  Signatures of X-ray reprocessing (reflection) out of an accretion disk are
commonly observed in the high-energy spectrum of accreting black holes, and can
be used to probe the strong gravity region around these objects. In this paper,
we extend previous work in the literature and we employ a full emission model
for relativistic reflection in non-Kerr spacetime to demonstrate an approach
that tests the Kerr black hole hypothesis. We analyze a composite spectrum
obtained with the Proportional Counter Array in the Rossi X-ray Timing Explorer
(RXTE), of the stellar-mass black hole GX 339-4 in its brightest hard state.
With a remarkable sensitivity of ~0.1% and 40 million counts in the 3-45 keV
band to capture the faint features in the reflection spectrum, we demonstrate
that it is possible with existing data and an adequate model to place
constraints on the black hole spin $a_*$ and the deformation parameter that
quantifies the departure from the Kerr metric. Our measurement obtained with
the best fit model, which should be regarded as principally a proof of concept,
is $a_*=0.92^{+0.07}_{-0.12}$ and $\alpha_{13}=-0.76^{+0.78}_{-0.60}$ with a
90% confidence level and is consistent with the hypothesis that the compact
object in GX 339-4 is a Kerr black hole. We also discuss how the physical model
choice and the emissivity profile adopted could make an impact on the
constraints of $\alpha_{13}$ and spin. To enable Kerr metric test using X-ray
reflection spectroscopy, it is essential to improve our astrophysical
understanding of accreting black holes, e.g., the natures of accretion flow and
corona.
