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Stat210B: Theoretical Statistics Lecture Date: April 26, 2007
More Functional Delta Method; Quantiles
Lecturer: Michael I. Jordan Scribe: Chris Haulk
1 Motivation
Last lecture we developed a functional delta method that using the notion of Gateaux derivative. With
φ′ P (δx − P ) =
d dt
φ((1 − t)P + tδx)
t=
= IFφ,P (x)
we write
φ(Pn) − φ(P ) =
n
i
IFφ,P (Xi) + Rn
and hope to show that EP [IFφ,P (X)] = 0, VarP [IFφ,P (X)] = γ^2 , and
nRn = op(1). Then the CLT gives √ n(φ(Pn) − φ(P )) d −→ N (0, γ^2 ).
Showing that EP [IFφ,P (X)] = 0 should not be too hard, and calculating a variance at some point probably cannot be avoided if we want to show asymptotic normality of
√ n(φ(Pn)^ −^ φ(P^ )). However, showing that nRn = op(1) may be difficult, depending on φ and P. For a delta method that avoids this last step, we will modify our notion of derivative.
2 Delta Method via Hadamard Differentiability
Let D and E be normed linear spaces and suppose φ : Dφ → E where Dφ ⊂ D. We say that φ is Hadamard differentiable at θ if there is a continuous, linear map φ′ θ : D → E such that ∣ ∣ ∣ ∣
φ(θ + tht) − φ(θ) t − φ′ θ(h)
E
→ 0 as t ↓ 0
for every sequence ht → h such that θ + tht ∈ Dφ for all sufficiently small t. If it is possible to define φ′ θ only on a subset D 0 ⊂ D and the sequences ht above are restricted to have limits h in D 0 , then φ is said to be Hadamard differentiable tangentially to D 0.
Theorem 1 (Delta Method). (van der Vaart, 1998, 20.8) Let D and E be normed linear spaces. Let φ : Dφ ⊂ D → E be Hadamard differentiable at θ tangentially to D 0. Let Tn : Ωn → Dφ be maps such that
rn(Tn − θ) d −→ T for some sequence of numbers rn → ∞ and a random element T that takes values in D 0.
Then rn(φ(Tn) − φ(θ)) d −→ φ′ θ(T ).
Proof. Define gn(h) = rn(φ(θ + h/rn) − φ(θ)) for h ∈ {h : θ + h/rn ∈ Dφ}. By Hadamard differentiability,
gn′^ (hn′^ ) → φ′ θ(h) for every subsequence hn′^ → h ∈ D 0. Therefore gn(rn(Tn − θ)) d −→ φ′ θ(T ) by the extended continuous mapping theorem 18.11.
2 More Functional Delta Method; Quantiles
3 Applications and Examples
Last lecture we used the Gateaux functional delta method to prove asymptotic normality of the Mann- Whitney test statistic. We will prove this fact again using the Hadamard version of the functional delta method.
Lemma 2. (van der Vaart, 1998, Lemma 20.10) Let φ : [0, 1] → R be twice continuously differentiable. Then the function (F 1 , F 2 ) 7 →
φ(F 1 )dF 2 is Hadamard-differentiable at every pair of functions (F 1 , F 2 ) such that Fi ∈ D[−∞, ∞] and Fi has bounded variation. The derivative is
(h 1 , h 2 ) 7 → h 2 φ ◦ F 1 |∞−∞ −
h 2 dφ ◦ F 1 +
φ′(F 1 )h 1 dF 2.
Here, h denotes the left-continuous version of h.
Proof. See text.
Now suppose at time ν we observe two independent random samples X 1 , ..., Xmν Y 1 , ..., Ynν from distributions F and G, respectively. Let Nν = mν + nν and suppose m/N → λ ∈ (0, 1) as ν → ∞. By Donsker’s theorem and Slutsky’s lemma, √ N (Fm − F, Gm − G) d −→
GF
λ
GG
1 − λ
for independent Brownian bridges GF and GG. Let φ(x) = x and apply Lemma 20.10 together with the functional delta method to see
√ N
FmdGn −
F dG
−→ −d
GG−
1 − λ
dF +
GF
λ
dG.
That the limit distribution is Gaussian follows from a generalization of a well-known result for finite di- mensional processes, namely that continuous linear transformations of Gaussian processes are Gaussian. Alternatively, note that Thm. 20.8 implies that the limit variable is the limit in distribution of
N (Gn − G) dF +
N (Fm − F )dG,
rewrite the expression above as a difference of scaled, centered sums, and apply the usual CLT.
4 Quantiles
The quantile function F −^1 : (0, 1) → R of a cumulative distribution function F is
F −^1 (p) = inf{x : F (x) ≥ p}.
The quantile function has some nice properties:
Lemma 3. (van der Vaart, 1998, Lemma 21.1) For 0 < p < 1 and x ∈ R,
- F −^1 (p) ≤ x iff p ≤ F (x)
- F ◦ F −^1 (p) ≥ p
- F −^1 ◦ F (x) ≤ x
