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Weak convergence of the residual empirical process in explosive autoregression. (English) Zbl 0695.60042

The following explosive autoregressive model of order 1 is considered: one observes r.v’s \(\{X_ i\}\) satisfying \[ X_ 0=0,\quad X_ i=\rho X_{i-1}+\epsilon_ i,\quad i\geq 1, \] where \(| \rho | >1\) and \(\{\epsilon_ i\}\) are i.i.d. r.v’s with distribution F. Let \({\hat \rho}\) be an estimator of \(\rho\) based on \(X_ 1,...,X_ n\) and let \(V_ n(y,{\hat \rho})\) be the empirical process given by \[ V_ n(y,{\hat \rho})=n^{-1/2}\sum^{n}_{i=1}I(X_ i-{\hat \rho}X_{i- 1}\leq y),\quad y\in R. \] The main result of the paper states that if i) E \(log^+| \epsilon_ 1| <\infty\), ii) F has uniformly bounded derivative \(f>0\) a.e., and iii) \(| \rho^ n({\hat \rho}-\rho)| =o_ P(n^{1/2})\), then \[ \sup_{y}| V_ n(y,{\hat \rho})-V_ n(y,\rho)| \to 0\quad in\quad probability. \] Consequently, \(V_ n(.,{\hat \rho})-n^{1/2}F(.)\Rightarrow B(F(.))\), where B is the Brownian bridge on [0,1]. This observation enables to solve the problem of testing \(H_ 0:\) \(F=F_ 0\), by means of the statistic \[ T_ n=\sup_{y}| V_ n(y,{\hat \rho})-n^{1/2}F_ 0(y)|. \] Then, under \(H_ 0\), \(T_ n\Rightarrow \sup_{0\leq u\leq 1}| B(u)|\), so that the test of \(H_ 0\) based on \(T_ n\) is asymptotically distribution-free. Some extensions of the above model which can be reduced to the previous case are also discussed. The proof of the main result is based on some ideas of E. Giné and J. Zinn [Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037)] and an exponential inequality for stopped bounded martingale-differences by S. Levental [J. Theor. Probab. 2, No.3, 271-287 (1989; Zbl 0681.60023)].
Reviewer: A.M.Zapala

MSC:

60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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