*2.2. Limiting Fluxes, Fluences, and Times*

The EP method can overcome selection bias for a particular variable of interest, but it must first be determined if the variable is dependent or independent of redshift.

It is then necessary to define limiting values for each of the variables. In Dainotti et al. [7], it was demonstrated that a good choice of limiting times and luminosities retains at least 90% of the total sample. For time variables, log *T* ∗ <sup>90</sup> and log *T* ∗ a,radio (in units of seconds), we define a general form for the limiting values as *<sup>T</sup>*min (1+*z*) , where *T*min is the minimum observed time. We need to choose a compromise between a limit which is representative of the population of data points, but still retains most of the sample size. It has been shown in Monte Carlo simulations in [7] that such a strategy with limiting values is accurate. For log *T* ∗ <sup>90</sup>, we find the best limiting duration to be log *T* ∗ 90min,obs <sup>=</sup> <sup>−</sup>0.54 s, with a limiting boundary defined as <sup>−</sup>0.30 (1+*z*) s, which excludes 5/80 (<10%) GRBs. The limiting line for log *T* ∗ <sup>90</sup> is shifted at a higher value to be allow the sample data to be representative of the whole population. For log *T* ∗ a,radio, we

find the limiting time to be the observed minimum log *T* ∗ a,radiomin,obs = 4.94 s, thus defining the boundary as log *T* ∗ a,radio <sup>=</sup> 4.94 (1+*z*) s, which does not exclude any data points.

For the isotropic energy, we instead define the limiting energy according to the methodology of Dainotti et al. [7], in which the limiting fluence should be representative of the population while including at least 90% of the sample. We use the following formula:

$$E\_{\rm iso,lim} = 4\pi D\_L^2(z) S\_{\rm lim\ \prime} \tag{8}$$

where *<sup>S</sup>*lim is the fluence limit. For our sample, we define *<sup>S</sup>*lim as 6.3 <sup>×</sup> <sup>10</sup>−<sup>8</sup> erg cm−<sup>2</sup> . Applying this limit excludes 8/80 GRBs, which is 10% of our sample. In all the method described here we use GRBs that have log *E*iso > log *E*iso,lim, log *T* ∗ <sup>90</sup> > log *T* ∗ <sup>90</sup>, log *T*a,radio > log *T*a,radio,lim, and log *L*a,radio > log *L*a,radio,lim. For the luminosity, however, a caveat should be posed when we consider the total distribution of the parent population of GRBs with and without redshift (see [48]).

Using the method presented in Dainotti et al. [48], we compare the parent sample of all GRBs with observed radio afterglow and known peak flux to a smaller "subsample" of GRBs with known peak flux and known redshift. We then apply cuts to the data by defining limiting fluxes at regular intervals. Considering only the data with values above the limiting fluxes, fluences and time, we conduct a two-sample Kolmogorov–Smirnov (KS) test between the data of the total sample and the data for which the limiting cuts have been applied to determine the distribution of the probability that the subsample was drawn from the parent sample, as well as the geometric distance between the two samples as determined by the KS test. We take the limiting flux to be the value of *f* lim where the probability as a function of limiting flux reaches a plateau in which the probability that two samples are drawn by the same population is 100%. In our sample, we find this limit to be log *f* lim <sup>=</sup> <sup>−</sup>17.2. We define the flux throughout our analysis in units of erg cm−<sup>2</sup> s.

We show the distribution of the parent sample and subsample in the left panel of Figure 2, with the limiting line shown in red. We plot this probability as a function of flux limit (blue), as well as the distance between the distributions (orange), in the right panel of Figure 2.

**Figure 2.** (**Left**): peak flux distribution for "parent" sample and subsample with known redshift. Limiting flux shown in red. (**Right**): plot of probability (blue) and distance between samples as given by the KS test (orange) as a function of flux limit. Limiting line *f* lim shown in red.
