New probability distributions in astrophysics: I. The truncated generalized gamma

The gamma function is a good approximation to the luminosity function of astrophysical objects, and a truncated gamma distribution would permit a more rigorous analysis. This paper examines the generalized gamma distribution (GG) and then introduces the scale and the new double truncation. The magnitude version of the truncated GG distribution with scale is adopted in order to fit the luminosity function (LF) for galaxies or quasars. The new truncated GG LF is applied to the five bands of SDSS galaxies, to the 2dF QSO Redshift Survey in the range of redshifts between 0.3 and 0.5, and to the COSMOS QSOs in the range of redshifts between 3.7 and 4.7. The average absolute magnitude versus redshifts for SDSS galaxies and QSOs of 2dF was modeled adopting a redshift dependence for the lower and upper absolute magnitude of the new truncated GG LF.


Introduction
The generalized gamma distribution (GG) was introduced by [1] and carefully analyzed by [2]. The GG has three-parameters and the techniques to find them is a matter of research, see among others [3,4,5]. A significant role in astrophysics is played by the luminosity function (LF) for galaxies and we present some models, among others, the Schechter LF, see [6], a two-component Schechter-like LF, see [7], and the double Schechter LF with five parameters, see [8]. Another approach starts from a given statistical distribution for which the probability density function (PDF) is known. We know that for a given PDF, f (L), where L is the luminosity. A data oriented LF, Ψ (L), is obtained by adopting Ψ * which is the normalization to the number of galaxies in a volume of 1 M pc 3 which means ∞ 0 Ψ (L)dL = Ψ * .
The above line of research allows exploring the LF for galaxies in the framework of well studied PDFs. Some examples are represented by the mass-luminosity relationship, see [9], some models connected with the generalized gamma (GG) distribution, see [10,11], the truncated beta LF, see [12], the lognormal LF, see [13], the truncated lognormal LF, see [14], and the Lindley LF, see [15]. This paper brings up the GG and introduces the scale in Section 2. The new double truncation for the GG and the GG with scale is introduced in Section 3. The derivation of the truncated GG LF is done in Section 4. Section 5 contains the application of the GG LF to galaxies and quasars as well the fit of the averaged absolute magnitude for QSOs as function of the redshift.

The generalized gamma distribution
Let X be a random variable defined in [0, ∞]; the GG (PDF), f (x), is where is the gamma function, with a > 0, b > 0 and c > 0. The above PDF can be obtained by setting the location parameter equal to zero in the four parameters GG as given by [16],pag. 113. The GG PDF scale as exp(−x c ) and the gamma PDF as exp(−x): the introduction of the parameter c increases the flexibility of the GG. The GG family includes several subfamilies, including the exponential PDF when a = 1 and c = 1, the gamma PDF when c = 1 and the Weibull PDF when b = 1 .
The distribution function (DF), F (x), is The DF, DF (x; a, b, c, x l , x u ), is where

The scale
The truncated GG PDF with scale requires the evaluation of the following integral, I ts , which is The constant of integration, K s (a, b, c, x l , x u ), is and as a consequence the truncated GG PDF with scale is, f ts (x; a, b, c, x l , x u ), The average value, µ s (a, b, c, x l , x u ) is where (40)

The luminosity function
In this section we present the Schechter LF, we derive the GG LF, we introduce the double truncation in the LF and we develop the adopted statistics.

The Schechter LF
The Schechter LF, introduced by [6], is here α sets the slope for low values of L, L * is the characteristic luminosity and Φ * is the normalization. The luminosity density, j, is The equivalent distribution in absolute magnitude is where M * is the characteristic magnitude as derived from the data. We now introduce the parameter h which is

Generalized gamma LF
We replace in the GG with scale, see equation (14) x with L (the luminosity), b with L * (the characteristic luminosity) and we insert Ψ * which is the normalization to the number of galaxies in a volume of 1 M pc 3 The magnitude version is where M is the absolute magnitude and M * is the characteristic magnitude. This four parameters LF, which was introduced in [10], was applied to the Sloan Digital Sky Survey (SDSS) in five different bands and to the near infrared band of the 2MASS Redshift Survey (2MRS), see [11].

Double truncation for the LF
We replace in the truncated GG with scale, see equation (35), x with L (the luminosity), b with L * (the characteristic luminosity), x l with L l (lower luminosity), x u with L u (upper luminosity), and we insert the normalization, Ψ * , where K s is given by equation (34). The luminosity density for the truncated GG LF, j T , has now a finite range of existence and is The

The adopted statistics
The unknown parameters of the LF can be found through the Levenberg-Marquardt method (subroutine MRQMIN in [18]) but the first derivative of the LF with respect to the unknown parameters should be provided. In the case of the truncated GG LF, see equation (49), the first derivative with respect to the unknown parameters has a complicated expression, so we used the numerical first derivative. The merit function χ 2 can be computed by where n is number of datapoints and the two indexes theo and astr stand for theoretical and astronomical, respectively. The residual sum of squares (RSS) is where y(i) theo is the theoretical value and y(i) astr is the astronomical value. Particular attention should be paid to the number of unknown parameters in the LF: three for the Schechter function (formula (43)) and four for formula (49). The reduced merit function χ 2 red can be computed by where N F = n − k, n being the number of datapoints and k the number of parameters. The Akaike information criterion (AIC), see [19] , is defined by where L is the likelihood function and k the number of free parameters in the model. We assume a Gaussian distribution for the errors and the likelihood function can be derived by the χ 2 statistic L ∝ exp(− χ 2 2 ) where χ 2 has been computed by equation (53), see [20], [21]. Now AIC becomes AIC = 2k + χ 2 . (57) The Bayesian information criterion (BIC), see [22], is where L is the likelihood function, k the number of free parameters in the model and n the number of observations. The phrase "better fit" used in the following means that the three statistical indicators : χ 2 , AIC and BIC are smaller for the considered LF than for the Schechter function .

Astrophysical Applications
In this section we apply the truncated GG LF to the SDSS galaxies and to QSOs. The introduction of the redshift dependence for lower and upper absolute magnitude allows to model the average absolute magnitude versus redshift for QSOs.

SDSS galaxies
In order to perform a test we selected the data of the Sloan Digital Sky Survey (SDSS) which has five bands u * (λ = 3550Å), g * (λ = 4770Å), r * (λ = 6230Å), i * (λ = 7620Å) and z * (λ = 9130Å) with λ denoting the wavelength of the CCD camera, see [23]. The data of the astronomical LF are reported in [24] and are available at https://cosmo.nyu.edu/blanton/lf.html. The numerical values of the four parameters a, c, M * and Ψ * are given in Table 1. The Schechter function, the new four parameters function as represented by formula (49) and the data are reported in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5, where bands u * , g * , r * , i * and z * are considered. Table 2 presents the luminosity density evaluated with the Schechter LF, j, and with the truncated GG LF, j T . The range of existence in the truncated case is finite rather than infinite and therefore the luminosity density is always smaller than in the standard case.

Luminosity function for QSOs
For our first example, we selected the catalog of the 2dF QSO Redshift Survey (2QZ), which contains 22431 redshifts of QSOs with 18.25 < b J < 20.85, see [25]. We processed them as explained in [26]. A     typical example of the observed LF for QSOs when 0.3 < z < 0.5 as well the fit with the four parameters truncated GG LF is presented in Figure 6 with data as in Table 3.  In the second example we explored the faint LF for quasars in the range of redshifts 3.7 < z < 4.7 as given in Figure 4 of [27]. The results are displayed in Figure 7 with data as in Table 4.

Average absolute magnitude versus redshift
The first application is about galaxies: we processed the SDSS Photometric Catalogue DR 12, see [28], which contains 10450256 galaxies (elliptical + spiral) with redshift and rest frame u absolute magnitude. The lower absolute magnitude is fixed at M l = −30 and the upper absolute magnitude is the maximum