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