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First- and second-order delta method for random vectors

Statement

Let $d, k \geq 1$ , let $r_{n} > 0$ be deterministic with $r_{n} \to \infty$ , and let $T_{n}$ be $R^{d}$ -valued random vectors with target $θ \in R^{d}$ . Suppose $Z_{n} = r_{n} (T_{n} - θ) \Rightarrow Z$ for an $R^{d}$ -valued random vector $Z$ . All vector norms below are Euclidean.

If $g : R^{d} \to R^{k}$ is Borel and there is a linear map $A : R^{d} \to R^{k}$ such that $g (θ + h) - g (θ) = A h + ρ_{1} (h), \frac{∥ ρ_{1} (h) ∥}{∥ h ∥} ⟶ 0 (h \to 0, h \neq 0),$ then $r_{n} {g (T_{n}) - g (θ)} \Rightarrow A Z .$

If, in addition, $A = 0$ and there is a continuous symmetric bilinear map $H : R^{d} \times R^{d} \to R^{k}$ such that $g (θ + h) - g (θ) = \frac{1}{2} H [h, h] + ρ_{2} (h), \frac{∥ ρ_{2} (h) ∥}{∥ h ∥^{2}} ⟶ 0 (h \to 0, h \neq 0),$ then $r_{n}^{2} {g (T_{n}) - g (θ)} \Rightarrow \frac{1}{2} H [Z, Z] .$ For scalar-valued $g$ , $H$ is the usual Hessian bilinear form. The second-order premise is a pointwise Peano expansion; continuous second derivatives on a neighborhood are not required.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Stochastic order and convergence notation for finite-dimensional random vectors: $O_{P} (1)$ means bounded in probability by tail probabilities, and $o_{P} (1)$ means convergence in probability to zero.

[F2]

Finite-dimensional stochastic-order calculus and tightness: finite-dimensional weak convergence implies $O_{P} (1)$ .

[F3]

Finite-dimensional stochastic-order calculus and tightness: a fixed continuous bilinear map sends an $o_{P} (1)$ factor times an $O_{P} (1)$ factor to $o_{P} (1)$ .

[F4]

Vector Slutsky theorem: if $X_{n} \Rightarrow X$ and $Y_{n} - c = o_{P} (1)$ , then $(X_{n}, Y_{n}) \Rightarrow (X, c)$ , and continuous maps preserve this limit.

[F5]

The real Frechet derivative on real and complex matrix spaces with the Frobenius norm: on finite-dimensional real matrix spaces, differentiability means a real-linear first-order approximation with a remainder whose norm divided by the input norm tends to zero. Identify $R^{d}$ with real column matrices and use the Euclidean/Frobenius norm equality.

[F6]

Convergence in distribution of random elements: $X_{n} \Rightarrow X$ means weak convergence of their laws in the common metric target; the random elements need not share probability spaces.

Proof

Given: The objects and hypotheses in the statement.

1.1F2F6given

By [F6], $Z_{n} \Rightarrow Z$ is weak convergence in $R^{d}$ . By [F2], this implies $Z_{n} = O_{P} (1)$ .

2.1F1F3step 1.1givenalgebra

Since $r_{n}^{- 1} \to 0$ , the deterministic sequence $r_{n}^{- 1}$ is $o_{P} (1)$ . Applying the bilinear product rule in [F3] to scalar multiplication gives $T_{n} - θ = r_{n}^{- 1} Z_{n} = o_{P} (1)$ .

3.1F1F2step 1.1step 2.1givenchoosealgebraF5

Fix $ε, η > 0$ . By [F2] choose $M > 0$ with ${lim sup}_{n} P (∥ Z_{n} ∥ > M) \leq η$ . The first-order remainder limit in the statement gives $δ > 0$ such that $∥ ρ_{1} (h) ∥ \leq (ε / M) ∥ h ∥$ when $0 < ∥ h ∥ < δ$ . Step 2.1 gives $P (∥ T_{n} - θ ∥ \geq δ) \to 0$ , so the limiting upper probability that $r_{n} ∥ ρ_{1} (T_{n} - θ) ∥ > ε$ is at most $η$ . Letting $η ↓ 0$ proves $r_{n} ρ_{1} (T_{n} - θ) = o_{P} (1)$ . Substitution in the first expansion yields $r_{n} {g (T_{n}) - g (θ)} = A Z_{n} + o_{P} (1)$ .

3.2F1F2step 1.1step 2.1givenchoosealgebra

Fix $ε, η > 0$ . By [F2] choose $M > 0$ with ${lim sup}_{n} P (∥ Z_{n} ∥ > M) \leq η$ . The second-order remainder limit in the statement gives $δ > 0$ such that $∥ ρ_{2} (h) ∥ \leq (ε / M^{2}) ∥ h ∥^{2}$ for $0 < ∥ h ∥ < δ$ . Step 2.1 puts $T_{n} - θ$ inside this neighborhood with probability tending to one. Outside the two tail events, $r_{n}^{2} ∥ ρ_{2} (T_{n} - θ) ∥ \leq ε$ because $r_{n} ∥ T_{n} - θ ∥ = ∥ Z_{n} ∥$ . Thus $r_{n}^{2} ρ_{2} (T_{n} - θ) = o_{P} (1)$ ; since $A = 0$ , substitution in the second expansion gives $r_{n}^{2} {g (T_{n}) - g (θ)} = \frac{1}{2} H [Z_{n}, Z_{n}] + o_{P} (1)$ .

4.1F4givenstep 3.1

The map $(x, e) \mapsto A x + e$ is continuous. Apply [F4] to $Z_{n} \Rightarrow Z$ and the $o_{P} (1)$ error in step 3.1 to obtain the first-order limit $A Z$ .

5.1F4givenstep 3.2∎

The quadratic map $z \mapsto \frac{1}{2} H [z, z]$ is continuous. Apply [F4] to its value at $Z_{n}$ and the error in step 3.2 to obtain the second-order limit $\frac{1}{2} H [Z, Z]$ .

Remarks

The first-order result is choice-free and does not require $Z$ to be Gaussian. If $A Z = 0$ almost surely, its first-order limit is degenerate; the second-order statement gives a different scale only when its stated expansion holds.

Depends on

Used by

Dependency tree · two levels

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 16, PDF pp. 9–12 (standard reference, not scraped)
John Duchi, Theory of Statistics, Stats 300B Lecture 2, PDF pp. 2–4 (standard reference, not scraped)