# On Hypotheses Testing for

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```					On Hypotheses Testing for
Ergodic Diﬀusion Process

Yu.A. Kutoyants
Laboratoire de Statistique & Processus,
e
Universit´ du Maine,
e
72085 Le Mans, C´dex 9,
FRANCE

20 December, 2002

1
Model

We observe a trajectory X T = {Xt, 0 ≤ t ≤ T }
of diﬀusion process

dXt = S(Xt) dt + σ (Xt) dWt,           X0 ,    t ≥ 0.
The functions S(·) and σ(·) > 0 are localy
bounded and such that the weak solution of
the equation exists, is unique and the condi-
tions
x             z S(y)
exp   −2        2
dy     dz → ±∞,        x → ±∞
0             0 σ(y)
and
∞    1               z S(y)
G (S ) =                 exp   2          dy   dz < ∞
−∞ σ (z )2           0 σ(y) 2

are fulﬁlled.

2
The process {Xt, t ≥ 0} is recurrent positive
with the invariant distribution
x        1                     z S(y)
F (x) =                        exp    2         dy   dz.
−∞ G (S ) σ (z )2             0 σ(y)2

Remind the empirical distribution function (EDF)
1 T
ˆ
F T ( x) =     χ      dt
T 0 {Xt<x}
and local time estimator (LTE) of the invariant
density
1        T
◦
f T (x) =                    sgn (x − Xt) dXt +
T σ (x)2 0
|XT − x| − |X0 − x|
+
T σ (x)2

3
The behavior of the likelihood
T S (Xt)
ln L(X T ) = f (X0)              2
dXt−
0 σ (Xt)
1     T S (Xt) 2
−                      dt
2 0      σ (Xt)
is deﬁned by the properties of the integral
2
1 T    S (Xt)
dt =
T 0    σ (Xt)
∞           2
S ( x)
=                     f ◦ (x) dx
−∞ σ (x)             T
∞           2
S (x )
=                      ˆ
dF T ( x ) .
−∞ σ (x)

4
Moreover, we can replace the stochastic inte-
gral by ordinary one

ln L(X T ) = − ln G (S ) + H(X0, S) + H(XT , S)
                               
2
T       S(x)           S(x)
−                     +           σ(x)2 f ◦ (x) dx
2       σ(x)           σ(x)2            T

= − ln G (S ) + H(X0, S) + H(XT , S)
                               
2
T       S(x)           S(x)
−                     +           σ(x)2 dFT (x)
ˆ
2       σ(x)           σ(x)2
where
x S(v)
H(x) =         dv.
0 σ(v)2
Hence
X0, XT , f ◦ (x), x ∈ R
T
and
ˆ
X0, XT , FT (x), x ∈ R
are suﬃcient statistics.

5
One-sided parametric alternative

A. Contiguous parametric alternatives (Regu-
lar case)
dXt = S (ϑ, Xt) dt + σ (Xt) dWt,         X0 .
Hypotheses testing problem
H0 :          ϑ = ϑ0 ,
H1 :          ϑ > ϑ0,
u
or if we put ϑ = ϑ0 + √ , then
T
H0 :            u = 0,
H1 :            u > 0,

P. Regularity condition: The process is er-
godic under H0 and
˙
S(ϑ0 + h, ·) − S(ϑ0, ·) − h S (ϑ0, ·)
= o (h )
σ (·)
where
g ( · ) 2 = E ϑ0 g (ξ ) 2 .

6
We say that a test φ∗ (X ) ∈ Kε is locally asymp-
T
totically uniformly most powerful in the class
Kε if for any other test φT (X ) ∈ Kε we have:
for any K > 0

lim   inf βT u, φ∗ − βT (u, φT ) ≥ 0.
T
T →∞ 0≤u≤K
Fisher information
∞ ˙            2
S ( ϑ0 , x )
I ( ϑ0 ) =                   f (ϑ0, x) dx,
−∞   σ ( x)
and for t ∈ (T, T + 1] we put

σ(Xt) = 1,    ˙
S ( ϑ0 , Xt ) =   T I ( ϑ0 ) ,
dXt − S (ϑ0, Xt) dt = dWt   ˜

7
Then we introduce the stopping time
                                               
˙               2
   1 τ       S ( ϑ0 , Xt )                     
τT = inf τ :                               dt ≥ I (ϑ0)
    T 0         σ (Xt)                          

∆τ ϑ0, X T =
1        ˙
τT S (ϑ0 , Xt)
=√                  2 [
dXt − S (ϑ0, Xt) dt] .
T 0      σ (Xt)
Theorem. Let the condition P be fulﬁlled and
I (ϑ0) > 0 then the test

φ∗ (X ) = χ
T             ∆τ ϑ0 ,X T ≥zε I(ϑ0 )1/2

is locally asymptotically uniformly most pow-
erful in the class Kε and

βT u, φ∗ = P ζ > zε − u I (ϑ0)1/2 + o (1) ,
T
where ζ ∼ N (0, 1).

8
B. Contiguous parametric alternatives (Non-
regular case)

We observe the ergodic diﬀusion process

dXt = S (ϑ, Xt) dt+σ (Xt) dWt,     X0 ,   0≤t≤T
and we would like to test hypotheses

H0 :      ϑ = ϑ0 ,
H1 :      ϑ > ϑ0
in the situations when the regularity condition
P is not fulﬁlled. Particularly we study three
models. The ﬁrst one corresponds to change-
point testing, the second to delay testing and
the third to cusp testing.

9
Change-point testing

Example: dXt = − sgn (Xt − ϑ) dt + dWt,

Suppose the trend coeﬃcient S (ϑ, x) is a dis-
continuous along the curve

{x∗ (θ) , θ ∈ [α, β ]}
function, i.e.

r (ϑ) = S (ϑ, x∗ (ϑ) +) − S (ϑ, x∗ (ϑ) −) = 0
Let us put
2
r (ϑ )
Γ ϑ = | x ∗ ( ϑ) |
˙                           f (ϑ, x∗ (θ))
σ (x∗ (ϑ))

10
We test the hypotheses

H0 :        ϑ = ϑ0 ,
u
H1 :        ϑ = ϑ0 +     ,   u > 0,
T

Introduce three independent random variables:
u
ζ(u) ∼ N 2 Γ2 , uΓ2 , η ∼ Exp (1) and η(u)
ϑ
0     ϑ 0
with distribution function v = uΓ2
ϑ   0
√
x      v
Fη(u) (x) = Φ √ +        +
v    2
√
−x 1 − Φ  x       v
+e           √ −            ,   x ≥ 0,
v    2

11
We study the likelihood ratio test (LRT) based
on the statistic

L X T = sup L ϑ, ϑ0, X T
ˆ
ϑ>ϑ0

Proposition The LRT

φT X T = χ
ˆ
LT X T > 1
ˆ
ε

belongs to Kε , is consistent and for any local
alternative ϑ = ϑ0 + T −1u, u > 0, the power
function
1
ˆ
βT u, φ = P ζ(u) + max [η(u), η ] > ln    + o (1) .
ε

12
Windows

Suppose that we know that under alternative
ϑ ∈ (ϑ0, β], then we can introduce the window

A = [x∗ (ϑ0) , x∗ (β )]
and modify the likelihood ratio as follows
T h (ϑ, Xt)2
ln L∗
T    ϑ, ϑ0, X T   =−               χ
2 {Xt∈A}
dt
0 2σ (Xt)
T h (ϑ, Xt)
+              χ
2 {Xt∈A} [
dXt − S (ϑ0, Xt) dt]
0 σ (Xt)
where h (ϑ, x) = S (ϑ, x) − S (ϑ0, x). Then we
put
L∗ X T = sup L∗ ϑ, ϑ0, X T
ˆT            T
ϑ>ϑ0

13
Proposition. The LRT

φ∗ X T = χ
ˆ
T            L∗ X T > 1
ˆ
T      ε

belongs to Kε , is consistent and for any local
alternative ϑ = ϑ0 + T −1u, u > 0, the power
function
1
ˆ∗ = P ζ(u) + max [η(u), η ] > ln
βT u, φ                                    + o (1) .
ε

This means that having observations in A only
we have no loss of information.

14
It is possible to construct the test with the
same asymptotic properties by the observa-
tions in the windows of decreasing to zero width
(as in estimation theory). Let

dXt = − sgn (Xt − ϑ) dt + dWt.
√
Using observations Xt, 0 ≤ t ≤ T we con-
struct a consistent estimator, say,
√
1    T
¯√ = √
ϑ T          Xt dt −→ ϑ
T 0
and then put

AT = ϑ√T − T −1/8, ϑ√T + T −1/8 .
¯             ¯

The LRT based on the statistic L∗ X T where
ˆ
T
we replace A by AT has the same asymptotic
properties as the LRT φT X T , i.e;, it belongs
ˆ
to Kε and is consistent.

15
Delay testing

The observed process is

dXt = −γ Xt−ϑ dt + dWt,        ˆ
X0,       0≤t≤T
and we test the hypotheses

H0 :       ϑ = ϑ0 ,
u
H1 :       ϑ = ϑ0 +      , u > 0,
T
where γ > 0 is known, the parameter ϑ ∈
[ϑ0, β ) , 0 ≤ ϑ0 < β < π/2γ.

Proposition. The LRT

φT X T = χ
ˆ
LT X T > 1
ˆ
ε

belongs to Kε , is consistent and for any local
alternative ϑ = ϑ0 + T −1u, u > 0, the power
function
1
ˆ
βT u, φ = P ζ(u) + max [η(u), η ] > ln    + o (1) .
ε
where we have to put Γϑ0 = γ.
16
Cusp testing

Let the observed process be

dXt = − sgn (Xt − ϑ) |Xt − ϑ|κ dt + dWt,
where κ ∈ (0, 1/2). The contiguous alterna-
tives are given by the following hypotheses test-
ing problem

H 0 : ϑ = ϑ0 ,
u
H 1 : ϑ = ϑ0 +       1 ,            u>0
T 2κ+1
Let us denote yε the (1 − ε)-quantile (P {η < yε} =
1 − ε) of the random variable
                         
2κ+1
|u|
η = sup W H (u) −                  ,
u≥0                      2

where W H (·) is a fractional Brownian motion
with Hurst constant H = κ + 1/2.

17
Let us introduce the process (v ≥ 0)

H v − |v − u|2κ+1 − |˜|2κ+1
˜         u
˜
Y (v, u) = W ( )
2
κ+1/2
˜
where u = u Γϑ        u and
0

1   Γ 1+κ Γ 1−κ
Γ2 =
ϑ              √   2      1 − cos(πκ) ,
G (ϑ) 22κ−1 π(2κ + 1)

Proposition. The LRT

φT X T = χ
ˆ                              ∈ Kε
ˆ
ln L X T ≥yε
T

is consistent and its power function for the lo-
cal alternatives ϑ = ϑ0 + T −1/(2κ+1)u is

ˆ
βT u, φT = P sup Y (v, u) ≥ yε + o (1) .
˜
v≥0

18
As in the change-point testing we need not to
have all observations and can use only that in
the window
A = [ ϑ0 , β ] .

The LRT based on the modiﬁed LR statistic
L∗ X T will have the same asymptotic prop-
ˆ
T
erties as if A = R.

19
Goodness-of-ﬁt tests

Let the observed process be

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

Fix some function S0 (·) and consider the prob-
lem

H0 :        S (·) = S0(·)
H1 :        S (·) = S0(·)

Then it is easy to see that if

sup |S (x) − S0 (x)| > 0,
x
then
sup fS (x) − fS (x) > 0
x             0

and
sup FS (x) − FS (x) > 0.
x             0

20
Therefore, it is possible to construct the goodness-
of-ﬁt tests based on the statistics
√
T = sup T f ◦ (x) − f (x)
δT X                   T       S
x                    0

and
√
γT   XT   = sup           ˆ
T FT (x) − FS (x) .
x                    0

Condition
A0. The function S (·) satisﬁes the condition

lim   sgn (x) S (x) < 0
|x|→∞

21
Let us introduce a Gaussian process η (x) with
E η (x) = 0 and the covariance function

RS (x, y ) = 4 fS (x) fS (y )
0               0     0
                                      
 χ{ξ>x} − FS 0 (ξ ) χ{ξ>y} − FS 0 (ξ ) 
ES                          2
.
0                 f S (ξ )
                                       
0

Denote by yε the value deﬁned by the equation

P sup |η (x)| > yε = ε
x
Proposition. Let the condition A0 be fulﬁlled,
then the test

φ∗ X T = χ
T            δ (X T )>yε
T

belongs to Kε and is consistent.

22

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