The estimate of the distribution of noise in autoregressive scheme

M. V. Boldin

Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 805-810

Résumé

Let $u_j=\beta_1u_{j-1}+\dots+\beta_qu_{j-q}+\varepsilon_j$ ($j=1,\dots,n$) аге $n$ observations of autoregressive scheme, where $\beta_1,\dots,\beta_q$ are unknown nonrandom parameters and $\varepsilon_j$ are independent identically distributed random variables with zero mean, finite variance and unknown distribution function $G(x)$. The estimate $\widehat G_n(x)$ of $G(x)$ is considered. It is proved that $\sqrt n[\widehat G_n(G^{-1}(t))-t]$ converges weakly to the Brownian bridge when $u\to\infty$. The result is used in the testing of the hypotheses on $G(x)$.