Constructing a bivariate distribution function with given marginals and correlation: application to the galaxy luminosity function

Abstract

We show an analytic method to construct a bivariate distribution function (DF) with given marginal distributions and correlation coefficient. We introduce a convenient mathematical tool, called a copula, to connect two DFs with any prescribed dependence structure. If the correlation of two variables is weak (Pearson's correlation coefficient $|\rho |<1/3$), the Farlie-Gumbel-Morgenstern (FGM) copula provides an intuitive and natural way for constructing such a bivariate DF. When the linear correlation is stronger, the FGM copula cannot work anymore. In this case, we propose to use a Gaussian copula, which connects two given marginals and directly related to the linear correlation coefficient between two variables. Using the copulas, we constructed the BLFs and discuss its statistical properties. Especially, we focused on the FUV--FIR BLF, since these two luminosities are related to the star formation (SF) activity. Though both the FUV and FIR are related to the SF activity, the univariate LFs have a very different functional form: former is well described by the Schechter function whilst the latter has a much more extended power-law like luminous end. We constructed the FUV-FIR BLFs by the FGM and Gaussian copulas with different strength of correlation, and examined their statistical properties. Then, we discuss some further possible applications of the BLF: the problem of a multiband flux-limited sample selection, the construction of the SF rate (SFR) function, and the construction of the stellar mass of galaxies ($M_{\ast }$)--specific SFR ($SFR/M_{\ast }$) relation. The copulas turned out to be a very useful tool to investigate all these issues, especially for including the complicated selection effects.

14 pages, 5 figures, accepted for publication in MNRAS.

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