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Global MISE upper bound for the KDE

Statement

Assume ${A C}_{ω}$ . Let $X_{1}, \dots, X_{n}$ be iid with a Borel density version $f \in W^{2, 2} (R^{d})$ . Let $K : R^{d} \to R$ be Borel, $K \in L^{1} \cap L^{2}$ , $\int K (u) d u = 1$ , $\int u_{j} K (u) d u = 0$ for every $j$ , and $M_{2} (K) : = \int ∣ K (u) ∣ ∥ u ∥^{2} d u < \infty$ . For every $h > 0$ define $MISE ({\hat{f}}_{h}) : = E \int_{R^{d}} ({\hat{f}}_{h} (x) - f (x))^{2} d x .$ Then $MISE ({\hat{f}}_{h}) \leq \frac{h^{4}}{4} M_{2} (K)^{2} ∥ D^{2} f ∥_{2, F}^{2} + \frac{∥ K ∥_{2}^{2}}{n h^{d}} .$ In particular, $h_{n} = n^{- 1 / (d + 4)}$ gives $MISE ({\hat{f}}_{h_{n}}) = O (n^{- 4 / (d + 4)})$ , where $O (n^{- 4 / (d + 4)})$ means bounded by a finite constant times $n^{- 4 / (d + 4)}$ for all sufficiently large $n$ . This is an upper rate under the displayed hypotheses; it is not a minimax lower bound.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n, d \geq 1$ , iid observations with density version $f \in W^{2, 2} (R^{d})$ , a Borel kernel with the displayed integrability and moment conditions, and $h > 0$ .

[A1]

The Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The estimate is the finite sum $(n h^{d})^{- 1} \sum_{i} K ((x - X_{i}) / h)$ , and $f$ is a selected Borel density version (Density targets and the kernel density estimator).

[F2]

The estimator is jointly Borel in the sample and evaluation point (The KDE is jointly measurable in sample and evaluation point); arithmetic operations preserve measurability (Arithmetic and lattice operations preserve measurability whenever they are defined).

[F3]

The convolution bias satisfies $∥ S_{h} f - f ∥_{2} \leq h^{2} M_{2} (K) ∥ D^{2} f ∥_{2, F} / 2$ (Second-order kernel convolution has an L² bias bound).

[F4]

Iid random elements are independent and identically distributed (Identical distribution and IID families).

[F5]

Measurable functions preserve independence, and expectations of integrable products of independent real random variables factor (Measurable coordinatewise functions preserve independence, Expectations factor over finite products of independent random variables).

[F6]

Tonelli interchanges nonnegative integrals on the product of a probability space and $R^{d}$ with Lebesgue measure; the factors are sigma-finite (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, The product measure of two sigma-finite measure spaces, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique).

[F7]

For nonnegative measurable integrands, expectation under $P_{f} = f d λ_{d}$ is integration against $f$ (Change of variables for expectation, Integrating against a density agrees with integrating the product).

[F8]

Under ${A C}_{ω}$ , the C $^{1}$ affine substitutions used below satisfy the nonnegative and $L^{1}$ change-of-variables formulas (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F9]

A nonnegative measurable function with finite integral is finite almost everywhere (A nonnegative measurable function with finite integral is finite almost everywhere).

[F10]

For a square-integrable variable $Z$ , $Var (Z) = E [(Z - E Z)^{2}]$ (Moments, variance, and covariance on a probability space).

[F11]

Expectation is linear on integrable variables (The Lebesgue integral is linear on $L^{1} (μ)$ ).

Proof

1.1givenF1F2F6

For fixed $h$ , [F2] makes $(z, x) \mapsto {\hat{f}}_{h} (x)$ Borel, and [F1] makes $f (x)$ Borel as a function of $(z, x)$ . Arithmetic measurability in [F2] therefore makes the squared error product-measurable and nonnegative. Tonelli [F6] identifies MISE with the integral over $x$ of its expected pointwise squared error, even before finiteness has been established.

2.1givenA1F4F6F7F8F9F11step 1.1

Write $q_{h} (x, z) = h^{- d} K ((x - z) / h)$ . By Tonelli and the substitution $z + h u$ in [F8], $\int_{R^{d}} E ∣ q_{h} (x, X_{1}) ∣ d x = ∥ K ∥_{1}, \int_{R^{d}} E [q_{h} (x, X_{1})^{2}] d x = h^{- d} ∥ K ∥_{2}^{2} .$ Consequently, [F9] makes the summands integrable and square-integrable for almost every $x$ . For such $x$ , put $g (z) = q_{h} (x, z)$ and $g^{+} = \max (g, 0)$ , $g^{-} = \max (- g, 0)$ . The nonnegative density formula [F7] applied to $∣ g ∣$ shows $\int ∣ g ∣ f < \infty$ . Thus both parts have finite density integrals, and [F7] applied separately to them, followed by subtraction using [F11], gives $E q_{h} (x, X_{1}) = \int g (z) f (z) d z$ . Signed substitution [F8] then identifies this integral with $S_{h} f (x)$ , and identical laws plus finite-sum linearity [F11] give $E {\hat{f}}_{h} (x) = S_{h} f (x)$ .

3.1step 2.1F4F5F10F11algebra

At almost every $x$ from step 2.1, the variables $q_{h} (x, X_{i})$ are iid and square-integrable. Set $W_{i} = q_{h} (x, X_{i}) - E q_{h} (x, X_{i})$ . These are independent, integrable and square-integrable by [F5] and step 2.1; for $i \neq j$ , [F5] makes $W_{i} W_{j}$ integrable and gives $E [W_{i} W_{j}] = E W_{i} E W_{j} = 0$ . Expanding the finite square in the variance definition [F10] and applying [F11] gives $Var ({\hat{f}}_{h} (x)) = n^{- 2} \sum_{i} E W_{i}^{2} = Var (q_{h} (x, X_{1})) / n$ . Expanding one centered square using [F10]–[F11] gives $Var (q_{h} (x, X_{1})) = E [q_{h} (x, X_{1})^{2}] - (E q_{h} (x, X_{1}))^{2} \leq E [q_{h} (x, X_{1})^{2}]$ . Integrating and using step 2.1 bounds the integrated variance by $∥ K ∥_{2}^{2} / (n h^{d})$ .

3.2step 2.1F3

For almost every $x$ , step 2.1 identifies the bias with $S_{h} f (x) - f (x)$ . The $L^{2}$ bound [F3] therefore gives $\int_{R^{d}} (E {\hat{f}}_{h} (x) - f (x))^{2} d x \leq \frac{h^{4}}{4} M_{2} (K)^{2} ∥ D^{2} f ∥_{2, F}^{2} .$

4.1F10F11step 1.1step 3.1step 3.2algebra∎

At almost every $x$ , expand ${\hat{f}}_{h} (x) - f (x) = ({\hat{f}}_{h} (x) - E {\hat{f}}_{h} (x)) + (E {\hat{f}}_{h} (x) - f (x))$ . The centered term has expectation zero, so [F10]–[F11] give $E [({\hat{f}}_{h} (x) - f (x))^{2}] = Var ({\hat{f}}_{h} (x)) + (E {\hat{f}}_{h} (x) - f (x))^{2} .$ Integrate this nonnegative identity using step 1.1, then apply steps 3.1 and 3.2 to obtain the displayed MISE bound. For $h_{n} = n^{- 1 / (d + 4)}$ , both $h_{n}^{4}$ and $(n h_{n}^{d})^{- 1}$ equal $n^{- 4 / (d + 4)}$ , which proves the stated upper rate.

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Sources

Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT OpenCourseWare 14.385, Fall 2007, integrated MSE calculation and multivariate rate, PDF pp. 5–7 (standard reference, not scraped)
Adrian Bowman and Ludger Evers, Nonparametric Smoothing, APTS 2013–14 course notes, §1.2.1, PDF p. 8 (standard reference, not scraped)