Weak convergence of the residual empirical process in explosive autoregression
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 \textit{E. Giné} and \textit{J. Zinn} [Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037)] and an exponential inequality for stopped bounded martingale-differences by \textit{S. Levental} [J. Theor. Probab. 2, No.3, 271-287 (1989; Zbl 0681.60023)].
scientific article; zbMATH DE number 4138864