The merit function ${\chi }^2$ is computed as

${\chi }^2=\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{L{F}_{theo}-L{F}_{astr}}{{\sigma }_{L{F}_{astr}}}\right)}^2,$(14)

where $n$ is the number of bins for the LF of the galaxies and the two indices $theo$ and $astr$ stand for “theoretical” and “astronomical”, respectively. The residual sum of squares (RSS) is

$\text{RSS}=\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\left(y{\left(i\right)}_{theo}-y{\left(i\right)}_{astr}\right)}^2,$(15)

where $y{\left(i\right)}_{theo}$ is the theoretical value and $y{\left(i\right)}_{astr}$ is the astronomical value.

A reduced merit function ${\chi }_{red}^2$ is evaluated by

${\chi }_{red}^2={\chi }^2/NF,$(16)

where $NF=n-k$ is the number of degrees of freedom and $k$ is the number of parameters. The goodness of the fit can be expressed by the probability $Q$, see Equation (15.2.12) in [16] , which involves the number of degrees of freedom and ${\chi }^2$. According to [16] , the fit “may be acceptable” if $Q>0.001$. The Akaike information criterion (AIC), see [17] , is defined by

$\text{AIC}=2k-2\ln \left(L\right),$(17)

where $L$ is the likelihood function and $k$ is the number of free parameters in the model. We assume a Gaussian distribution for the errors, then the likelihood

function can be derived from the ${\chi }^2$ statistic $L\propto \text{exp}\left(-\frac{{\chi }^2}{2}\right)$ where ${\chi }^2$ has been computed by Equation (14), see [18] [19] . Now the AIC becomes

$\text{AIC}=2k+{\chi }^2.$(18)

The mean percentage error (MPE) is defined as

$\text{MPE}=100\times \frac{1}{n}\times \underset{j=1}{\overset{n}{{\displaystyle \sum }}}\Vert \frac{y{\left(i\right)}_{theo}-y{\left(i\right)}_{astr}}{y{\left(i\right)}_{theo}}\Vert .$(19)

Let $L$ be a random variable taking values in the closed interval $\left[0,\infty \right]$. The Schechter LF of the galaxies, after [20] , is

$\Phi \left(L;{\Phi }^{*},\alpha ,L^{*}\right)\text{d}L=\left(\frac{{\Phi }^{*}}{L^{*}}\right){\left(\frac{L}{L^{*}}\right)}^{\alpha }\exp \left(-\frac{L}{L^{*}}\right)\text{d}L,$(20)

where $\alpha$ sets the slope for low values of $L$, $L^{\text{*}}$ is a characteristic luminosity, and ${\Phi }^{*}$ is the number of galaxies per Mpc³. The normalization is

${\displaystyle {\int }_0^{\infty }\Phi \left(L;{\Phi }^{*},\alpha ,L^{*}\right)\text{d}L}={\Phi }^{*}\Gamma \left(\alpha +1\right),$(21)

where

$\text{Γ}\left(z\right)={\displaystyle {\int }_0^{\infty }{\text{e}}^{-t}{t}^{z-1}\text{d}t},$(22)

is the gamma function. An equivalent form in absolute magnitude of the Schechter LF is

$\Phi \left(M;{\Phi }^{*},\alpha ,M^{*}\right)\text{d}M=0.921{\Phi }^{*}{10}^{0.4\left(\alpha +1\right)\left(M^{*}-M\right)}\exp \left(-{10}^{0.4\left(M^{*}-M\right)}\right)\text{d}M,$(23)

where $M^{\text{*}}$ is the characteristic magnitude.

We assume that the luminosity $L$ takes values in the interval $\left[L_l,L_u\right]$, where the indices $l$ and $u$ mean “lower” and “upper”; the truncated Schechter LF, $S_T$, is

$S_T\left(L;{\Psi }^{*},\alpha ,L^{*},L_l,L_u\right)=\frac{-{\left(\frac{L}{L^{*}}\right)}^{\alpha }{\text{e}}^{-\frac{L}{L^{*}}}{\Psi }^{*}\Gamma \left(\alpha +1\right)}{L^{*}\left(\Gamma \left(\alpha +1,\frac{L_u}{L^{*}}\right)-\Gamma \left(\alpha +1,\frac{L_l}{L^{*}}\right)\right)},$(24)

where $\text{Γ}\left(a,z\right)$ is the incomplete Gamma function defined as

$\text{Γ}\left(a,z\right)={\displaystyle {\int }_z^{\infty }{t}^{a-1}{\text{e}}^{-t}\text{d}t},$(25)

see [21] . The normalization is the same as for the Schechter LF, see Equation (21),

${\displaystyle {\int }_0^{\infty }{S}_T\left(L;{\Psi }^{*},\alpha ,L^{*},L_l,L_u\right)\text{d}L}={\Psi }^{*}\Gamma \left(\alpha +1\right).$(26)

The average value is

$\langle L\left({\Psi }^{*},\alpha ,L^{*},L_l,L_u\right)\rangle =\frac{N}{L^{*}\left(\Gamma \left(\alpha +1,\frac{L_u}{L^{*}}\right)-\Gamma \left(\alpha +1,\frac{L_l}{L^{*}}\right)\right)}$(27)

with

$\begin{array}{c}N={\Psi }^{*}(L^{*}{}^2\Gamma \left(\alpha +1,\frac{L_u}{L^{*}}\right)\alpha -L^{*}{}^2\Gamma \left(\alpha +1,\frac{L_l}{L^{*}}\right)\alpha +L^{*}{}^2\Gamma \left(\alpha +1,\frac{L_u}{L^{*}}\right)\\ -L^{*}{}^2\Gamma \left(\alpha +1,\frac{L_l}{L^{*}}\right)-L^{*}{}^{-\alpha +1}{\text{e}}^{-\frac{L_l}{L^{*}}}{L}_l^{\alpha +1}+L^{*}{}^{-\alpha +1}{\text{e}}^{-\frac{L_u}{L^{*}}}{L}_u^{\alpha +1})\Gamma \left(\alpha +1\right).\end{array}$(28)

The four luminosities $L,L_l,L^{\text{*}}$ and $L_u$ are connected with the absolute magnitudes $M$, $M_l$, $M_u$ and $M^{\text{*}}$ through the following relationship

$\frac{L}{L_{\odot }}={10}^{0.4\left(M_{\odot }-M\right)},\frac{L_l}{L_{\odot }}={10}^{0.4\left(M_{\odot }-M_u\right)},\frac{L^{*}}{L_{\odot }}={10}^{0.4\left(M_{\odot }-M^{*}\right)},\frac{L_u}{L_{\odot }}={10}^{0.4\left(M_{\odot }-M_l\right)}$(29)

where the indices $u$ and $l$ are inverted in the transformation from luminosity to absolute magnitude and $L_{\odot }$ and $M_{\odot }$ are the luminosity and absolute magnitude of the sun in the considered band. The equivalent form in absolute magnitude of the truncated Schechter LF is therefore

$\begin{array}{l}\Psi \left(M;{\Psi }^{*},\alpha ,M^{*},M_l,M_u\right)\text{d}M\\ =\frac{-0.4{\left({10}^{0.4M^{*}-0.4M}\right)}^{\alpha }{\text{e}}^{-{10}^{0.4M^{*}-0.4M}}{\Psi }^{*}\Gamma \left(\alpha +1\right){10}^{0.4M^{*}-0.4M}\left(\ln \left(2\right)+\ln \left(5\right)\right)}{\Gamma \left(\alpha +1,{10}^{-0.4M_l+0.4M^{*}}\right)-\Gamma \left(\alpha +1,{10}^{0.4M^{*}-0.4M_u}\right)}.\end{array}$(30)

The average absolute magnitude is

$\langle M\left({\Psi }^{*},\alpha ,L^{*},L_l,L_u\right)\rangle =\frac{{\displaystyle {\int }_{M_l}^{M_u}M}\left(M;{\Psi }^{*},\alpha ,L^{*},L_l,L_u\right)M\text{d}M}{{\displaystyle {\int }_{M_l}^{M_u}M}\left(M;{\Psi }^{*},\alpha ,L^{*},L_l,L_u\right)\text{d}M}.$(31)

More details can be found in [22] .

The magnitude version of the generalized gamma LF with truncation is

$\Psi \left(M;a,M^{*},c,M_l,M_u,{\Psi }^{*}\right)={\Psi }^{*}\frac{LD}{LN},$(32)

$LD=0.4c\left(c+1\right)a{\text{e}}^{-{\text{e}}^{0.921a\left(M^{\text{*}}-M\right)}+ac\left(0.921M^{\text{*}}-0.921M\right)}\left(\text{ln}\left(2\right)+\text{ln}\left(5\right)\right),$(33)

$\begin{array}{c}LN={\text{e}}^{-{\text{e}}^{-0.921a\left(M_l-M^{\text{*}}\right)}+\left(0.921M^{\text{*}}-0.921M_l\right)ca}c-{\text{e}}^{-{\text{e}}^{0.921a\left(M^{\text{*}}-M_u\right)}+ac\left(0.921M^{\text{*}}-0.921M_u\right)}c\\ +{\text{e}}^{-0.5{\text{e}}^{-0.921a\left(M_l-M^{\text{*}}\right)}+\left(-0.46M_l+0.46M^{\text{*}}\right)ca}{M}_{c/2,c/2+1/2}\left({\text{e}}^{-0.921a\left(M_l-M^{\text{*}}\right)}\right)\\ -{\text{e}}^{-0.5{\text{e}}^{0.921a\left(M^{\text{*}}-M_u\right)}+\left(0.46M^{\text{*}}-0.46M_u\right)ca}{M}_{c/2,c/2+1/2}\left({\text{e}}^{0.921a\left(M^{\text{*}}-M_u\right)}\right)\\ +{\text{e}}^{-{\text{e}}^{-0.921a\left(M_l-M^{\text{*}}\right)}+\left(0.921M^{\text{*}}-0.921M_l\right)ca}-{\text{e}}^{-{\text{e}}^{0.921a\left(M^{\text{*}}-M_u\right)}+ac\left(0.921M^{\text{*}}-0.921M_u\right)}.\end{array}$(34)

In the above formulae, $a$ and $c$ are two positive parameters, ${\Psi }^{*}$ is the number of galaxies per Mpc³, $M_l$, $M_u$ and $M^{\text{*}}$ are, respectively, the lower, the upper and the characteristic absolute magnitudes. The average absolute magnitude is

$\langle \Psi \left(M;a,M^{*},c,M_l,M_u,{\Psi }^{*}\right)\rangle =\frac{{\displaystyle {\int }_{M_l}^{M_u}\Psi }\left(M;a,M^{*},c,M_l,M_u,{\Psi }^{*}\right)M\text{d}M}{{\displaystyle {\int }_{M_l}^{M_u}\Psi }\left(M;a,M^{*},c,M_l,M_u,{\Psi }^{*}\right)\text{d}M}.$(35)

More details can be found in [23] .