Weighted composite quantile inference for nearly nonstationary autoregressive models

In this paper, we focus on the following nearly nonsationary autoregressive model: $$y_t = q_n y_{t-1}+u_t$$ y t = q n y t - 1 + u t , $$t=1,\ldots ,n$$ t = 1 , … , n , where $$q_n=1+c/k_n$$ q n = 1 + c / k n with c a non-zero constant and $$\{k_n, n\geqslant 1\}$$ { k n , n ⩾ 1 } a sequence of positive constants increasing to $$\infty $$ ∞ such that $$k_n=o(n)$$ k n = o ( n ) as $$n\rightarrow \infty $$ n → ∞ , and $$\{u_t, t\geqslant 1\}$$ { u t , t ⩾ 1 } is a sequence of independent and identically distributed random variables which are in the domain of attraction of the normal law with zero mean and possibly infinity variance. The weighted composite quantile estimate of $$q_n$$ q n is examined, and the corresponding limiting distributions under the cases of $$c>0$$ c > 0 and $$c