  It has recently been proposed that one sub-class of type Ia supernovae (SNe
Ia) is sufficiently both distinct and common to be classified separately from
the bulk of SNe Ia, with a suggested class name of "type Iax supernovae" (SNe
Iax), after SN 2002cx. We show that the population properties of this class can
be understood if the events originate from helium double-detonation
sub-Chandrasekhar mass explosions, in which a carbon--oxygen white dwarf (CO
WD) accumulates a helium layer from a non-degenerate helium star. We have
incorporated detailed binary evolution calculations for the progenitor systems
into a binary population synthesis model to obtain birthrates and delay times
for such events. The predicted Galactic event rate is $\sim$$0.6-1.8\times
10^{-3}\,{\rm yr}^{-1}$, in good agreement with the measured rates of SNe Iax.
In addition, predicted delay times are $\sim$70\,Myr$-$710\,Myr, consistent
with the fact that SNe Iax have so far only been discovered in late-type
galaxies. Based on the CO WD mass at explosion and previous detonation models,
we also estimate the distribution of resulting SN brightness ($-13 \gtrsim
M_{\rm bol} \gtrsim -19$ mag), which can reproduce the empirical diversity of
SNe Iax. We speculate on why binaries with non-degenerate donor stars might
lead to SNe Iax if similar systems with degenerate donors do not and suggest
that the higher mass of the helium layer necessary for ignition at the lower
accretion rates typically delivered from non-degenerate donors might provide
the explanation.
