(14), which prove the result. Remark 6. Similarly to the previous computation of the joint distribution of ( 1 , 2 , R(ˆ z)), one can show that under the alternative H1 P(( 1 , 2 , R(z)) 2 B) = (cst) Z T E(| det( e ⇤X(z) + R(z))|10< z 2 <X(z)1(X(z), z 2 ,R(z))2B )dz = (cst) Z B⇥T det( e ⇤l1 + r)10<l2<l1 1(l1 µ(z)) µ ,z(d(l2 , r)) ✓q µ0(z)T e ⇤ 1µ0(z) ◆ dl1dz where µ(·) = E(X(·)) and µ0 denotes the gradient of µ. We can now state our result when the variance is known. Theorem 6. Set 8r 2 S +, 8` > 0, Gr(`) := Z +1 ` det( e ⇤u + r) (u 1)du , where e ⇤ is the Hessian of the correlation function ⇢ of X at the origin. Under Assumptions (Anorm ) and (Adegen ), the test statistic SRice := GR(b z) ( 1) GR(b z) ( 2) ⇠ U([0, 1]) under the null H0 . Proof. Using Proposition 5, we know that the density of 1 at `1 and conditional to ( 2 , R(b z)) = (`2 , r) is equal to The independent part of the hessian e ⇤ = ⇢00(0) where and ⇢ covariance function of X X00(z) = e ⇤X(z) | {z } E ⇥ X00(z) (X(z),X0(z)) ⇤ +R(z) ˆ z s.t. X(ˆ z) = 1 where Yohann DE CASTRO
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