By Walter Philipp

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**Additional resources for Almost sure invariance principles for partial sums of weakly dependent random variables**

**Example text**

It follows from the deﬁnition of Sloc that Sloc = A⊥ loc , and by the assumptions on Ψ, we have Ψ = Sloc ⊕ Aloc . However the decomposition is associated with θ ∈ Θ. A more detailed discussion of the space Ψ can be found in Small and McLeish (1991). In general there is no guarantee of the existence of a complete locally Esuﬃcient subspace. However, if it does exist it is a closed linear subspace of Ψ. Completeness is dispensed with in the following deﬁnition. 4 A subset L of Ψ is locally E-suﬃcient if for φ ∈ Ψ, for all ψ ∈ L and θ ∈ Θ Eθ (φ (θ) ψ(θ)) = 0 implies that φ is locally E-ancillary.

N G∗ = σ −2 (xi − θ) i=1 where σ 2 = var(x1 ) = Ee21 . 729477 of (E G˙ ∗ )−2 E(G∗ )2 = 1 α−1/2 Γ(3/4) . 1708 times the corresponding one derived from U . 4. WEDDERBURN’S QUASI-LIKELIHOOD 25 third moments being assumed zero, in contrast to G∗ derived from n H= ai (xi − θ), ai ’s constant . i=1 From the above discussion, we see the ﬂexibility of being able to choose an appropriate space of estimating functions in the optimal estimating functions approach. 3 Generalized Estimating Equations Closely associated with Wedderburn’s quasi-likelihood is the moment based generalized estimating equation (GEE) method developed by Liang and Zeger (1986), and Prentice (1988).

Exercise 7. Let Y1 , Y2 , . . d. such that Yj = |Xj |, where Xj has a N (µ, 1) distribution. The situation arises, for example, if Xj is the diﬀerence between a matched pair of random variables whose control and treatment labels are lost. The sign of the parameter µ is unidentiﬁable so we may work with θ = |µ|, with parameter space Θ = [0, ∞). Show that the score function is n [Yj + tanh (θYj ) − θ] j=1 and observe that for θ > 0 the problem is a regular one and √ with d n (θˆn − θ) −→ N (0, i(θ)) 1 2 2 i(θ) = 1 − √ e− 2 θ 2π ∞ 1 2 y 2 e− 2 y sech (θy) dy.