Extreme Value Theory with Operator Norming Stilian Stoev ( - - PowerPoint PPT Presentation

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Extreme Value Theory with Operator Norming

Stilian Stoev (sstoev@umich.edu) University of Michigan, Ann Arbor Nov 9, 2012 Workshop on Spatial Extreme Value Theory and Properties of Max–Stable Processes In honor of the habilitation of Professeur Cl´ ement Dombry Joint work with Mark M. Meerschaert and Hans-Peter Scheffler.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

1

Some Limit Theory

2

Representation and Simulation

3

Testing for Hetero-Ouracity

4

Implementation and Applications

5

References

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Operator Normalized Exteremes

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Preliminaries

A r.vector X in Rd has an operator regularly varying law if nP{AnX ∈ ·}

v

− → φ(·), in Rd \ {0} (1) for a sequence of d × d matrices An → 0. To avoid trivialities the limit measure φ is assumed to be full, i.e. it’s support is not concentrated on a sub-space of Rd As shown in Ch. 6.1 of Meerschaert & Scheffler (2001), tφ(B) = φ(t−EB), for all B ∈ B(Rd \ {0}), t > 0, where t−E = e− log(t)E = ∞

n=0(−log(t))nE n/n!.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Preliminaries

A r.vector X in Rd has an operator regularly varying law if nP{AnX ∈ ·}

v

− → φ(·), in Rd \ {0} (1) for a sequence of d × d matrices An → 0. To avoid trivialities the limit measure φ is assumed to be full, i.e. it’s support is not concentrated on a sub-space of Rd As shown in Ch. 6.1 of Meerschaert & Scheffler (2001), tφ(B) = φ(t−EB), for all B ∈ B(Rd \ {0}), t > 0, where t−E = e− log(t)E = ∞

n=0(−log(t))nE n/n!.

Moreover, the An’s can be chosen so that A[tn]A−1

n

− → t−E, n → ∞.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Examples

scalar: E = 1/αId = diag(1/α, · · · , 1/α), α > 0. diagonal: E = diag(1/α1, · · · , 1/αd). more complicated: arbitrary E positive definite. heavy tailed: E has eigenvalues λ1, · · · , λd with positive real parts 0 < 1/α1 := Re(λ1) ≤ · · · ≤ 1/αd := Re(λd).

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Motivating question and some difficulties

Are there operator max-stable distributions and what are they?

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Motivating question and some difficulties

Are there operator max-stable distributions and what are they? There are some challenges...

• perator sum-stability: For each n, exists an operator (matrix)

An, such that An(X1 + · · · + Xn) d = X, (2) with Xi’s independent copies of X. In this case, An = n−E.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Motivating question and some difficulties

Are there operator max-stable distributions and what are they? There are some challenges...

• perator sum-stability: For each n, exists an operator (matrix)

An, such that An(X1 + · · · + Xn) d = X, (2) with Xi’s independent copies of X. In this case, An = n−E. Note that An commutes with the ‘+’ operation!

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Motivating question and some difficulties

Are there operator max-stable distributions and what are they? There are some challenges...

• perator sum-stability: For each n, exists an operator (matrix)

An, such that An(X1 + · · · + Xn) d = X, (2) with Xi’s independent copies of X. In this case, An = n−E. Note that An commutes with the ‘+’ operation!

• perator max-stability? For An = n−E, we cannot always write

An(X1 ∨ · · · ∨ Xn) = AnX1 ∨ · · · ∨ AnXn, since An may not commute with the ‘∨’ operation...

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Motivating question and some difficulties

Are there operator max-stable distributions and what are they? There are some challenges...

• perator sum-stability: For each n, exists an operator (matrix)

An, such that An(X1 + · · · + Xn) d = X, (2) with Xi’s independent copies of X. In this case, An = n−E. Note that An commutes with the ‘+’ operation!

• perator max-stability? For An = n−E, we cannot always write

An(X1 ∨ · · · ∨ Xn) = AnX1 ∨ · · · ∨ AnXn, since An may not commute with the ‘∨’ operation... There is no obvious direct analog of (2) for maxima that covers all stability exponents E...

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Motivating question and some difficulties

Are there operator max-stable distributions and what are they? There are some challenges...

• perator sum-stability: For each n, exists an operator (matrix)

An, such that An(X1 + · · · + Xn) d = X, (2) with Xi’s independent copies of X. In this case, An = n−E. Note that An commutes with the ‘+’ operation!

• perator max-stability? For An = n−E, we cannot always write

An(X1 ∨ · · · ∨ Xn) = AnX1 ∨ · · · ∨ AnXn, since An may not commute with the ‘∨’ operation... There is no obvious direct analog of (2) for maxima that covers all stability exponents E... One good approach is via point processes and directional extremes...

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Operator scaling and directional extremes

Let X1, · · · , Xn be i.i.d. with heavy-tailed operator RV(E) law. Convergence of point clouds under operator scaling: Nn := {AnX1, · · · , AnXn} ⇒ N, as n → ∞, where N is a PPP with operator-scaling intensity φ.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Operator scaling and directional extremes

Let X1, · · · , Xn be i.i.d. with heavy-tailed operator RV(E) law. Convergence of point clouds under operator scaling: Nn := {AnX1, · · · , AnXn} ⇒ N, as n → ∞, where N is a PPP with operator-scaling intensity φ. (directional extremes) Define the maxima along direction (angle) θ ∈ Rd \ {0}: Mn(θ) :=

n

• i=1

Xj, θ. Note that Mn(At

nθ) = n

• i=1

Xj, At

nθ = n

• i=1

AnXj, θ so we expect {Mn(At

nθ)}θ∈Rd to have a non-degenerate limit.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

A Limit Theorem

Theorem (Meerschaert, Scheffler, & S.) As n → ∞, we have {Mn(At

nθ)}θ∈Rd =

⇒ {Y (θ)}θ∈Rd, in the space C(Rd; R), where nP{AnX1 ∈ ·} v → φ(·), n → ∞.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

A Limit Theorem

Theorem (Meerschaert, Scheffler, & S.) As n → ∞, we have {Mn(At

nθ)}θ∈Rd =

⇒ {Y (θ)}θ∈Rd, in the space C(Rd; R), where nP{AnX1 ∈ ·} v → φ(·), n → ∞. Idea of proof: Consider the tail sets B(θ, r) = {x ∈ Rd : x, θ > r}, r > 0. Then P{Mn(At

nθ) ≤ r} = P{AnXi ∈ B(θ, r)c, 1 ≤ i ≤ n},

which is P{Nn ⊂ B(r, θ)c} − → P{N ⊂ B(r, θ)c} = exp{−φ(B(r, θ))}

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

A Limit Theorem

Theorem (Meerschaert, Scheffler, & S.) As n → ∞, we have {Mn(At

nθ)}θ∈Rd =

⇒ {Y (θ)}θ∈Rd, in the space C(Rd; R), where nP{AnX1 ∈ ·} v → φ(·), n → ∞. Idea of proof: Consider the tail sets B(θ, r) = {x ∈ Rd : x, θ > r}, r > 0. Then P{Mn(At

nθ) ≤ r} = P{AnXi ∈ B(θ, r)c, 1 ≤ i ≤ n},

which is P{Nn ⊂ B(r, θ)c} − → P{N ⊂ B(r, θ)c} = exp{−φ(B(r, θ))} = P{Y (θ) ≤ r}.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The limit process

The limit process Y : is non-negative, max-i.d. with f.d.d. P{Y (θ1) ≤ r1, · · · , Y (θm) ≤ rm} = exp

• − φ(

m

• j=1

B(rj, θj))

• ,

where B(r, θ) := {y ∈ Rd : y, θ > r}.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The limit process

The limit process Y : is non-negative, max-i.d. with f.d.d. P{Y (θ1) ≤ r1, · · · , Y (θm) ≤ rm} = exp

• − φ(

m

• j=1

B(rj, θj))

• ,

where B(r, θ) := {y ∈ Rd : y, θ > r}. has continuous paths. has the operator scaling or perhaps operator max-stability? property {Y1(θ) ∨ · · · ∨ Yn(θ)}θ∈Γ

d

= {Y (nE tθ)}θ∈Γ

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Representation

We have the disintegration formula φ(A) =

• S

∞ 1A(τ −Eθ)dτλ(dθ), where λ is the spectral measure on S = {x = 1}.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Representation

We have the disintegration formula φ(A) =

• S

∞ 1A(τ −Eθ)dτλ(dθ), where λ is the spectral measure on S = {x = 1}. Theorem (Meerschaert, Scheffler, & S. ) Y (θ) =

• i∈N

Γ−E

i

Λi, θ, θ ∈ Rd \ {0}, where 0 < Γ1 < Γ2 < · · · are the arrivals of a Poisson point process on (0, ∞) with rate λ(S) and the Λi’s are i.i.d. with distribution λ(·)/λ(S) on S.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Simulation

Consider the truncated maxima: Y (n)(θ) =

n

• i=1

Γ−E

i

Λi, θ, θ ∈ Rd \ {0}

• How well does Y (n) approximate Y ?

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Simulation

Consider the truncated maxima: Y (n)(θ) =

n

• i=1

Γ−E

i

Λi, θ, θ ∈ Rd \ {0}

• How well does Y (n) approximate Y ?

Theorem (Meerschaert, Scheffler, & S. ) For all K ⊂ S, ǫ > 0 and δ ∈ (0, 1), exists nδ,ǫ, so that ∀n ≥ nδ,ǫ, P{Y (θ) = Y (n)(θ), for some θ ∈ K} ≤ δn + P{ inf

θ∈K Y (θ) ≤ ǫ}.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Simulation

Consider the truncated maxima: Y (n)(θ) =

n

• i=1

Γ−E

i

Λi, θ, θ ∈ Rd \ {0}

• How well does Y (n) approximate Y ?

Theorem (Meerschaert, Scheffler, & S. ) For all K ⊂ S, ǫ > 0 and δ ∈ (0, 1), exists nδ,ǫ, so that ∀n ≥ nδ,ǫ, P{Y (θ) = Y (n)(θ), for some θ ∈ K} ≤ δn + P{ inf

θ∈K Y (θ) ≤ ǫ}.

Idea of the proof: Suppose that K = S and supp(λ) = S. Then, one can show that ξ := inf

θ∈K Y (θ) ≡ min θ∈S Y (θ) > 0,

almost surely.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Idea of the proof...

We need to show that: P{Y (θ) ≡ Y (n)(θ), ∀θ ∈ S} ≥ 1 − δn − P{ξ ≤ ǫ}, (n ≥ nδ,ǫ). Recall that Y (θ) =

∞

• i=1

Γ−E

i

Λi, θ.

• Over the event {ξ ≡ minθ Y (θ) > ǫ}, the terms Γ−E

i

Λi do not contribute to Y (θ), provided Γ−E

i

Λi ≤ Γ−E

i

≤ ǫ. (3)

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Idea of the proof...

We need to show that: P{Y (θ) ≡ Y (n)(θ), ∀θ ∈ S} ≥ 1 − δn − P{ξ ≤ ǫ}, (n ≥ nδ,ǫ). Recall that Y (θ) =

∞

• i=1

Γ−E

i

Λi, θ.

• Over the event {ξ ≡ minθ Y (θ) > ǫ}, the terms Γ−E

i

Λi do not contribute to Y (θ), provided Γ−E

i

Λi ≤ Γ−E

i

≤ ǫ. (3)

• By Thm 2.2.4 in [Meerschaert and Scheffler, 2001]

t−E ≤ Cat−a, for 0 < a < min Re(spec(E)). Thus, (3) follows if CaΓ−a

i

≤ ǫ ⇔ Γi ≥ (ǫ/Ca)−1/a.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Idea of the proof...

We need to show that: P{Y (θ) ≡ Y (n)(θ), ∀θ ∈ S} ≥ 1 − δn − P{ξ ≤ ǫ}, (n ≥ nδ,ǫ). Recall that Y (θ) =

∞

• i=1

Γ−E

i

Λi, θ.

• Over the event {ξ ≡ minθ Y (θ) > ǫ}, the terms Γ−E

i

Λi do not contribute to Y (θ), provided Γ−E

i

Λi ≤ Γ−E

i

≤ ǫ. (3)

• By Thm 2.2.4 in [Meerschaert and Scheffler, 2001]

t−E ≤ Cat−a, for 0 < a < min Re(spec(E)). Thus, (3) follows if CaΓ−a

i

≤ ǫ ⇔ Γi ≥ (ǫ/Ca)−1/a.

• A simple large deviation bound for Γi = E1 + · · · + Ei implies the

exponential bound in the theorem.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Illustration: E = diag(1, 1/3, 1/3)

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Hetero-ouracity Testing for hetero–ouracity tail (Eng.) – oυρα in Greek

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Hetero-ouracity Testing for hetero–ouracity tail (Eng.) – oυρα in Greek Problem formulation: Given are i.i.d. X1, · · · , Xn with heavy operator RV tails.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Hetero-ouracity Testing for hetero–ouracity tail (Eng.) – oυρα in Greek Problem formulation: Given are i.i.d. X1, · · · , Xn with heavy operator RV tails. Goal: test for the need of operator normalization, i.e. test for

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Hetero-ouracity Testing for hetero–ouracity tail (Eng.) – oυρα in Greek Problem formulation: Given are i.i.d. X1, · · · , Xn with heavy operator RV tails. Goal: test for the need of operator normalization, i.e. test for hetero-

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Hetero-ouracity Testing for hetero–ouracity tail (Eng.) – oυρα in Greek Problem formulation: Given are i.i.d. X1, · · · , Xn with heavy operator RV tails. Goal: test for the need of operator normalization, i.e. test for hetero-ouracity.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Key Asymptotic Results

Let Vn(θ) := max1≤i≤n |Xi, θ| and Y |·|(θ) := Y (θ) ∨ Y (−θ). The fact that {Mn(At

nθ)} ⇒ {Y (θ)} and the continuous mapping

theorem imply Fact (Meerschaert, Scheffler, & S. ) As n → ∞, we have {Vn(At

nθ)} ⇒ {Y |·|(θ)}, in C(S, R),

and hence

• max

θ=1 Vn(At nθ), min θ=1 Vn(At nθ)

• ⇒ (Y (max), Y (min)),

where Y (max) := maxθ=1 Y |·|(θ) and Y (min) := minθ=1 Y |·|(θ).

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Key Asymptotic Results (cont’d)

Recall Vn(θ) := max1≤i≤n |Xi, θ|, and let V (max)

n

:= max

θ=1 Vn(θ)

and V (min)

n

:= min

θ=1 Vn(θ).

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Key Asymptotic Results (cont’d)

Recall Vn(θ) := max1≤i≤n |Xi, θ|, and let V (max)

n

:= max

θ=1 Vn(θ)

and V (min)

n

:= min

θ=1 Vn(θ).

More delicate analysis due to Peter Scheffler yields: Fact (Meerschaert, Scheffler, & S. ) As n → ∞, we have log V (max)

n

log n

P

− → 1/αd (heaviest tail) and log V (min)

n

log n

P

− → 1/α1 (lightest tail) Note: The real parts of the eigenvalues of the RV exponent E are 0 < 1/α1 ≤ · · · ≤ 1/αd.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Test statistic

Consider the hypotheses H0 : anId = An (scalar norming) Ha : α1 > αd (different tails).

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Test statistic

Consider the hypotheses H0 : anId = An (scalar norming) Ha : α1 > αd (different tails). (under the null) anVn(θ) = Vn(Anθ) and by the key asymptotic results: an(V (max)

n

, V (min)

n

) ⇒ (Y (max), Y (min)), n → ∞.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Test statistic

Consider the hypotheses H0 : anId = An (scalar norming) Ha : α1 > αd (different tails). (under the null) anVn(θ) = Vn(Anθ) and by the key asymptotic results: an(V (max)

n

, V (min)

n

) ⇒ (Y (max), Y (min)), n → ∞. Therefore Tn := V (max)

n

V (min)

n

⇒ Y (max) Y (min) , n → ∞.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Test statistic

Consider the hypotheses H0 : anId = An (scalar norming) Ha : α1 > αd (different tails). (under the null) anVn(θ) = Vn(Anθ) and by the key asymptotic results: an(V (max)

n

, V (min)

n

) ⇒ (Y (max), Y (min)), n → ∞. Therefore Tn := V (max)

n

V (min)

n

⇒ Y (max) Y (min) , n → ∞. (Under the alternative) We also get Tn

P

− → ∞, n → ∞.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Implementation and Applications Implementation and Applications

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Implementation

Decision rule: Reject H0 at level β, if Tn > c1−β = F ←

Y (max)/Y (min)(1−β) := inf{x : FY (max)/Y (min)(x) ≥ 1−β}.

Otherwise, fail to reject.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Implementation

Decision rule: Reject H0 at level β, if Tn > c1−β = F ←

Y (max)/Y (min)(1−β) := inf{x : FY (max)/Y (min)(x) ≥ 1−β}.

Otherwise, fail to reject. Practical issues: FY (max)/Y (min) depends on λ and α. Even if λ and α are known, the distribution is intractable. The spectral measure λ and the common tail exponent α are unknown.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Implementation

Decision rule: Reject H0 at level β, if Tn > c1−β = F ←

Y (max)/Y (min)(1−β) := inf{x : FY (max)/Y (min)(x) ≥ 1−β}.

Otherwise, fail to reject. Practical issues: FY (max)/Y (min) depends on λ and α. Even if λ and α are known, the distribution is intractable. The spectral measure λ and the common tail exponent α are unknown. Plan: We will replace λ and α with consistent estimates. Then, we will sample from FY (max)/Y (min) thus obtaining parametric bootstrap type estimates of c1−β.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The spectral measure

Given an i.i.d. sample X1, · · · , Xn, define the empirical spectral measure:

• λn,u(A) :=

n

i=1 1A(Xi/Xi)1(u,∞)(Xi)

n

i=1 1(u,∞)(Xi)

, where u is a sufficiently large threshold.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The spectral measure

Given an i.i.d. sample X1, · · · , Xn, define the empirical spectral measure:

• λn,u(A) :=

n

i=1 1A(Xi/Xi)1(u,∞)(Xi)

n

i=1 1(u,∞)(Xi)

, where u is a sufficiently large threshold. Fact (Meerschaert, Scheffler, & S. ) Under the null, if unan → 0 as un → ∞,

• λn,u =

⇒ λ(·)/λ(S), in M+(S).

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The bootstrap: theory

Under H0, the limit process has the representation: Y (θ) =

• e

S

u, θ+Mα,λ(du), θ ∈ S= {x ∈ Rd : x = 1}, where Mα,λ is an α-Fr´ echet random sup-measure with control measure λ(du) on S.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The bootstrap: theory

Under H0, the limit process has the representation: Y (θ) =

• e

S

u, θ+Mα,λ(du), θ ∈ S= {x ∈ Rd : x = 1}, where Mα,λ is an α-Fr´ echet random sup-measure with control measure λ(du) on S. Given λn and αn, let Y ∗

n (θ) :=

• e

S

u, θ+M∗

• αn,

λn(du),

θ ∈ S, where M∗

• αn,

λn is defined on the probability space (Ω∗, F∗, P∗).

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The bootstrap: consistency and examples

Fact (Meerschaert, Scheffler, & S. ) If anun → 0 as un → ∞ and αn

P

→ α, then, as n → ∞, LP∗ Y ∗(max)

n

Y ∗(min)

n

• P

= ⇒ LP∗ Y ∗(max) Y ∗(min)

• ≡ LP

Y (max) Y (min)

• ,

where Y () = θ=1(Y (θ) ∨ Y (−θ)), ∈ {max, min}.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

The bootstrap: consistency and examples

Fact (Meerschaert, Scheffler, & S. ) If anun → 0 as un → ∞ and αn

P

→ α, then, as n → ∞, LP∗ Y ∗(max)

n

Y ∗(min)

n

• P

= ⇒ LP∗ Y ∗(max) Y ∗(min)

• ≡ LP

Y (max) Y (min)

• ,

where Y () = θ=1(Y (θ) ∨ Y (−θ)), ∈ {max, min}.

Corollary If c1−β(α, λ) is a continuity point of the quantile function F ←

Y (max)/Y (min), then

c1−β( αn, λn)

P

− → c1−β(α, λ), as n → ∞.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

An operator Pareto example

Let U ∼ Unif(0, 1) and Θ ∼ Unif(S) be independent. Set X = U−EΘ, where E = diag(1/α1, 1/α2, 1/α3).

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

An operator Pareto example

Let U ∼ Unif(0, 1) and Θ ∼ Unif(S) be independent. Set X = U−EΘ, where E = diag(1/α1, 1/α2, 1/α3). Setup: Let α1 = α2 = 1 and 1/α3 = a∗ ∈ {0.1, 0.5, 0.9, 1, 1.1, 2, 3}. We simulate n ∈ {100, 1000, 10000} independent realizations. Using the Poisson representation with the empirical spectral measure and un = 1, we get parametric bootstrap approximations of c1−β. Test and repeat independently 1000 times to get empirical Type I error and power.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Type I error and power

Table: Empirical rejection probabilities for our test (at level β = 0.1) under the null and six alternatives. n\ a∗ 0.1 0.5 0.9 1.0 1.1 2.0 3.0 100 1.000 0.380 0.093 0.083 0.083 0.311 0.528 1000 1.000 0.828 0.111 0.097 0.111 0.743 0.938 10000 1.000 0.998 0.143 0.110 0.145 0.947 0.955

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Some p-values

0.01 0.02 100 200 300 400 500 600 700 800 a*=0.5 0.5 1 50 100 150 a*=0.9 0.5 1 20 40 60 80 100 120 a*=1 0.5 1 50 100 150 200 250 300 a*=1.5

Figure: Histograms of the p-values based on 1000 replicates of the test. The first two panels (left to right) correspond to the alternatives α1 = a∗, α2 = α3 = 1, with a∗ = 0.5 and 0.9, respectively. The third panel corresponds to the null and the right-most panel to the alternative α1 = α2 = 1, and α3 = 1.5.

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Thank you for the patience!

Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References

Some References M.M. Meerschaert (1988) ‘Regular variation in Rk’

• Proc. Amer. Math. Soc. 102, 341-348.

M.M. Meerschaert & H.-P. Scheffler (2001) Limit distributions for sums

• f independent random vectors: Heavy tails in theory and practice.

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