Convergence of Multivariate Quantile Surfaces

Transcription

1 Convergence of Multivariate Quantile Surfaces Adil Ahidar Institut de Mathématiques de Toulouse - CERFACS August 30, 2013 Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

2 Outline 1 Introduction Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

3 Outline 1 Introduction 2 Definitions Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

4 Outline 1 Introduction 2 Definitions 3 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

5 Outline 1 Introduction 2 Definitions 3 Empirical quantile surfaces 4 Directional regularity assumptions Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

6 Outline 1 Introduction 2 Definitions 3 Empirical quantile surfaces 4 Directional regularity assumptions 5 Main Results Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

7 Outline 1 Introduction 2 Definitions 3 Empirical quantile surfaces 4 Directional regularity assumptions 5 Main Results 6 Conclusion Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

8 Introduction Introduction Let (X n ) be i.id. on R d, P = P X. Desirable properties for multidimensional quantile: Generalization of unidimensionnal quantile Estimability by the sample (X n ) non parametric Interpretability (Mods detection, Mass localisation, ) Admit ULLN, UCLT, SAP, Caracterize the law Calculability and Fast simulation Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

9 Introduction We will define quantile surfaces via a class of shapes General classes can be used (curves, manifolds,...) Here we focus on the hyperplans class Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

10 Definitions Definitions Let define the α-th quantile surfaces associated to P and seen from O R d. H the collection of all half-spaces and, the inner product in R d and { } H(O, u, y) = x R d : x O, u y H the halfspace standing at distance y R from O in direction u S d 1. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

11 Definitions Given α (1/2, 1) and a direction u S d 1 let Y α (O, u) = inf {y : P(H(O, u, y)) α} be the u-directional α-th quantile range and H α (u) = H(O, u, Y α (O, u)) be the u-directional α-th quantile halfspace. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

12 Definitions Remark that Y α (O, u) = F 1 X O,u (y) and thus Y α(o, u) is the α-th real quantile of the real random variable X O, u Definition (Multivariate quantile set) For α (1/2, 1], O R d and u S d 1 define the u-directional α-th quantile point seen from O to be Q α (O, u) = O + Y α (O, u)u (1) and the α-th quantile set seen from O to be the star-shaped collection of points Q α (O) = {Q α (O, u) : u S d 1 }. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

13 Definitions Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

14 Definitions Definition Let H α = {H α (u) : u S d 1 } = {H : H is a half-space, P(H) = α} C α = H H α H. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

15 Definitions Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

16 Empirical quantile surfaces Empirical quantile surfaces Let α = [α, α + ] (1/2, 1) and O R d. Define P n and P n,o,u as follows, P n = 1 δ Xi, P n,o,u = 1 δ n n Xi O,u, i n where δ x is the Dirac mass at x R d or x R. i n Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

17 Empirical quantile surfaces Definition For u S d 1 let Y n,α (O, u) = inf {y : P n (H(O, u, y)) α} = inf {y : P n,o,u ((, y)) α}. Define the u-directional α-th empirical quantile point seen from O to be Q n,α (O, u) = O + Y n,α (O, u)u and associate to this point the subjective α-th quantile half-space H n,α (u) = H(O, u, Y n,α (O, u)). Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

18 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

19 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

20 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

21 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

22 Empirical quantile surfaces... Let the α-th empirical quantile set seen from O be Q n,α (O) = {Q n,α (O, u) : u S d 1 }. The half-spaces indexed by points Q n,α (O, u) are collected into H n,α = {H n,α (u) : u S d 1 } from which we further define the random convex set C n,α = H H n,α H. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

23 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

24 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

25 Empirical quantile surfaces Proposition The sets C n,α and Q n,α (O) are closed surfaces and piecewise spherical. We also have { [ H n,α = H : H is a half-space, P n (H) α, α + d ]}. n Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

26 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

27 Empirical quantile surfaces Proposition The processes Q n Q and Y n Y do not depend on O and n(qn Q)(u, α) = n(y n Y)(u, α)u (2) is a R d -valued empirical process indexed by S d 1. when O moves we have Q α (O + δ, u) = Q α (O, u) + δ δ, u u and Q n,α (O + δ, u) = Q n,α (O, u) + δ δ, u u Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

28 Empirical quantile surfaces Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

29 Directional regularity assumptions Directional regularity assumptions Let = [α, α + ] 0 = ( α0, ) α+ 0, where 1/2 < α 0 < α 0 + < 1. Assumption A0 (A 0 ): for all u S d 1, for all open interval I F 1 X O,u ( ) P X O,u (I ) > 0 and for all x R P X O,u ({x}) = 0 Proposition Assume that d > 1 and (A 0 ) holds. Then the sets Q α (O), α, are closed surfaces with at most two connex components and are unions of convex surfaces. Moreover (u, α) Q(u, α) is continuous on S d 1. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

30 Directional regularity assumptions From now we suppose for all u S d 1, the random variable X, u has a density f X,u on F 1 X,u ( 0). This implies A 0 Notation h u = h X,u = f X,u F 1 X,u. (3) Assumption A1 The function (u, α) h(u, α) = h u (α) is continuous on S d 1 and Remark: 0 < m inf α 0 inf h u (α) sup u S d 1 α 0 sup h u (α) M <. u S d 1 Assumption (A 1 ) does not imply that P has a density on R d. However it implies that P(H) = 0 for all H H. Under (A 1 ) h u = f X O,u F 1 X O,u do not depend on O. Allows supp P with low dimension. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

31 Main Results Main Results Theorem (Uniform consistency) Under (A 1 ) we have lim sup sup n u S d 1 α Y n,α (O, u) Y α (O, u) = lim n Y n Y Sd 1 = 0 a.s. Corollary Under (A 1 ) we have lim sup n sup sup O D u S d 1 α lim sup n O D α Q α,n (O, u) Q α (O, u) = 0 sup d H (Q α,n (O), Q α (O)) = 0 lim sup d H (C α,n, C α ) = 0 n α a.s. a.s. a.s. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

32 Main Results To provide the UCLT we define the limiting gaussian process G P Definition Let B P be the P-Brownian bridge indexed by half-spaces, that is the zero mean Gaussian process on H with cov(b P (H), B P (H )) = P(H H ) P(H)P(H ), for (H, H ) H H For O,u fixed we have cov(b O,u (y), B O,u (y )) = min(f X O,u (y), F X O,u (y )) with notation B O,u (y) := B P (H(O, u, y)) F X O,u (y)f X O,u (y ). Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

33 Main Results Definition For (u, α) S d 1 we define G P (u, α) = B P(H(O, u, Y (O, u, α))) h(u, α) = B P(H α (u)). (4) h(u, α) For (u 1, α 1 ), (u 2, α 2 ) S d 1 we have cov(g P (u 1, α 1 ), G P (u 2, α 2 )) = P(H α 1 (u 1 ) H α2 (u 2 )) α 1 α 2. h(u 1, α 1 )h(u 2, α 2 ) and the correlation c(u 1, α 1, u 2, α 2 ) = corr(g P (u 1, α 1 ), G P,α (u 2, α 2 )) = P(H α 1 (u 1 ) H α2 (u 2 )) α 1 α 2 α1 (1 α 1 ) α 2 (1 α 2 ) (1 α 1 )(1 α 2 ) (α 1 α 2 )(1 α 1 α 2 ), α 1 α 2 (α 1 α 2 )(1 α 1 α 2 ) Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

34 Main Results Let B be the set of all bounded real functions on S d 1, endowed with the supremum norm. Theorem (Uniform central limit theorem) If P satisfies (A 1 ) then the sequence n(y n Y) weakly converges to G P on B. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

35 Main Results Corollary Finite dimensional marginal laws convergence Fix (O 1, u 1, α 1 ),..., (O k, u k, α k ) in R d S d 1. Under (A 1 ) we have n Y n (O 1, u 1, α 1 ) Y (O 1, u 1, α 1 )... Y n (O k, u k, α k ) Y (O k, u k, α k ) where the limiting covariance matrix Σ has coordinates law N (0 k, Σ) Σ i,j = P(H α i (u i ) H αj (u j )) α i α j. h αi (u i )h αj (u j ) Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

36 Main Results Assumption A2 (A 2 ) :(A 1 ) holds and h(u, α) = h u (α) is differentiable on S d 1 0 in variables (u, α) with uniformly bounded derivatives. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

37 Main Results Theorem (Uniform strong approximation with rate) if (A 1 ) then one can construct on the same probability space (Ω, T, P) an i.i.d. sequence X n with law P and G n versions of G P such that for O R d, α, u S d 1 Y n (O, u, α) = Y(O, u, α) + G n(u, α) n + Z n(u, α) n (5) where Z n = n(y n Y) G n is such that lim Z n n Sd 1 = lim n sup sup u S d 1 α Z n (u, α) = 0 a.s. Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

38 Main Results... If P moreover satisfies (A 2 ) then G n can be constructed such that for v 1 = v 2 = 1/4, w 1 = 1/2, w 2 > 1 and, if d 3, v d = 1/(3 + 4d), w d =..., there exists c θ (m, M, d) > 0 and n θ (m, M, d) > 0 such that we have, for all n > n θ, ( P Z n Sd 1 c (log n) w ) d θ n v 1 d n θ. (6) Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

39 Main Results Let Λ n = n(p n P) be the empirical process indexed by H and define E n (u, α) = Λ n (H α (u)) = n(p n (H α (u)) α) its restriction to H = H α = {H : H is a half-space, P(H) } α Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

40 Main Results Theorem (Bahadur-Kiefer type representation of multivariate quantiles) If P satisfies (A 1 )then we have lim n n(y n Y)h + E Sd 1 n = 0 a.s. (7) and if moreover satisfies (A 2 ) then it holds n(y n Y)h + E n Sd 1 = O a.s. ( (log n) w d n v d ). (8) Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

41 steps of the proof Main Results Step I: Approximate the empirical process F is pointwise measurable, uniformly bounded for ν 0 > 0 and any probability measure P Corollary N (ε ) PF 2, F, d P 1 ε ν 0, 0 < ε < 1 if F = {I C : C C}, C VC-class then ν 0 = VC 1 We can construct {X n } and {G n } on Ω such that (log n) P ( α τ ) 0 n G n F > c θ 1 n θ n τ where τ = ν 0, τ 0 = 4 + 5ν ν 0 Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

42 Main Results step II Discretization of the direction Control of the Modulus of continuity and deviation of the supremum of processes Given one direction and one quantile level α the process G P reduces to a standart Brownian bridge on R Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

43 Main Results For ξn U = n(αn U Bn U ) and ξ n = ξn U F, B n = Bn U F and y α = F 1 (α), ŷ = Fn 1 (α) We have ŷ = inf {y : F n (y) α} { = inf y : F (y) + α } n(y) F (y α ) n { = inf y : B n (y) n (F (y α ) F (y)) ξ } n(y) n inf { y : B(y) nf (y α ) (y y α ) } we have B(y) [B(y α ) ɛ, B(y α ) + ɛ] with large probability on [y α δ ɛ, y α δ ɛ ] Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

44 Main Results Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

45 Main Results B(y α ) ± ɛ = nh(y α ) (ŷ y α ) donc ŷ = B(y α) ± ɛ nh(yα ) + y α B(y α) nh(yα ) + y α Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

46 Conclusion Approximation (with speed) of the directional quantile via the Gaussian that approximate the empirical process Joint region of confidence Fast simulation of the quantile surface Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

47 Conclusion Extension to general classes (geodesics,...) Mass localisation via different classes and exploration with O New depth notion via Q α (O) mods detetction Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43

48 Conclusion thank you for your attention Adil Ahidar (Institut de Mathématiques de Toulouse Convergence - CERFACS) of Multivariate Quantile Surfaces August 30, / 43