  We measure the rates of type I X-ray bursts, as a function of the bolometric
luminosity, from a likely complete sample of 37 non-pulsing transients
(1996-2004). Our goals are to test the burst model for neutron stars and to
investigate whether black holes have event horizons. We find 135 type I bursts
in 3.7 Ms of exposure for the neutron-star group, and the burst rate function
is generally consistent with model predictions. However, for the black hole
groups (18 sources), there are no confirmed type I bursts in 6.5 Ms of
exposure, and the upper limits in the burst function are inconsistent with the
model predictions for heavy compact objects with a solid surface. There are
systematic spectral differences between the neutron-star and black-hole groups,
supporting the presumption that physical differences underly the sample
classifications. These results provide indirect evidence that black holes do
have event horizons.
