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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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null hypothesis, hypothesis testing, test statistic, alternative hypothesis, type i error, sample mean, hypothesis test, significance level, type ii error, population mean, critical value, critical region, confidence interval, degrees of freedom, standard deviation

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posted: | 8/28/2010 |

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