In this paper, the authors obtain the endpoint estimates for the following non-standard commutator defined by \[ T_\alpha^Af(x)=\int_{\mathbb R^n}\frac {\Omega(x-y)}{|x-y|^{n-\alpha+m-1}} R_m(A;x,y)f(y) dy \] and its variant \[ \bar T_\alpha^Af(x)=\int_{\mathbb R^n}\frac {\Omega(x-y)}{|x-y|^{n-\alpha+m-1}} Q_m(A;x,y)f(y) dy, \] where \(0\leq\alpha <n\), \(\Omega\in \text{ Lip}_1(S^{n-1})\) is homogeneous of degree zero, \(m\in{\mathbb N}\), \(A\) has derivatives of order \(m-1\) in \(\text{ BMO}(\mathbb R^n)\), \[ R_m(A;x,y)=A(x)-\sum_{|\gamma|<m}\frac 1{\gamma!}D^\gamma A(y)(x-y)^\gamma, \] and \[ Q_m(A;x,y)=R_{m-1}(A;x,y)-\sum_{|\gamma|=m-1}\frac 1{\gamma!}D^\gamma A(x) (x-y)^\gamma. \] If \(m=1\), \(T^A_\alpha\) degenerates into the classical commutator of the fractional or singular integral with the BMO function \(A\). The results of the authors indicate that \(T^A_\alpha\) and \(\bar T^A_\alpha\) have better properties when \(m\geq 2\) than the classical commutator. Moreover, the authors also show that these operators are not bounded in certain cases.