x)^{2} μ_{2} > 0$ , since it is a polynomial in the entries. A zero determinant lies outside a fixed neighborhood of this positive limit, so its probability tends to zero.

Depends on

Used by

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Sources

Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT course notes (standard reference, not scraped)
Whitney K. Newey, Locally Linear Regression, MIT course notes (standard reference, not scraped)