Marked empirical processes for non-stationary time series

الأصل · EN

Consider a first-order autoregressive process Xᵢ=βXᵢ₋₁+εᵢ, where εᵢ=G(ηᵢ,ηᵢ₋₁,) and ηᵢ,i are i.i.d. random variables. Motivated by two important issues for the inference of this model, namely, the quantile inference for H₀: β=1, and the goodness-of-fit for the unit root model, the notion of the marked empirical process αₙ(x)=1/n∑ᵢ₌₁ⁿg(Xᵢ/aₙ)I(εᵢ≤ x),x is investigated in this paper. Herein, g(·) is a continuous function on R and {aₙ} is a sequence of self-normalizing constants. As the innovation {εᵢ} is usually not observable, the residual marked empirical process αₙ(x)=1/n∑ᵢ₌₁ⁿg(Xᵢ/aₙ)I(εᵢłeq x),x, is considered instead, where εᵢ=Xᵢ-βXᵢ₋₁ and β is a consistent estimate of β. In particular, via the martingale decomposition of stationary process and the stochastic integral result of Jakubowski (Ann. Probab. 24 (1996) 2141-2153), the limit distributions of αₙ(x) and αₙ(x) are established when {εᵢ} is a short-memory process. Furthermore, by virtue of the results of Wu (Bernoulli 95 (2003) 809-831) and Ho and Hsing (Ann. Statist. 24 (1996) 992-1024) of empirical process and the integral result of Mikosch and Norvaiša (Bernoulli 6 (2000) 401-434) and Young (Acta Math. 67 (1936) 251-282), the limit distributions of αₙ(x) and αₙ(x) are also derived when {εᵢ} is a long-memory process.