用给定的边值和相关性构造二元分布函数：应用于星系光度函数
Constructing a bivariate distribution function with given marginals and correlation: application to the galaxy luminosity function
摘要：
给出了一种构造具有给定边缘分布和相关系数的二元分布函数的解析方法。我们引入一个方便的数学工具，称为Copula，连接两个DF与任何规定的依赖结构。如果两个变量的相关性很弱（Pearson相关系数$|ρ| <1/3 $），Farlie-Gumbel-Morgenstern（FGM）copula提供了一种直观和自然的方法来构建这样的二元DF。当线性相关性较强时，FGM Copula函数不再起作用。在这种情况下，我们建议使用高斯copula，它连接两个给定的边缘，并直接关系到两个变量之间的线性相关系数。利用Copula函数，我们构造了BLF，并讨论了它的统计性质。特别是，我们专注于FUV-FIR BLF，因为这两个光度与恒星形成（SF）活动有关。虽然FUV和FIR都与SF活动有关，但单变量LF具有非常不同的函数形式：前者由Schechter函数很好地描述，而后者具有更扩展的幂律，如发光端。我们构造了FGM和高斯Copula不同强度的FUV-FIR BLF，并研究了它们的统计特性。然后，我们讨论了BLF的一些进一步的可能的应用：多波段通量有限的样本选择问题，SF率（SFR）函数的建设，和建设的恒星质量的星系（$M_*$）-特定的SFR（$SFR/M_*$）关系。Copula函数被证明是研究所有这些问题的一个非常有用的工具，特别是在包括复杂的选择效应方面。
英文摘要：
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 $|ρ| <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_*$)--specific SFR ($SFR/M_*$) relation. The copulas turned out to be a very useful tool to investigate all these issues, especially for including the complicated selection effects.
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