Reconstruction of luminosity function from flux-limited samples

The properties of the progenitors of gamma-ray bursts (GRBs) and of their environment are encoded in their luminosity function and cosmic formation rate. They are usually recovered from a flux-limited sample based on Lynden-Bell's $c^{-}$ method. However, this method is based on the assumption that the luminosity is independent of the redshift. Observationally, if correlated, people use nonparametric $\tau$ statistical method to remove this correlation through the transformation, $L^{\prime}=L/g(z)$, where $z$ is the burst redshift, and $g(z)=(1+z)^{k}$ parameterizes the underlying luminosity evolution. However, the application of this method to different observations could result in very different luminosity functions. By the means of Monte Carlo simulation, in this paper, we demonstrate that the origin of an observed correlation, measured by the $\tau$ statistical method, is a complex combination of multiple factors when the underlying data are correlated. Thus, in this case, it is difficult to unbiasedly reconstruct the underlying population distribution from a truncated sample, unless the detailed information of the intrinsic correlation is accurately known in advance. In addition, we argue that an intrinsic correlation between luminosity function and formation rate is unlikely eliminated by a misconfigured transformation, and the $g(z)$, derived from a truncated sample with the $\tau$ statistical method, does not necessarily represent its underlying luminosity evolution.