Special Models

Cusp Estimation

Let the observed process be

dXt = a |Xt − ϑ|κ dt + h (Xt) dt + dWt, X0, 0 ≤ t ≤ T,

where κ ∈ (0, 1/2).

The MLE ϑT and BE (for quadratic loss function) ϑT are defined bythe usual relations

L(ϑT , XT

)= sup

θ∈ΘL

(θ,XT

),

ϑT =∫

Θ

θ p(θ|XT

)dθ, p

(ϑ|XT

)=

p(θ) L(θ, XT

)∫Θ

p(v) L (v, XT ) dv.

1

Introduce the process

Z (u) = exp{

WH (u)− 12|u|2H

}, u ∈ IR,

where WH (·) is fBm, H = κ + 1/2 and random variables

Z (u) = supu∈IR

Z (u) , u =

∫IR

u Z(u) du

∫IR

Z(v) dv.

and the constant

Γ2ϑ =

2a2

G (ϑ)Γ(1 + κ

)Γ(

12 − κ

)

22κ√

π(2κ + 1)[1− cos(πκ)

],

2

Put γϑ = Γ1/Hϑ . The lower bound for all estimators ϑT

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

T 1/HEϑ

(ϑT − ϑ

)2 ≥ Eu2

γ2ϑ0

,

The MLE and BE are consistent, have the following limits indistribution

Lϑ

{T 1/2H

(ϑT − ϑ

)}=⇒ Lϑ

{u

γϑ

},

Lϑ

{T 1/2H

(ϑT − ϑ

)}=⇒ Lϑ

{u

γϑ

},

and we have the convergence of moments too. Moreover, the BE areasymptotically efficient.

3

Introduce the normalized LR

ZT (u) =L

(ϑ + T−1/2Hu,XT

)

L (ϑ,XT ),

where u ∈ UT =(T 1/2H (α− ϑ) , T 1/2H (β − ϑ)

)and show that

• (ZT (u1) , . . . , ZT (uk)) ⇒ (Zϑ (u1) , . . . , Zϑ (uk)),

• Eϑ

∣∣∣ZT (u2)1/2 − ZT (u1)

1/2∣∣∣2

≤ C |u2 − u1|2H ,

• Pϑ

{ZT (u) > e−κ|u|2H

}≤ CN

|u|N

Here Zϑ (u) = Z (γ (ϑ)u).

4

The limit distribution of the MLE we obtained by the “usual” way.

Pϑ0

{T 1/2H

(ϑT − ϑ0

)< x

}=

= P

{sup

T 1/2H(θ−ϑ0)<x

L(ϑ,XT

)> sup

T 1/2H(θ−ϑ0)≥x

L(ϑ,XT

)}

= P

{sup

T 1/2H(θ−ϑ0)<x

L(ϑ, XT

)

L (ϑ0, XT )> sup

T 1/2H(θ−ϑ0)≥x

L(ϑ,XT

)

L (ϑ0, XT )

}

= P{

supu<x

ZT (u) > supu≥x

ZT (u)}→ P

{supu<x

Zϑ (u) > supu≥x

Zϑ (u)}

= P(

u

γ (ϑ0)< x

), i.e. T 1/2H

(ϑT − ϑ0

)=⇒ u

γ (ϑ0).

where we put ϑ = ϑ0 + T−1/2Hu.

5

ZT (u) ⇒ Zϑ (u)? Let us put δt (u) = S(ϑ + T−1/2Hu, Xt

)

−S (ϑ, Xt) . Then

ln ZT (u) =∫ T

0

δt (u) dWt − 12

∫ T

0

δt (u)2 dt.

For the ordinary integral (ϕT = T−1/2H)

JT =∫ T

0

δt (u)2 dt = a2

∫ T

0

(|Xt − ϑ− ϕT u|κ − |Xt − ϑ|κ)2 dt+o (1) .

Further∫ T

0

(|Xt − ϑ− ϕT u|κ − |Xt − ϑ|κ)2 dt

= T

∫ ∞

−∞(|x− ϑ− ϕT u|κ − |x− ϑ|κ)2 f◦T (x) dx

= T

∫ ∞

−∞(|x− ϑ− ϕT u|κ − |x− ϑ|κ)2 f (ϑ, x) dx + o (1)

6

We change the variable x = ϑ + sϕT u

a2T

∫ ∞

−∞(|x− ϑ− ϕT u|κ − |x− ϑ|κ)2 f (ϑ, x) dx

= |u|2κ+1a2ϕ2κ+1

T T f (ϑ, ϑ)∫ ∞

−∞(|s− 1|κ − |s|κ)2 ds + o (1)

−→ Γ2ϑ |u|2κ+1

, i.e. JT → Γ2ϑ |u|2H

,

because ϕ2κ+1T T = 1. By the CLT

∫ T

0

δt (u) dWt =⇒ N(0, Γ2

ϑ |u|2H)∼ WH (γ (ϑ) u) .

7

Delay Estimation

The observed process is

dXt = −γ Xt−ϑ dt + dWt, X0 (s) , −τ ≤ s ≤ 0,

where γ > 0, 0 < ϑ < π/2γ. Note that

Xt−ϑ = X0 + γ

∫ t−ϑ

0

Xs−ϑ ds + Wt−ϑ

Therefore the trend is as smooth w.r.t. ϑ as Wiener process w.r.t.time (nondiferentiable). The Gaussian process Xt has ergodicproperties.

8

Introduce the stochastic process

Z (u) = exp{

W (u)− |u|2

}, u ∈ IR

and the random variables

u = arg supu∈IR

Z (u) , u =

∫IR

u Z (u) du∫IR

Z (u) du.

Note that

Eu2 = 26, Eu2 = σ20 , σ2

0 = 16ζ (3) ' 19.3

9

The lower bound for all estimators ϑT is

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

T 2Eϑ

(ϑT − ϑ

)2 ≥ Eu2

γ2,

The MLE and BE are consistent, have the following limits indistribution

T(ϑT − ϑ

)=⇒ u

γ, T

(ϑT − ϑ

)=⇒ u

γ,

Moreover, the BE are asymptotically efficient. Introduce

ZT (u) =L

(ϑ + T−1u,XT

)

L (ϑ,XT )=⇒ Z (γu) ?

10

For the MLE we have once more :

Pϑ0

{T

(ϑT − ϑ0

)< x

}=

= P

{sup

T (θ−ϑ0)<x

L(ϑ, XT

)> sup

T (θ−ϑ0)≥x

L(ϑ,XT

)}

= P

{sup

T (θ−ϑ0)<x

L(ϑ,XT

)

L (ϑ0, XT )> sup

T (θ−ϑ0)≥x

L(ϑ,XT

)

L (ϑ0, XT )

}

= P{

supu<x

ZT (u) > supu≥x

ZT (u)}→ P

{supu<x

Z (γu) > supu≥x

Z (γu)}

= P(

u

γ< x

), i.e. T

(ϑT − ϑ0

)=⇒ u

γ.

where we put ϑ = ϑ0 + T−1u.

11

Why ZT (u) ⇒ Z (γu)? Put δt (u) = Xt−ϑ−u/T −Xt−ϑ. Then

δt(u) = −γ

∫ t−ϑ− uT

t−ϑ

Xs−ϑ ds +(Wt−ϑ− u

T−Wt−ϑ

)

=γ u

TXt−2ϑ (1 + o(1)) +

1√T

Wt(u).

Further ∫ T

0

δt (u)2 dt ∼ 1T

∫ T

0

Wt (u)2 dt −→ |u| .

Hence, by CLT∫ T

0

(Xt−ϑ−u/T −Xt−ϑ

)dWt =⇒ N (0, |u|) ∼ W (u)

12

Change-point estimation

Suppose that the observed diffusion process

dXt = S (ϑ,Xt) dt + σ (Xt) dWt, X0, 0 ≤ t ≤ T,

where the trend coefficient S (θ, x) is discontinuous function along thetwo curves

{x

(1)∗ (θ) , θ ∈ [α, β]

},

{x

(2)∗ (θ) , θ ∈ [α, β]

}.

Example. Let ϑ ∈ (α, β), α > 1

dXt = −Xt sgn(X2

t − 2 ϑXt + 1)

dt + σ dWt.

13

Γ2ϑ =

2∑

i=1

∣∣∣x(i)∗ (ϑ)

∣∣∣S+ (ϑ)− S− (ϑ)

σ(x

(i)∗ (ϑ)

)

2

fϑ,

where S± (ϑ) = S(ϑ, x

(i)∗ (ϑ)±

), fϑ = f

(ϑ, x

(i)∗ (ϑ)

),

x(i)∗ (ϑ) =

dx(i)∗

dϑ(ϑ) , ϑ ∈ [α, β]

and introduce the process

Z (u) = exp{

W (u)− |u|2

}, u ∈ IR,

where W (u) , u ∈ IR is two-sided Wiener process. The r.v.’s u and u

are defined with the help of Z (·).

14

Low bound

limδ→0

limT→∞

sup|ϑ−ϑ0|<δ

T 2Eϑ

(ϑT − ϑ

)2 ≥ Eu2

Γ2ϑ0

,

Theorem 1 The MLE ϑT and the BE ϑT are uniformly on

compacts K ⊂ Θ consistent, their limit distributions are

T(ϑT − ϑ

)=⇒ u

Γϑ, T

(ϑT − ϑ

)=⇒ u

Γϑ,

the moments converge and the estimator ϑT is asymptotically

efficient.

The normalized LR is

ZT (u) =L

(ϑ + T−1u,XT

)

L (ϑ,XT )=⇒ Z (Γϑu) ?

15

As usual to show that T(ϑT − ϑ0

)=⇒ u Γ−1

ϑ we write

Pϑ0

{T

(ϑT − ϑ0

)< x

}= P

{supu<x

ZT (u) > supu≥x

ZT (u)}

−→ P{

supu<x

Z (Γϑu) > supu≥x

Z (Γϑu)}

= P(

u

Γϑ< x

).

Examples.

dXt = − sgn (Xt − ϑ) dt + dWt, 0 ≤ t ≤ T,

dXt = −Xt

(a + b 1I{ϑ<Xt<c+ϑ}

)dt + dWt,

dXt = −Xt

(a + b 1I{ϑ1<Xt<ϑ2}

)dt + dWt.

16

Simple switching with f (ϑ, x) = e−2|x−ϑ|

dXt = − sgn (Xt − ϑ) dt + dWt

ln ZT (u) = −∫ T

0

1I{ϑ≤Xt≤ϑ+u/T} dWt − 12

∫ T

0

1I{ϑ≤Xt≤ϑ+u/T} dt

We have (u > 0)∫ T

0

1I{ϑ≤Xt≤ϑ+u/T} dt = T

∫ ∞

−∞1I{ϑ≤x≤ϑ+u/T}f◦T (x) dx

= T

∫ ϑ+u/T

ϑ

f◦T (x) dx → u f (ϑ, ϑ) = u

and ∫ T

0

1I{ϑ≤Xt≤ϑ+u/T} dWt =⇒ N (0, |u|) ∼ W (u)

17

Nonparametric estimation

Diffusion process

dXt = S(Xt) dt + σ(Xt) dWt, X0, t ≥ 0.

The functions S(·) and σ(·) > 0 are continuous and S (·) is unknown.

Conditions:∫ x

0

exp{−2

∫ y

0

S(v)σ2(v)

dv

}dy → ±∞, x → ±∞

and

G (S) =∫ ∞

−∞

1σ(x)2

exp{

2∫ x

0

S(v)σ2(v)

dv

}dx < ∞

The process Xt is ergodic.

18

The invariant density function

fS(x) =1

G(S)σ(x)2exp

{2

∫ x

0

S(v)σ2(v)

dv

}, FS (x) =

∫ x

−∞fS(y) dy.

Main Problems:

• Distribution function estimation FS (x)

• Invariant density estimation fS (x)

• Trend coefficient estimation S (x)

Fix some S∗(·) and introduce the set

Vδ = {S(·) : supx∈IR

|S(x)− S∗(x)| ≤ δ}

19

Distribution function estimation.Empirical distribution function is consistent

FT (x) =1T

∫ T

0

1I{Xt<x} dt −→ FS (x)

and asymptotically normal√

T(F

T(x)− F (x)

)=⇒ N

(0, IF (S, x)−1

)

where the Fisher information IF (S, x) = dF (S, x)−2 ,

dF (S, x)2 = 4 ES

(F (ξ ∧ x)− F (ξ)F (x)

σ (ξ) f (ξ)

)2

< ∞

20

It follows from the representation

√T

(F

T(x)− F (x)

)=

2√T

∫ XT

X0

F (v ∧ x)− F (v)F (x)σ (v)2 f (v)

dv

− 2√T

∫ T

0

F (Xt ∧ x)− F (Xt)F (x)σ (Xt) f (Xt)

dWt.

Is it possible to estimate better? Lower bound:

limδ→0

limT→∞

supS(·)∈Vδ

T ES

(FT (x)− F

S(x)

)2

≥ IF (S∗, x)−1.

21

The first step is

supS(·)∈Vδ

T ES

(FT (x)− F

S(x)

)2

≥ sup|ϑ−ϑ0|<δ

Eϑ

(FT (x)− F

ϑ(x)

)2

where the parametric family is

dXt = [S∗(Xt) + (ϑ− ϑ0)ψ(Xt) σ(Xt)2] dt + σ(Xt) dWt,

The DF can be expanded by ϑ− ϑ∗

Fϑ(x) = F (x) + 2 (ϑ− ϑ0) E

{[1I{ξ<x} − F (x)]

∫ ξ

0

ψ (y) dy

}+ o(1),

Introduce the class

K =

{ψ(·) : E

{[1I{ξ<x} − F (x)]

∫ ξ

0

ψ (y) dy

}=

12

}.

Then with ϑ0 = F (x) obtain Fϑ(x) = ϑ + o (1)

22

sup|ϑ−ϑ0|<δ

TEϑ

(FT (x)− F

ϑ(x)

)2

= sup|ϑ−ϑ0|<δ

TEϑ

(ϑT − ϑ

)2 ≥ I−1ψ

where Iψ = Eψ (ξ)2 σ (ξ)2. We have to find the least favorable Iψ∗ :

infψ(·)∈K

Iψ = Iψ∗ .

We obtain the inequality

Iψ ≥{

4E(

[1− F (ξ ∨ x)]F (ξ ∧ x)σ(ξ) f(ξ)

)2}−1

= IF (S∗, x)

Hence

Iψ∗ =

{4E

([1− F (ξ ∨ x)]F (ξ ∧ x)

σ(ξ) f(ξ)

)2}−1

= IF (S∗, x)

and EDF is asimptotically efficient.

23