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Keeps the experiences in quantum chance and similar issues.
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Also when n = 50 the power of the tests is higher than that with n = 20. Furthermore, the power performance of the tests are less affected by the dimension of variable. 4 Appendix: Proofs of Theorems In this section, we only present the proofs of theorems about elliptical symmetry testing. Similar arguments can be applied to prove the theorems for reflection symmetry testing; we omit them here. 4). When the shape matrix ˆ applying the triangle identity, is replaced by the sample covariance matrix Σ, we have √ ˆ nPn (sin(taτ A(X − µ))) √ τ = nPn (sin(ta A(X − µ))) cos(taτ (Aˆ − A)(X − µ))) √ + n(Pn (cos(taτ A(X − µ))) sin(taτ (Aˆ − A)(X − µ))) =: In1 (t, a) + In2 (t, a).
Use the above algorithm except replacing A by its estimator Aˆ = Σ When µ is unknown, the situation is not so simple. This is different from that of Chapter 2 because of the use of a different test statistic. In order to ensure the equivalence between the conditional empirical process below and its unconditional counterpart, we shall use the following fact to construct conditional empirical process. It can be derived by the triangle identity and ¯ that uniformly on t ∈ I and a ∈ S d Pn X = X √ ˆ ¯ − X))) nPn (sin(taτ A(X √ ˆ ˆ n (X − µ)) = n Pn (sin(taτ A(X − µ))) cos(taτ AP √ ˆ ˆ n (X − µ)) − µ))) sin(taτ AP − n Pn (cos(taτ A(X √ = nPn (sin(taτ A(X − µ)) √ − n Pn (cos(taτ A(X − µ))) sin(taτ APn (X − µ)) + op (1).
1) in the Skorohod space Dd [−∞, ∞], where B is a vector of Gaussian processes (B1 , · · · Bd )τ with the covariance function cov(Bi (t), Bi (s)) = Fε (min(t, s)) − Fε (t)Fε (s), Fε (t) and fε (t) are respectively the distribution and density functions of ε, and N is a random vector with a normal distribution N (0, σ 2 V ) with V = E(V1 V1τ ). 2) (i) where (Σ −1/2 (X − E(X)))i is the i-th component of Σ −1/2 (X − E(X)). The process convergence implies that Tn converges in distribution to T = supa aτ (I(t)I(t)τ )dFε (t) a.