Eustasio del Barrio - Uni Ulm

A contamination model for approximate stochastic order

Eustasio del Barrio

Universidad de Valladolid. IMUVA.

3rd Workshop on Analysis, Geometry and Probability - Universitat Ulm

28th September - 2dn October 2015, Ulm

Eustasio del Barrio Testing approximate stochastic order 1 / 37

Outline

Outline

1 Stochastic order, testability and relaxed versions of s. o.

2 Inference for approximate stochastic order

3 Implementation, simulation & data example


Stochastic order, testability and relaxed versions of s. o. The stochastic order model

130 140 150 160

0.0

0.2

0.4

0.6

0.8

1.0

π(Fn,Gm) = 0.128409 π(Gm,Fn) = 0.009342

heights

ED

F's

Gm Age 10 Boys Fn Age 10 Girls

Data: National Health and Nutrition Examination SurveyEmpirical d.f.’s for boys and girls at age 10.

Are girls taller than boys?

Stochastic order (Lehmann, 1955): P,Q probs. on R with d.f.’s F , G

P ≤st Q if F (x) ≥ G(x), x ∈ R

For NHANES data, P10 ≤st Q10?Eustasio del Barrio Testing approximate stochastic order 3 / 37

Stochastic order, testability and relaxed versions of s. o. Testing stochastic order

Common testing problems in literature (P ≤st Q ≡ F ≤st G)

a) H0: F = G vs Ha: F <st G

b) H0: F ≤st G vs Ha: F 6≤st G

c) H0: F 6≤st G vs Ha: F ≤st G

Problem a)

focus on statistical evidence for strict relation

assumes stochastic order holds

both H0 and Ha can be false (here focus on b), c))

Problem b) ‘testing for stochastic dominance’(McFadden, 1989; Mosler, 1995; Anderson, 1996, Davidson & Duclos, 2000;Linton et al., 2005, 2010,. . . )

goodness-of-fit problem

absence of evidence against s. o. as minimal requirement for a)

lack of evidence against H0 not evidence for F ≤st G



Common testing problems in literature (P ≤st Q ≡ F ≤st G)

a) H0: F = G vs Ha: F <st G

b) H0: F ≤st G vs Ha: F 6≤st G

c) H0: F 6≤st G vs Ha: F ≤st G

Problem a)

focus on statistical evidence for strict relation

assumes stochastic order holds

both H0 and Ha can be false (here focus on b), c))

Problem b) ‘testing for stochastic dominance’(McFadden, 1989; Mosler, 1995; Anderson, 1996, Davidson & Duclos, 2000;Linton et al., 2005, 2010,. . . )

goodness-of-fit problem

absence of evidence against s. o. as minimal requirement for a)

lack of evidence against H0 not evidence for F ≤st G



Problem c) H0: F 6<st G vs Ha: F <st G: assessing stochastic order

rejection provides convincing evidence of F <st G

Unfortunately, no good test for b) exists:

Assume X1, . . . , Xn i.i.d. F <st G

Φ an α-level test (EHΦ(X1, . . . , Xn) ≤ α, H ∈ H0)

Take xm s.t. G(xm) > 1− 1m , Hm s.t. Hm(xm) = 0

Set Fm = (1− 1m )F + 1

mHm; Fm 6<st G

α ≥ EFmΦ(X1, . . . , Xn) ≥ (1− 1m )nEFΦ(X1, . . . , Xn)

Take m→∞

‘no data’ test (reject H0 with prob α regardless data) is UMP!

Berger, 1988 (one-sample); Davidson & Duclos, 2013 (two-sample)



Problem c) H0: F 6<st G vs Ha: F <st G: assessing stochastic order

rejection provides convincing evidence of F <st G

Unfortunately, no good test for b) exists:

Assume X1, . . . , Xn i.i.d. F <st G

Φ an α-level test (EHΦ(X1, . . . , Xn) ≤ α, H ∈ H0)

Take xm s.t. G(xm) > 1− 1m , Hm s.t. Hm(xm) = 0

Set Fm = (1− 1m )F + 1

mHm; Fm 6<st G

α ≥ EFmΦ(X1, . . . , Xn) ≥ (1− 1m )nEFΦ(X1, . . . , Xn)

Take m→∞

‘no data’ test (reject H0 with prob α regardless data) is UMP!

Berger, 1988 (one-sample); Davidson & Duclos, 2013 (two-sample)


Stochastic order, testability and relaxed versions of s. o. Uniformly consistent tests

Uniformly consistent tests

X1, X2, . . . i.i.d. P with values in X

A0,n (A1,n) acceptance (critical) set for Hn against Kn based on X1, . . . , Xn

Test uniformly (exponentially) consistent if for some r, r′ > 0

supP∈Hn

Pn(A1,n) ≤ e−rn, supP∈Kn

Pn(A0,n) ≤ e−r′n

Consider the testing problem

H : P = P0 vs K : d(P, P0) > δ

If d dominates dTV and P0 not discrete, no uniformly consistent test of H vs K(Barron, 1989)


Stochastic order, testability and relaxed versions of s. o. Uniformly consistent tests

Here we propose

A relaxed version of stochastic order for which we can expect to getstatistical evidence

A consistent procedure for gathering that evidence

Which is exponentially uniformly consistent (with due corrections)

Some of our relaxations does hold

Deviation from stochastic order measured through required level ofrelaxation

Easy interpretation


Stochastic order, testability and relaxed versions of s. o. Relaxations of stochastic order

A relaxation of stochastic order (Arcones et al., 2002)

θ(P,Q) := P[X ≤ Y ]

X,Y independent r.v.’s with laws P,Q, resp.

P ≤sp Q (stochastically precedes) if θ(P,Q) ≥ 12

Stochastic ordering implies stochastic precedence: if P ≤st Q

P(X ≤ Y ) =

∫(1−G(x−))dF (x) ≥

∫(1− F (x−))dF (x) = P(X ≤ X ′) ≥ 1

2,

X ′ independent copy of X

Stochastic precedence a less restrictive assumption

But P ≤sp Q equivalent to median(Y −X) ≥ 0, roughly change in location(different, but similar nature as E(Y −X) ≥ 0)




θ(P,Q) := P[X ≤ Y ]




P(X ≤ Y ) =

∫(1−G(x−))dF (x) ≥

∫(1− F (x−))dF (x) = P(X ≤ X ′) ≥ 1

2,







θ(P,Q) := P[X ≤ Y ]




P(X ≤ Y ) =

∫(1−G(x−))dF (x) ≥

∫(1− F (x−))dF (x) = P(X ≤ X ′) ≥ 1

2,



But P ≤sp Q equivalent to median(Y −X) ≥ 0, roughly change in location

(different, but similar nature as E(Y −X) ≥ 0)




θ(P,Q) := P[X ≤ Y ]




P(X ≤ Y ) =

∫(1−G(x−))dF (x) ≥

∫(1− F (x−))dF (x) = P(X ≤ X ′) ≥ 1

2,





Stochastic order, testability and relaxed versions of s. o. Tolerance zones around false models

False model assessment

Assume model F is false (X ∼ P , P /∈ F)

Is model F an adequate approximation for the data? for P?

Pθ ∈ F is an adequate approximation for the data, X1, . . . , Xn, if a typicalsample of size n from Pθ looks like the data

Data features (Davies, 1995)Credibility indices (Lindsay & Liu, 2009)

Pθ ∈ F gives an adequate description of P if d(P, Pθ) ≤ τd = χ2− distance (Hodges & Lehmann, 1954)d = Euclidean distance (Dette & Munk, 2003)d = smallest π such that P = (1− π)Pθ + πR (Rudas et al. (1994);Ae-dB-C-M, 2008, 2010, 2011, 2012; Liu & Lindsay, 2009; Cerioli etal., 2012)

Choice of τ a hard issue

Interpretation of τ simpler for the π-index


Stochastic order, testability and relaxed versions of s. o. Essential model validation

Essential model validation

Observe data X ∼ P , test H1

P = (1− α)R+ αP for some R ∈ FObserve ind. samples X ∼ P , Y ∼ Q, test H2

P = (1− α)R+ αP

Q = (1− α)S + αQ for some (R,S) ∈ FRelated problem of interest: Find α0 = minimal α s.t. null model holds

Example: the similarity model (AE-dB-C-M, 2012)

P and Q α-similar, α ∈ [0, 1) if ∃ prob, R, s.t.{P = (1− α)R+ αP

Q = (1− α)R+ αQ

P , Q α-similar, ⇔ dTV (P,Q) ≤ α (dTV (P,Q) = supA |P (A)−Q(A)|)

Here F = {(R,R)} and

α0,sim(P,Q) = dTV (P,Q)






P = (1− α)R+ αP




Q = (1− α)R+ αQ









P = (1− α)R+ αP




Q = (1− α)R+ αQ









P = (1− α)R+ αP




Q = (1− α)R+ αQ






Approximate stochastic order: P ≤st,α Q if

P = (1− α)R+ αP

Q = (1− α)S + αQ for some R ≤st S

(F = {(R,S) : R ≤st S})

(maybe s. o. too restrictive, but core of distribution fits model)

Interest on minimal contamination level s.t. stochastic order model holds

α0(P,Q) := inf{α : P ≤st,α Q}




P = (1− α)R+ αP


(F = {(R,S) : R ≤st S})







P = (1− α)R+ αP


(F = {(R,S) : R ≤st S})







P = (1− α)R+ αP


(F = {(R,S) : R ≤st S})





Stochastic order, testability and relaxed versions of s. o. Trimming methods in essential model validation

Trimmed Distributions

(X , β) measurable space; P(X , β) prob. measures on (X , β), P ∈ P(X , β)

Rα(P ) =

{R ∈ P(X , β) : R� P,

dR

dP≤ 1

1− αP -a.s.

}

Proposition

(a) Rα(P ) is a convex set; α1 ≤ α2 ⇒ Rα1(P ) ⊂ Rα2

(P )

(b) If α < 1 and (X , β) complete separable metric space then Rα(P ) compactfor weak convergence.

(c) R ∈ Rα(P ) iff P = (1− α)R+ αP


Stochastic order, testability and relaxed versions of s. o. Essential model validation & trimming

Null models in essential model validation expressable in terms of trimmings

Observe X ∼ P , test H1

P = (1− α)R+ αP for some R ∈ F

H1 holds iff Rα(P ) ∩ F 6= ∅

Observe indep. X ∼ P , Y ∼ Q test H2

P = (1− α)R+ αP

Q = (1− α)S + αQ for some (R,S) ∈ F

H2 holds iff (Rα(P )×Rα(Q)) ∩ F 6= ∅

If R(P ),F closed for metric d

H1 holds iff d(Rα(P ),F) = 0

H2 holds iff d(Rα(P )×Rα(Q),F) = 0


Stochastic order, testability and relaxed versions of s. o. Approximate stochastic order & trimming

Trimmings mix well with stochastic order:

For any P ∃ Pα, Pα in Rπ(P ) s. t.

Pα ≤st R ≤st Pα for every R ∈ Rα(P )

Pα, Pα easily computable

Recall P ≤st,α Q iff ∃ P ∈ Rα(P ), Q ∈ Rα(Q), s.t. P ≤st Q

Hence P ≤st,α Q iff Pα ≤st Qα

Conclude from thisα0(P,Q) = sup

x(G(x)− F (x))

Equivalently,P ≤st,α Q ⇔ α0(P,Q) ≤ α










x(G(x)− F (x))











x(G(x)− F (x))











x(G(x)− F (x))











x(G(x)− F (x))











x(G(x)− F (x))




Example 1. P = N(µ, σ), Q = N(ν, σ), µ > ν

Now Q ≤st P and,

α0(P,Q) = (supx

(G(x)− F (x)) = 2Φ(µ−ν2σ

)− 1.

µ− ν = 0.1σ ⇒ P ≤st,0.04 Q

µ− ν = 0.25σ ⇒ P ≤st,0.0987 Q

µ− ν = 0.5σ ⇒ P ≤st,0.1915 Q

µ− ν = σ ⇒ P ≤st,0.3413 Q

Example 2. P = N(µ, σ), Q = N(ν, τ)

Here α0(P,Q) depends on µ, ν, σ, τ

θ(P,Q) = µ− ν (Arcones et al.,2002)

For µ = ν, P ≤sp Q regardless σ, τ

But N(0, σ) takes values greater than N(0, 0) 50% of times!




Now Q ≤st P and,

α0(P,Q) = (supx

(G(x)− F (x)) = 2Φ(µ−ν2σ

)− 1.

µ− ν = 0.1σ ⇒ P ≤st,0.04 Q

µ− ν = 0.25σ ⇒ P ≤st,0.0987 Q

µ− ν = 0.5σ ⇒ P ≤st,0.1915 Q

µ− ν = σ ⇒ P ≤st,0.3413 Q









Now Q ≤st P and,

α0(P,Q) = (supx

(G(x)− F (x)) = 2Φ(µ−ν2σ

)− 1.

µ− ν = 0.1σ ⇒ P ≤st,0.04 Q

µ− ν = 0.25σ ⇒ P ≤st,0.0987 Q

µ− ν = 0.5σ ⇒ P ≤st,0.1915 Q

µ− ν = σ ⇒ P ≤st,0.3413 Q









Now Q ≤st P and,

α0(P,Q) = (supx

(G(x)− F (x)) = 2Φ(µ−ν2σ

)− 1.

µ− ν = 0.1σ ⇒ P ≤st,0.04 Q

µ− ν = 0.25σ ⇒ P ≤st,0.0987 Q

µ− ν = 0.5σ ⇒ P ≤st,0.1915 Q

µ− ν = σ ⇒ P ≤st,0.3413 Q







Inference for approximate stochastic order Estimation & testing

Inference in approximate stochastic order models

Assume X1, ..., Xn i.i.d. P ; Y1, ..., Ym i.i.d. Q, independent samples

Goals

(a) For a fixed α, test H0 : P ≤st,α Q vs. H0 : P 6≤st,α Q

(b) For a fixed α, test H0 : P 6≤st,α Q vs. H0 : P ≤st,α Q

(c) Estimation/confidence intervals/confidence bounds for α0(P,Q)

Recall P ≤st,α Q⇔ α0(P,Q) ≤ α; reformulate (a), (b) as

(a) H0 : α0(P,Q) ≤ α vs. Ha : α0(P,Q) > α (testing against approximate s.o.)

(b’) H0 : α0(P,Q) ≥ α vs. Ha : α0(P,Q) < α (testing for approximate s.o.)





Goals











Goals








Inference for approximate stochastic order Asymptotic theory

Assume F and G continuous; n = m→∞

Fn, Gn empirical d.f.’s

Theorem

α0(Fn, Gn) →a.s.

α0(F,G),

√n(α0(Fn, Gn)− α0(F,G))→

wB(F,G)

withB(F,G) = sup

x∈Γ(F,G)

(B1(F (x))−B2(G(x))),

B1, B2 independent Brownian Bridges;

Γ(F,G) := {x ∈ R : F (x)−G(x) = α0(F,G)}

A bootstrap version also available, but slow approximation (support estimation)



Quantiles of B(F,G) depend on F,G in a complex way

Define Bα = B(U(α, 1 + α), U(0, 1)), 0 ≤ α ≤ 1

Bα = supα≤t≤1

(B1(t)−B2(t− α))

P (B0 >√

2t) = e−t2/2, α = 0, 0.1, . . . , 0.5



Quantiles of B(F,G) depend on F,G in a complex way

Define Bα = B(U(α, 1 + α), U(0, 1)), 0 ≤ α ≤ 1

Bα = supα≤t≤1

(B1(t)−B2(t− α))

P (B0 >√

2t) = e−t2/2, α = 0, 0.1, . . . , 0.5



Bounds for asymptotic quantiles

Kβ(F,G) (resp. Kβ(α)) β-quantile of B(F,G) (resp. Bα)

Kβ(F,G) ≤ Kβ(α(F,G)), β ∈ (0, 1)

If β ∈ (0, 12 ]

σ(F,G, α(F,G))Φ−1(β) ≤ Kβ(F,G),

where σ(F,G, α(F,G)) = mint∈T (F,G,α(F,G)) σt,

T (F,G, α(F,G)) = {t : ∃x s.t. F (x) = t, G(x) = t− α(F,G)} and

σ2t = t(1− t) + (t− α(F,G))(1− t+ α(F,G))


Inference for approximate stochastic order Testing against essential stochastic order

Testing against essential stochastic order

ConsiderH0 : α0(F,G) ≤ α, vs. Ha : α0(F,G) > α

(equivalently, H0 : F ≤st,α G vs. Ha : F 6≤st,α G)

Reject H0 if √n(α0(Fn, Gn)− α) > K1−β(α),

K1−β(α) = 1− β quantile of B(α)

Theorem

limn→∞

sup(F,G)∈H0

PF,G(√n(α0(Fn, Gn)− α) > K1−β(α))

= limn→∞

PF0,G0(√n(α0(Fn, Gn)− α) > K1−β(α)) = β,

F0 ≡ U(α, 1 + α), G0 ≡ U(0, 1)








Theorem

limn→∞

sup(F,G)∈H0

PF,G(√n(α0(Fn, Gn)− α) > K1−β(α))

= limn→∞

PF0,G0(√n(α0(Fn, Gn)− α) > K1−β(α)) = β,

F0 ≡ U(α, 1 + α), G0 ≡ U(0, 1)








Theorem

limn→∞

sup(F,G)∈H0

PF,G(√n(α0(Fn, Gn)− α) > K1−β(α))

= limn→∞

PF0,G0(√n(α0(Fn, Gn)− α) > K1−β(α)) = β,

F0 ≡ U(α, 1 + α), G0 ≡ U(0, 1)








Theorem

limn→∞

sup(F,G)∈H0

PF,G(√n(α0(Fn, Gn)− α) > K1−β(α))

= limn→∞

PF0,G0(√n(α0(Fn, Gn)− α) > K1−β(α)) = β,

F0 ≡ U(α, 1 + α), G0 ≡ U(0, 1)



Nonasymptotic bounds

Theorem

If α0(F,G) < α and K1−β(α) ≥ 0 then

PF,G(√n(α0(Fn, Gn)− α) > K1−β(α)) ≤ 2e−n(α−α0(F,G))2 .

If α0(F,G) > α then

PF,G(√n(α0(Fn, Gn)− α) ≤ K1−β(α)) ≤ e−2(

√n(α−α0(F,G))−K1−β(α))2 .

Test is u.e.c. for H ′0 : α0(F,G) ≤ α′ vs. H ′a : α0(F,G) > α′′ if α′ < α < α′′

Compute sample sizes to guarantee given power against fixed alternatives




Theorem

If α0(F,G) < α and K1−β(α) ≥ 0 then

PF,G(√n(α0(Fn, Gn)− α) > K1−β(α)) ≤ 2e−n(α−α0(F,G))2 .


PF,G(√n(α0(Fn, Gn)− α) ≤ K1−β(α)) ≤ e−2(

√n(α−α0(F,G))−K1−β(α))2 .

Test is u.e.c. for H ′0 : α0(F,G) ≤ α′ vs. H ′a : α0(F,G) > α′′ if α′ < α < α′′

Compute sample sizes to guarantee given power against fixed alternatives


Inference for approximate stochastic order Testing for essential stochastic order

Testing for essential stochastic order

ConsiderH0 : α0(F,G) ≥ α, vs. Ha : α0(F,G) < α

(equivalently, H0 : F 6≤st,α′ G if α′ < α vs. Ha : F ≤st,α′ G for some α′ < α)

Reject H0 if √n(α0(Fn, Gn)− α) < σαΦ−1(β),

where σ2α = 1−α2

2 , (assume β < 12 )

Theorem

limn→∞

sup(F,G)∈H0

PF,G(√n(α0(Fn, Gn)− α) < σαΦ−1(β))

= limn→∞

PF0,G0(√n(α0(Fn, Gn)− α) < σαΦ−1(β)) = β,

F0 ≡ 1−α2 U(0, 1+α

2 ) + 1+α2 U( 1+α

2 , 1 + α(1−α)2 ), G0 ≡ U(0, 1)








2 , (assume β < 12 )

Theorem

limn→∞

sup(F,G)∈H0


= limn→∞


F0 ≡ 1−α2 U(0, 1+α

2 ) + 1+α2 U( 1+α

2 , 1 + α(1−α)2 ), G0 ≡ U(0, 1)








2 , (assume β < 12 )

Theorem

limn→∞

sup(F,G)∈H0


= limn→∞


F0 ≡ 1−α2 U(0, 1+α

2 ) + 1+α2 U( 1+α

2 , 1 + α(1−α)2 ), G0 ≡ U(0, 1)








2 , (assume β < 12 )

Theorem

limn→∞

sup(F,G)∈H0


= limn→∞


F0 ≡ 1−α2 U(0, 1+α

2 ) + 1+α2 U( 1+α

2 , 1 + α(1−α)2 ), G0 ≡ U(0, 1)




Theorem


PF,G(√n(α0(Fn, Gn)− α) < σαΦ−1(β)) ≤ e−2n(α−α0(F,G))2

If α0(F,G) < α and n(α− α0(F,G))2 ≥ log 2 −σαΦ−1(β),

PF,G(√n(α0(F,G)− α) ≥ σαΦ−1(β)) ≤ 2e−(σαΦ−1(β)+

√n(α−α0(F,G)))2

Test is u.e.c. for H ′0 : α0(F,G) ≥ α′ vs. H ′a : α0(F,G) < α′′ if α′′ < α < α′

Compare to case H0 : F 6≤st G vs. Ha : F ≤st G

Try to assess F ≤st G up to α = 0.05 contamination; β = 0.05

Want to detect alternatives with α0(F,G) ≤ 0.01 with power 0.9

Take n = 8143




Theorem





√n(α−α0(F,G)))2





Take n = 8143




Theorem





√n(α−α0(F,G)))2





Take n = 8143




Theorem





√n(α−α0(F,G)))2





Take n = 8143




Theorem





√n(α−α0(F,G)))2





Take n = 8143




Theorem





√n(α−α0(F,G)))2





Take n = 8143


Inference for approximate stochastic order Confidence bounds

Confidence bounds

Instead of testing for/against contaminated stochastic order, report upper/lowerbounds for true contamination level, α0(F,G)

For β < 12 ,

α0(Fn, Gn)−√nσnΦ−1(β)

σ2n = mint:Fn(t)−Gn(t)=α0(Fn,Gn)) σ

2t ,

σ2t = t(1− t) + (t− α0(Fn, Gn))(1− t+ α0(Fn, Gn))

is an upper bound with asymptotic confidence level at least 1− β

Better use bias corrected α0(Fn, Gn)BOOT

α0(Fn, Gn)−√nK1−β(α0(Fn, Gn))

is a lower confidence bound for α0(F,G) with asymptotic confidence level 1− βQuantiles K1−β(α0(Fn, Gn)) numerically approximated


Inference for approximate stochastic order Paired sampling

Dependent data

Often X = pre-treatment, Y = post-treatment measurement

(X,Y ) ∼ H with marginals F and G

Has patient improved with treatment? ⇔ F ≤st G?

As before, H0 : F 6≤st G vs. Ha : F ≤st G not testable

Consider instead H0 : α0(F,G) ≥ α, vs. Ha : α0(F,G) < α

(X1, Y1), . . . , (Xn, Yn) i.i.d. random vectors with common joint d.f. H

H(x, y) = C(F (x), G(y)), C copula

α0(F,G) depends only on marginals; α0(Fn, Gn) consistent estimator

Distribution of α0(Fn, Gn) does depend on C:

√n (α0(Fn, Gn)− α0(F,G))→w sup

x:G(x)−F (x)=α0(F,G)

BC(G(x), F (x)),

BC centered Gaussian on [0, 1]2, covarianceKC((s, t), (s′, t′)) = s ∧ s′ + t ∧ t′ − (s− t)(s′ − t′)− C(t′, s)− C(t, s′)



Testing for approximate stochastic order, dependent data

Now

α(1− α) ≤ Var(BC(t, t− α))≤1− α2 − 2|t− 1+α2 |, t ∈ [α, 1]

equality for antimonotone coupling C(s, t) = (s+ t− 1)+

KC,β(F,G) β-quantile of rhs in limit distribution; for β ∈ (0, 12 )

KC,β(F,G) ≥ (1− α0(F,G)2)1/2Φ−1(β)

Similar to independent case, set σ2α = 1− α2; reject α0(F,G) ≥ α if

√n(α0(Fn, Gn)− α) < σαΦ−1(β)



Testing for approximate stochastic order, dependent data

As before, test uniformly asymptotically consistent:

limn→∞

supH∈H0

PH[√n(α0(Fn, Gn)− α) < σαΦ−1(β)

]= lim

n→∞PH[√n(α0(Fn, Gn)− α) < σπ0

Φ−1(β)]

= β,

H joint d.f. with marginals F ≡ 1−α2 U(0, 1+α

2 ) + 1+α2 U( 1+α

2 , 1 + α(1−α)2 ),

G ≡ U(0, 1) and copula C(s, t) = (s+ t− 1)+.



Nonasymptotic bounds: if α0(F,G) > α then

PH[√n(α0(Fn, Gn)− α < σαΦ−1(β)

]≤ e−n2 (α−α0(Fn,Gn)2 ,

If α0(F,G) < α and n(α− α0(F,G))2 ≥ 2 log 2− σαΦ−1(β),

PH[√n(α0(Fn, Gn)− α) ≥ σαΦ−1(β)

]≤ 2e

−n2 [(α−α0(F,G))+√

2σα√n

Φ−1(β)]2.

Independent vs. dependent setup

In independent setup rejection of H0 : α0(F,G) ≥ α when

√n(α0(Fn, Gn)− α) < σα√

2Φ−1(β).

Under dependence, the extra√

2 factor allows tocontrol uniformly type I error probability



Nonasymptotic bounds: if α0(F,G) > α then

PH[√n(α0(Fn, Gn)− α < σαΦ−1(β)

]≤ e−n2 (α−α0(Fn,Gn)2 ,

If α0(F,G) < α and n(α− α0(F,G))2 ≥ 2 log 2− σαΦ−1(β),

PH[√n(α0(Fn, Gn)− α) ≥ σαΦ−1(β)

]≤ 2e

−n2 [(α−α0(F,G))+√

2σα√n

Φ−1(β)]2.

Independent vs. dependent setup

In independent setup rejection of H0 : α0(F,G) ≥ α when

√n(α0(Fn, Gn)− α) < σα√

2Φ−1(β).

Under dependence, the extra√

2 factor allows tocontrol uniformly type I error probability


Implementation, simulation & data example Implementation issues

Testing for essential stochastic order: finite sample performance

Consider again H0 : α0(F,G) ≥ α, vs. Ha : α0(F,G) < α

Rejection of H0 if √n(α0(Fn, Gn)− α) < σαΦ−1(β),

σ2α = 1−α2

2 , asympt. of level β; type I-type II error probs. exponentially → 0

σα from worst case choice; possible improvement from estimated σn

A more important improvement: E(α0(Fn, Gn)) ≥ α0(F,G); estimate bias by

biasBOOT(α0(Fn, Gn)) := E∗(α0(F ∗n , G∗n))− α0(Fn, Gn)

Defineαn,BOOT := α0(Fn, Gn)− biasBOOT(α0(Fn, Gn)).

Reject H0 if √n(αn,BOOT − α) < σnΦ−1(β),

Test asympt. of level β






σ2α = 1−α2













σ2α = 1−α2













σ2α = 1−α2













σ2α = 1−α2













σ2α = 1−α2







Test asympt. of level βEustasio del Barrio Testing approximate stochastic order 29 / 37

Implementation, simulation & data example Simulation setup

Fα,a ≡ U(a, 1 + a); Fα,b ≡ 1−α2 U(0, 1+α

2 ) + 1+α2 U( 1+α

2 , 1 + α(1−α)2 )

G ≡ U(0, 1)

Fα,a Fα,b

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Fα,a, G worst choice in test against essential s.o.

Fα,b, G worst choice in test for essential s.o.


Implementation, simulation & data example Simulation results


Table : Observed rejection frequencies. H0 : α0(F,G) ≥ α vs. Ha : α0(F,G) < αG = U(0, 1), m = n; reject if

√n(α0(Fn, Gn)− α) < σαΦ−1(0.05)

α n F0.1,a F0.1,b F0.05,a F0.05,b F0.01,a F0.01,b F0

0.01

50 0.000 0.000 0.000 0.000 0.000 0.000 0.000100 0.000 0.000 0.000 0.000 0.000 0.000 0.000500 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1000 0.000 0.000 0.000 0.000 0.000 0.000 0.0005000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.05

50 0.000 0.000 0.000 0.000 0.000 0.000 0.000100 0.000 0.000 0.000 0.000 0.000 0.000 0.000500 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1000 0.000 0.000 0.000 0.000 0.016 0.071 0.1835000 0.000 0.000 0.000 0.008 0.924 0.939 0.995

0.1

50 0.000 0.000 0.000 0.000 0.000 0.000 0.000100 0.000 0.000 0.000 0.000 0.000 0.000 0.000500 0.000 0.001 0.010 0.152 0.532 0.589 0.698

1000 0.000 0.011 0.155 0.491 0.946 0.952 0.9795000 0.000 0.025 0.996 1.000 1.000 1.000 1.000

Nonasymptotic estimate n = 8143





√n(αn,BOOT − α) < σnΦ−1(0.05)

α n F0.1,a F0.1,b F0.05,a F0.05,b F0.01,a F0.01,b F0

0.01

50 0.000 0.003 0.001 0.012 0.019 0.034 0.038100 0.000 0.001 0.000 0.005 0.015 0.027 0.028500 0.000 0.000 0.000 0.001 0.016 0.028 0.049

1000 0.000 0.000 0.000 0.000 0.013 0.030 0.0635000 0.000 0.000 0.000 0.000 0.019 0.038 0.136

0.05

50 0.000 0.009 0.015 0.042 0.062 0.067 0.093100 0.000 0.006 0.008 0.038 0.065 0.098 0.106500 0.000 0.003 0.007 0.058 0.208 0.220 0.332

1000 0.000 0.001 0.009 0.039 0.415 0.426 0.5665000 0.000 0.000 0.003 0.057 0.978 0.987 1.000

0.1

50 0.003 0.030 0.054 0.089 0.137 0.138 0.134100 0.007 0.052 0.076 0.121 0.246 0.250 0.266500 0.007 0.040 0.337 0.387 0.801 0.830 0.876

1000 0.005 0.056 0.589 0.661 0.976 0.985 0.9975000 0.003 0.057 0.999 1.000 1.000 1.000 1.000




Table : Observed rejection frequencies. H0 : α0(F,G) ≤ α vs. Ha : α0(F,G) > αG = U(0, 1), m = n; reject if

√n(α0(Fn, Gn)− α) > K0.95(α)

α n F0 F0.01,b F0.01,a F0.05,b F0.05,a F0.1,b F0.1,a

0.01

50 0.045 0.039 0.052 0.060 0.111 0.159 0.302100 0.022 0.031 0.040 0.066 0.107 0.199 0.450500 0.021 0.026 0.045 0.210 0.477 0.951 1.000

1000 0.011 0.028 0.047 0.441 0.774 1.000 1.0005000 0.002 0.017 0.049 1.000 1.000 1.000 1.000

0.05

50 0.015 0.010 0.016 0.023 0.052 0.056 0.142100 0.004 0.007 0.009 0.014 0.031 0.069 0.186500 0.000 0.001 0.002 0.009 0.047 0.211 0.664

1000 0.000 0.000 0.000 0.001 0.060 0.606 0.9545000 0.000 0.000 0.000 0.001 0.040 1.000 1.000

0.1

50 0.001 0.002 0.005 0.004 0.007 0.009 0.027100 0.000 0.003 0.000 0.001 0.002 0.010 0.031500 0.000 0.000 0.000 0.000 0.001 0.002 0.056

1000 0.000 0.000 0.000 0.000 0.000 0.001 0.0355000 0.000 0.000 0.000 0.000 0.000 0.000 0.056



Testing for essential stochastic order, dependent case


√n(α0(Fn, Gn − α) < σnΦ−1(0.05)

α n H0.1,a H0.1,b H0.05,a H0.05,b H0.01,a H0.01,b H0

0.01

50 0.000 0.006 0.000 0.026 0.003 0.084 0.019100 0.000 0.001 0.000 0.006 0.003 0.042 0.009500 0.000 0.000 0.000 0.000 0.000 0.022 0.009

1000 0.000 0.000 0.000 0.000 0.000 0.028 0.0125000 0.000 0.000 0.000 0.000 0.000 0.043 0.049

0.05

50 0.000 0.006 0.000 0.031 0.008 0.090 0.025100 0.000 0.010 0.000 0.034 0.020 0.140 0.028500 0.000 0.001 0.000 0.061 0.094 0.210 0.167

1000 0.000 0.000 0.000 0.076 0.209 0.295 0.3525000 0.000 0.000 0.000 0.047 0.956 0.839 0.999

0.1

50 0.000 0.044 0.013 0.128 0.071 0.191 0.085100 0.001 0.053 0.028 0.130 0.116 0.240 0.153500 0.000 0.040 0.155 0.263 0.640 0.567 0.783

1000 0.000 0.044 0.357 0.397 0.956 0.869 0.9915000 0.000 0.048 0.999 0.973 1.000 1.000 1.000

Hπ,a independent marginals U(π, 1 + π), U(0, 1); H0 = H0,a

Hπ,b marginals F ≡ 1−π2 U(0, 1+π

2 ) + 1+π2 U( 1+π

2 , 1 + π(1−π)2 ), G ≡ U(0, 1),

copula C(s, t) = (s+ t− 1)+


Implementation, simulation & data example Case study

Data: National Health and Nutrition Examination Survey

Evolution with age of the heights of boys and girls

Sample sizes by age (boys, top)2 3 4 5 6 7 8 9 10 11 12 13 14 15

796 632 633 563 557 582 579 543 556 556 735 728 704 716776 563 620 567 542 564 572 579 536 587 733 757 764 665

130 140 150 160

0.0

0.2

0.4

0.6

0.8

1.0

π(Fn,Gm) = 0.128409 π(Gm,Fn) = 0.009342

heights

ED

F's

Gm Age 10 Boys Fn Age 10 Girls


Implementation, simulation & data example Case study

95%-Upper bounds by age for α0(F a, Ga) (top row) and α0(Ga, F a) (bottom)

2 3 4 5 6 7 8 9 10 11 12 13 140.00 0.00 0.00 0.01 0.00 0.04 0.02 0.04 0.16 0.16 0.15 0.05 0.010.17 0.09 0.14 0.18 0.15 0.11 0.13 0.08 0.03 0.01 0.07 0.28 0.47

5 10 15

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Upper 95% confidence bounds for the stochastic dominance levels

age

Statistical evidence that girls are taller than boys at 10-11


Conclusions

Conclusions

Trimmed stochastic order models capture adequately deviations from exactstochastic order

Provided valid inference models/methods

Valid testing procedures with controlled error probabilities

Nonasymptotic bounds; uniformly exponentially consistent tests

Good finite sample performance through bootstrap correction

Thanks for your attention!


Conclusions

Conclusions

Trimmed stochastic order models capture adequately deviations from exactstochastic order

Provided valid inference models/methods

Valid testing procedures with controlled error probabilities

Nonasymptotic bounds; uniformly exponentially consistent tests

Good finite sample performance through bootstrap correction

Thanks for your attention!


Documents

Eustasio del Barrio - Uni Ulm