We present a comprehensive study to measure the locations of \gamma -ray bursts ( GRBs ) relative to their host galaxies .
In total , we find the offsets of 20 long-duration GRBs from their apparent host galaxy centers utilizing ground-based images from Palomar and Keck and space-based images from the Hubble Space Telescope ( HST ) .
We discuss in detail how a host galaxy is assigned to an individual GRB and the robustness of the assignment process .
The median projected angular ( physical ) offset is 0.17 arcsec ( 1.3 kpc ) .
The median offset normalized by the individual host half-light radii is 0.98 suggesting a strong connection of GRB locations with the UV light of their hosts .
This provides strong observational evidence for the connection of GRBs to star-formation .

We further compare the observed offset distribution with the predicted burst locations of leading stellar-mass progenitor models .
In particular , we compare the observed offset distribution with an exponential disk , a model for the location of collapsars and promptly bursting binaries ( e.g .
helium star–black hole binaries ) .
The statistical comparison shows good agreement given the simplicity of the model , with the Kolmogorov-Smirnov probability that the observed offsets derive from the model distribution of P _ { KS } = 0.45 .
We also compare the observed GRB offsets with the expected offset distribution of delayed merging remnant progenitors ( black hole–neutron star and neutron star–neutron star binaries ) .
We find that delayed merging remnant progenitors , insofar as the predicted offset distributions from population synthesis studies are representative , can be ruled out at the 2 \times 10 ^ { -3 } level .
This is arguably the strongest observational constraint yet against delayed merging remnants as the progenitors of long-duration GRBs .
In the course of this study , we have also discovered the putative host galaxies of GRB 990510 and GRB 990308 in archival HST data .