tokens: of SNe Iax, which are in the range 12+log(O/H)=∼8.16–8.87 (Magee et al. 2017 ; Lyman et al. 2018 ), somehow lower than the solar abundance (Caffau et al. 2008 ). This indicates that the site of SNe Iax may have relatively lower metallicity compared with other subtypes of SNe Ia, but comparable to that of SNe II or SNe Ibc. Note that Lyman et al. ( 2018 ) recently carried out an analysis of metallicity for the host galaxies and explosion sites of a large sample of SNe Iax through both integral-field and long-slit spectroscopy and they also found that the SN Iax explosion site metallicity distribution is similar to that of core-collapse SNe. We also measure the fractional flux of the SN site, which is the fraction for which the total host light in a pixel is fainter than or equal to the light in the pixel at the location of the SN. This parameter can be used to trace the difference/similarity between luminosity at SN locations and the light of the hosts (Fruchter et al. 2006 ). As the fractional flux is independent of galaxy morphology and size, it is thus a better parameter to quantify the correlation between SN progenitor environments and the light of their hosts (i.e. Wang et al. 2013 ). Such an analysis can be performed for 21 SNe Iax of our sample by using the host-galaxy images from SDSS DR12 (Alam et al. 2015 ) and Panstarrs (Chambers et al. 2017 ); the results are also tabulated in columns (9)–(11) of Table 5 . The distribution of the fractional fluxes of our SN Iax sample in the ugr bands is shown in Fig. 14 . The fractional flux distributions of SNe Ia, SNe Ibc and SNe II are overplotted for comparison and the measurements of these samples are taken from Wang et al. ( 2013 ). One can see that the progenitor populations of SNe Iax belong to the brighter regions of the host galaxies, similar to the case for SNe Ibc (especially in the g and r bands), suggesting that SNe Iax may arise in larger star-forming regions that produce more massive stars. This is also evidenced by the Kolmogorov–Smirnov (KS) test results, as tabulated in Table 6 . One can see that SNe Iax and SNe Ia trace the host-galaxy light differently, with a very low probability (i.e. P = 0.04 in r ) that they come from the same stellar populations. SNe Iax also show some differences from SNe II in the light distribution of progenitor populations, with KS test results of P = 0.13 in g and P = 0.17 in r , respectively. In the u band, however, we notice that SNe Iax seem to show large differences from all kinds of SNe in the fractional flux distribution. This is likely related to the relatively larger uncertainty in the SDSS u -band photometry. As for the comparison between SNe Iax and SNe Ibc, P values of 0.33 obtained in the g band and 0.37 in the r band indicate that the hypothesis that they share a similar fractional flux (and hence environment) distribution cannot be rejected. Figure 14. Histogram distribution of the fractional flux of the host-galaxy light at the location
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