  We consider a pure-jump stable Cox-Ingersoll-Ross ($\alpha$-stable CIR)
process driven by a non-symmetric stable L{\'e}vy process with jump activity
$\alpha$ $\in$ (1, 2) and we address the joint estimation of drift, scaling and
jump activity parameters from high-frequency observations of the process on a
fixed time period. We first prove the existence of a consistent, rate optimal
and asymptotically conditionally gaussian estimator based on an approximation
of the likelihood function. Moreover, uniqueness of the drift estimators is
established assuming that the scaling coefficient and the jump activity are
known or consistently estimated. Next we propose easy-toimplement preliminary
estimators of all parameters and we improve them by a one-step procedure.
