  We provide an estimate for the essential norm of a weighted composition
operator $W_{\psi,\varphi}\colon f\mapsto \psi(f\circ\varphi)$ acting on the
space $BMOA$ in terms of the weight function $\psi$ and the $n$-th power
$\varphi^n$ of the analytic self-map $\varphi$ of the open unit disc
$\mathbb{D}$. We also provide a new estimate for the norm of the weighted
composition operator on $BMOA$.
