Anderson (The Annals of Mathematical Statistics 30(3):676–687, 1959) studied the limiting distribution of the least square estimator for explosive AR(1) process under the independent and identically distributed (iid) condition on error i.e., \(X_t=\rho X_{t-1}+e_t\) where \(\rho >1\) and \(e_t\) is iid error with \(Ee=0\) and \(Ee^2<\infty \). This paper is mainly concerned about the limiting distribution of the least square estimator of \(\rho \), that is \(\hat{\rho }\), when errors are not identically distributed. In addition, we provide an approximate description of the limiting distribution of \(\sum _{j=0}^{n-1}\rho ^{-j}e_{n-j}\) when \(\rho >1\) as \(n\rightarrow \infty \).

中文翻译：

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具有独立但分布不均的错误的爆炸性AR（1）流程

Anderson（《数学统计年鉴》 30（3）：676–687，1959年）研究了在误差为独立且均等分布（iid）的条件下，爆炸AR（1）过程的最小二乘估计的极限分布，即\（ X_t = \ rho X_ {t-1} + e_t \）其中\（\ rho> 1 \）和\（e_t \）是iid错误，其中\（Ee = 0 \）和\（Ee ^ 2 <\ infty \ ）。本文主要关注\（\ rho \）的最小二乘估计的极限分布，即误差不完全相同时的\（\ hat {\ rho} \）。此外，我们提供\（\ sum _ {j = 0} ^ {n-1} \ rho ^ {-j} e_ {nj} \）的极限分布的近似描述当\（\ rho> 1 \）为\（n \ rightarrow \ infty \）时。