Let $X$ be a random variable taking values $x$ in the interval $\left[0,\infty \right]$; the first definition for the lognormal PDF, following [6] or formula (14.2) in [7] , is

$f\left(x:m,\sigma \right)=\frac{1}{x\sigma \sqrt{2\pi }}\exp \frac{-{\left[\ln \left(x/m\right)\right]}^2}{2{\sigma }^2}.$(2)

The average value, $E\left(m,\sigma \right)$, is

$E\left(m,\sigma \right)=m{\text{e}}^{\frac{{\sigma }^2}{2}},$(3)

and the distribution function, $F\left(x:m,\sigma \right)$, is given by

$F\left(x:m,\sigma \right)=\frac{1}{2}-\frac{\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(m\right)-\text{ln}\left(x\right)\right)}{2\sigma }\right)}{2}.$(4)

The second definition is

$f\left(x:\mu ,\sigma \right)=\frac{1}{x\sigma \sqrt{2\pi }}\exp \frac{-{\left(\ln x-\mu \right)}^2}{2{\sigma }^2},$(5)

where $m=\exp \mu$ and $\mu =\ln m$.

Let $X$ be a random variable defined in $\left[x_l,x_u\right]$; the truncated lognormal PDF ( $f_T$) is based on the first definition of the lognormal as given by Equation (2)

$\begin{array}{l}{f}_T\left(x:m,\sigma ,x_l,x_u\right)\\ =\frac{\sqrt{2}{\text{e}}^{-\frac{1}{2}\frac{1}{{\sigma }^2}{\left(\text{ln}\left(\frac{x}{m}\right)\right)}^2}}{-\sqrt{\pi }\sigma \left(\text{erf}\left(\frac{1}{2}\frac{\sqrt{2}}{\sigma }\text{ln}\left(\frac{x_l}{m}\right)\right)-\text{erf}\left(\frac{1}{2}\frac{\sqrt{2}}{\sigma }\text{ln}\left(\frac{x_u}{m}\right)\right)\right)x},\end{array}$(6)

where $m$ is the scale parameter, $\sigma$ is the shape parameter, $x_l$ denotes the minimal value, and $x_u$ denotes the maximal value. The introduction of the following coefficients allows a compact notation:

$a_1=\frac{1}{2}\frac{\sqrt{2}\left(-{\sigma }^2+\text{ln}\left(x_l\right)-\text{ln}\left(m\right)\right)}{\sigma },$

$a_2=\frac{1}{2}\frac{\sqrt{2}\left({\sigma }^2+\text{ln}\left(m\right)-\text{ln}\left(x_u\right)\right)}{\sigma },$

$a_3=\frac{1}{2}\frac{\sqrt{2}\left(\text{ln}\left(x_l\right)-\text{ln}\left(m\right)\right)}{\sigma },$

$a_4=\frac{1}{2}\frac{\sqrt{2}\left(-\text{ln}\left(x_u\right)+\text{ln}\left(m\right)\right)}{\sigma },$

$a_5=\frac{1}{2}\frac{\sqrt{2}\left(-2{\sigma }^2+\text{ln}\left(x_l\right)-\text{ln}\left(m\right)\right)}{\sigma },$

$a_6=\frac{1}{2}\frac{\sqrt{2}\left(2{\sigma }^2+\text{ln}\left(m\right)-\text{ln}\left(x_u\right)\right)}{\sigma },$

$a_7=\frac{1}{2}\frac{\sqrt{2}\left(-2{\sigma }^2+\text{ln}\left(x_u\right)-\text{ln}\left(m\right)\right)}{\sigma },$

$a_8=\frac{1}{2}\frac{\sqrt{2}\left(\text{ln}\left(x_u\right)-\text{ln}\left(m\right)\right)}{\sigma }.$

In this compact notation, the PDF is

$f_T\left(x:m,\sigma ,x_l,x_u\right)=\frac{-\sqrt{2}{\text{e}}^{-\frac{1}{2}\frac{1}{{\sigma }^2}{\left(\text{ln}\left(\frac{x}{m}\right)\right)}^2}}{\sqrt{\pi }\sigma \left(\text{erf}\left(a_3\right)-\text{erf}\left(a_8\right)\right)x},$(7)

the DF is

$F_T\left(x:m,\sigma ,x_l,x_u\right)=\frac{-\text{erf}\left(\frac{1}{2}\frac{\sqrt{2}}{\sigma }\text{ln}\left(\frac{x}{m}\right)\right)+\text{erf}\left(a_3\right)}{\text{erf}\left(a_3\right)-\text{erf}\left(a_8\right)},$(8)

and the mean, $E_T\left(m,\sigma ,x_l,x_u\right)$, is

$E_T\left(m,\sigma ,x_l,x_u\right)=\frac{{\text{e}}^{\frac{1}{2}{\sigma }^2}m\left(\text{erf}\left(a_1\right)+\text{erf}\left(a_2\right)\right)}{\text{erf}\left(a_3\right)+\text{erf}\left(a_4\right)}.$(9)

More details can be found in [8] .

The modified lognormal distribution with a power-law (MLP) defined in the interval $\left[0,\infty \right]$ has the PDF

$f\left(x:\alpha ,\sigma ,\mu \right)=\frac{x^{-1-\alpha }\text{erfc}\left(\frac{\sqrt{2}\left(\alpha \sigma -\frac{\text{ln}\left(x\right)-\mu }{\sigma }\right)}{2}\right)\alpha }{2{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }},$(10)

where $\text{erfc}\left(x\right)$ is the complementary error function

$\text{erfc}\left(x\right)x=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_x^{\infty }{\text{e}}^{-t^2}\text{d}t}=1-\text{erf}\left(x\right),$(11)

see the handbook [9] . Its DF is

$\begin{array}{c}F\left(x:\alpha ,\sigma ,\mu \right)=-\frac{\text{erfc}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x\right)+\mu \right)}{2\sigma }\right)x^{-\alpha }{\text{e}}^{\frac{1}{2}{\alpha }^2{\sigma }^2+\alpha \mu }}{2}\\ +1-\frac{\text{erfc}\left(\frac{\sqrt{2}\left(\text{ln}\left(x\right)-\mu \right)}{2\sigma }\right)}{2},\end{array}$(12)

see formula (16) in [10] . The first moment, or mean, $E\left(\alpha ,\sigma ,\mu \right)$, is defined for $\alpha >1$:

$E\left(\alpha ,\sigma ,\mu \right)=\frac{\alpha {\text{e}}^{\frac{{\sigma }^2}{2}+\mu }}{\alpha -1},$(13)

see formula (19) in [10] .

The variance, $Var\left(\alpha ,\sigma ,\mu \right)$, is defined for $\alpha >2$:

$Var\left(\alpha ,\sigma ,\mu \right)=-\frac{{\alpha }^2{\left({\text{e}}^{\frac{{\sigma }^2}{2}+\mu }\right)}^2}{{\left(\alpha -1\right)}^2}+\frac{\alpha {\text{e}}^{2{\sigma }^2+2\mu }}{\alpha -2},$(14)

see formula (21) in [10] . The mode should be evaluated numerically. The random generation of the variate $X$ of the truncated MLP is accomplished by solving the following nonlinear equation in $x$:

$F\left(x:\alpha ,\sigma ,\mu \right)=R,$(15)

where $R$ is the unit rectangular variate. More details can be found in [11] .

The truncated version of the MLP defined in the interval $\left[x_l,x_u\right]$ has the following PDF:

$f_T\left(x:\alpha ,\sigma ,\mu ,x_l,x_u\right)=\frac{x^{-1-\alpha }\text{erfc}\left(\frac{\sqrt{2}\left(\alpha \sigma -\frac{\text{ln}\left(x\right)-\mu }{\sigma }\right)}{2}\right)}{A},$(16)

where

$\begin{array}{c}A=\frac{1}{\alpha }\times [{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_u\right)-\mu \right)}{2\sigma }\right)\\ -{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_l\right)-\mu \right)}{2\sigma }\right)\\ -x_l^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ +x_u^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right)+x_l^{-\alpha }-x_u^{-\alpha }].\end{array}$(17)

The DF is

$\begin{array}{c}{F}_T\left(x:\alpha ,\sigma ,\mu ,x_l,x_u\right)=\frac{1}{\alpha A}\times [-x_l^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ +{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x\right)-\mu \right)}{2\sigma }\right)\\ -{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_l\right)-\mu \right)}{2\sigma }\right)\\ +x^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x\right)+\mu \right)}{2\sigma }\right)-x^{-\alpha }+x_l^{-\alpha }].\end{array}$(18)

The first moment, or average value, is

$\begin{array}{l}E\left(x:\alpha ,\sigma ,\mu ,x_l,x_u\right)\\ =\frac{1}{A\left(-1+\alpha \right)}\times [{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu +\frac{1}{2}{\sigma }^2+\mu }\text{erf}\left(\frac{\sqrt{2}\left({\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ -{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu +\frac{1}{2}{\sigma }^2+\mu }\text{erf}\left(\frac{\sqrt{2}\left({\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right)\\ -x_l^{1-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ +x_u^{1-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right)+x_l^{1-\alpha }-x_u^{1-\alpha }].\end{array}$(19)

More details can be found in [11] .

The Kolmogorov-Smirnov test (K-S), see [12] - [14] , does not require the data to be binned. The K-S test, as implemented by the FORTRAN subroutine KSONE in [15] , given ordered statistics $x_i$, $i=1,\cdots ,N$, where $N$ defines the number of elements in the sample, finds the maximum distance, $D$, between the theoretical and the astronomical DFs. The significance is

$Q_{KS}\left(\lambda \right)=2\left({\displaystyle \underset{j=1}{\overset{\infty }{\sum }}{\left(-1\right)}^{j-1}{\text{e}}^{-2j^2{\lambda }^2}}\right),$(20)

and the significance level to have an observed $D$ is

$P_{KS}=Probability\left(D>D_{observed}\right)=Q_{KS}\left(Q_{KS}\left(\sqrt{N}+0.12+\frac{0.11}{\sqrt{N}}\right)D\right),$(21)

see formulas 14.3.5 and 14.3.9 in [15] . If $P_{KS}\ge 0.1$, then the goodness of the fit is believable.

The first catalog of cosmic voids can be found in [16] , where the effective radius of the voids, $R_{eff}$, has been derived to be

$R_{eff}=18.23h^{-1}\text{Mpc}$ Pam et al. 2012.(22)

The second catalog is that with radii up to redshift $0.12h^{-1}$ Mpc in (SDSS-DR7), see [17] ,

$R_{eff}=11.85h^{-1}\text{Mpc}$ Varela et al. 2012.(23)

The third catalog is that of the Baryon Oscillation Spectroscopic Survey, see [18] ,

$R_{eff}=57.53h^{-1}\text{Mpc}$ Mao et al. 2017.(24)

The fourth catalog is that of the VAST void catalog for SDSS DR7 which uses three algorithms, VoidFinder, V²/VIDE and V²/REVOLVER in the framework of two cosmologies, Planck2018 and WMAP5, see [1] . The data with Cartesian coordinates and effective radius $R_{eff}$ in Mpc are available at the following address https://zenodo.org/records/11043278 . A graphical display of the voids is reported in Figure 1 .

The statistics for the lognormal distribution of the VAST catalog are given in Table 1 , for the truncated lognormal distribution in Table 2 , for the MLP distribution in Table 3 and for the truncated MLP distribution in Table 4 .

We now report the best results for the three cases here analysed as empirical and theoretical PDFs: Figure 2 reports the WMAP5 and VoidFinder data in the case of the truncated MLP distribution, Figure 3 reports the Planck 2018 and V²/VIDE data in the case of the truncated lognormal distribution and Figure 4 reports the Planck 2018 and V²/REVOLVER data in the case of the truncated lognormal distribution.

Let $X$ be a random variable taking values $x$ in the interval $\left[0,\infty \right]$; the gamma PDF is

$f\left(x;b,c\right)=\frac{{\left(\frac{x}{b}\right)}^{c-1}{\text{e}}^{-\frac{x}{b}}}{b\Gamma \left(c\right)}$(25)

where

$\Gamma \left(z\right)={\displaystyle {\int }_0^{\infty }{\text{e}}^{-t}{t}^{z-1}\text{d}t},$(26)

is the gamma function, $b>0$ is the scale parameter and $c>0$ is the shape parameter, see formula (17.23) in [7] . Its expected value is

$\mu \left(x;b,c\right)=bc.$(27)

We now re-scale the above PDF in such a way that the expected value is 1 and the gamma variate, $H\left(x;c\right)$, ( [5] ) is obtained

$H\left(x;c\right)=\frac{c}{\Gamma \left(c\right)}{\left(cx\right)}^{c-1}\exp \left(-cx\right),$(28)

where $0\le x<\infty$, $c>0$. The DF of the Kiang function is

$F\left(x;c\right)=1-\frac{\Gamma \left(c,cx\right)}{\Gamma \left(c\right)},$(29)

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

$\Gamma \left(a,z\right)={\displaystyle {\int }_z^{\infty }{t}^{a-1}{\text{e}}^{-t}\text{d}t},$(30)

see [9] . The Kiang PDF has a mean, $\mu$, of

$\mu =1,$(31)

the variance is

${\sigma }^2=\frac{1}{c},$(32)

the skewness is

$\text{skewness}=\frac{2}{\sqrt{c}},$(33)

the kurtosis is

$\text{kurtosis}=\frac{3c+6}{c},$(34)

and the mode is at

$\text{mode}=\frac{c-1}{c}.$(35)

An approximate expression for the median can be obtained by an order 3 Taylor series for the DF about the mean

$\text{median}=-\frac{-2c^c{\text{e}}^{-c}+\sqrt{{\left(c^c\right)}^2{\left({\text{e}}^{-c}\right)}^2-2{\text{e}}^{-c}{c}^c\Gamma \left(c,c\right)+{\text{e}}^{-c}{c}^c\Gamma \left(c\right)}}{c^c{\text{e}}^{-c}}$(36)

The percent error of the above median is 0.013% at $c=2$.

In the case of a 1D Poissonian Voronoi tessellation (PVT), $c=2$ is an exact analytical result, but $c$ is supposed to be 4 or 6 for 2D or 3D PVTs, respectively, the so called Kiang conjecture [5] . The value $c=6$ in 3D was successively refined to 5.5 due to a change in the generator of random numbers [19] , or to 5.78 as deduced by [20] . A first method to derive the numerical value of $c$ is to equalize the variance of the sample, var, with the theoretical variance

$\text{var}={\sigma }^2.$(37)

A second method to derive the numerical value of $c$ is the maximum likelihood (MLE) method, which maximizes

$\Lambda =nc\ln \left(c\right)-n\ln \left(\Gamma \left(c\right)\right)+{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(\ln \left(x_j\right)c-c{x}_j-\ln \left(x_j\right)\right)},$(38)

where $n$ is the number of elements in the sample $x_j$. The value of $c$ is found solving the following non-linear equation

$\frac{\partial \Lambda }{\partial c}=n\ln \left(c\right)+n-n\Psi \left(c\right)+{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(\ln \left(x_j\right)-x_j\right)}=0,$(39)

where $\Psi \left(z\right)$ is the digamma or Psi function defined as

$\Psi \left(z\right)={\Gamma }^{\prime }\left(z\right)/\Gamma \left(z\right),$(40)

where $ℛz>0$, see [9] .

Figure 5 reports a typical evaluation of the volume distribution of the Voronoi diagrams in the presence of Poissonian seeds.

We approximate the volume of each cell of the Voronoi diagrams, $v$, with

$v=\frac{4}{3}\pi R^3.$(41)

The Kiang function in volumes is

$H\left(v;c\right)=\frac{c{\left(cv\right)}^{c-1}{\text{e}}^{-cv}}{\Gamma \left(c\right)},$(42)

and that in radius, $R$,

$H\left(R;c\right)=\frac{4c{\left(\frac{4c\pi R^3}{3}\right)}^{c-1}{\text{e}}^{-\frac{4c\pi R^3}{3}}\pi R^2}{\Gamma \left(c\right)},$(43)

where $c$ is the variable to be found from the data of the volumes. We now introduce the scale $b$: the scaled PDF is

$f\left(R;b,c\right)=\frac{4c{\left(\frac{4c\pi R^3}{3}\right)}^{c-1}{\text{e}}^{-\frac{4c\pi R^3}{3}}\pi R^2}{\Gamma \left(c\right)},$(44)

and the scaled DF is

$F\left(R;b,c\right)=1+\frac{4^c{\pi }^c{3}^{-c}{\left(R^3\right)}^c{b}^{-3c}{c}^c{\text{e}}^{-\frac{4c\pi R^3}{3b^3}}-\Gamma \left(c+1,\frac{4c\pi R^3}{3b^3}\right)}{c\Gamma \left(c\right)}.$(45)

The above distribution has a mean

$\mu \left(b,c\right)=b\Gamma \left(c+\frac{1}{3}\right)6^{1/3},$(46)

variance

${\sigma }^2\left(b,c\right)=b^2\left(\Gamma \left(c+\frac{2}{3}\right)\Gamma \left(c\right)-\Gamma {\left(c+\frac{1}{3}\right)}^2\right)6^{2/3},$(47)

skewness

$\text{skewness}=\Gamma {\left(c\right)}^{3}c-3\Gamma \left(c+\frac{2}{3}\right)\Gamma \left(c+\frac{1}{3}\right)\Gamma \left(c\right)+2\Gamma {\left(c+\frac{1}{3}\right)}^3,$(48)

and kurtosis

$\text{kurtosis}=\frac{N}{D},$(49)

where

$\begin{array}{c}N=3\Gamma \left(c+\frac{4}{3}\right)\times (-2{\left(c+\frac{1}{3}\right)}^2\Gamma \left(c+\frac{4}{3}\right)\Gamma \left(c\right)\Gamma \left(c+\frac{2}{3}\right)\\ +\Gamma {\left(c+\frac{4}{3}\right)}^3+{\left(c+\frac{1}{3}\right)}^3\left(c-\frac{1}{9}\right)\Gamma {\left(c\right)}^3),\end{array}$(50)

and

$D={\left({\left(c+\frac{1}{3}\right)}^2\Gamma \left(c\right)\Gamma \left(c+\frac{2}{3}\right)-\Gamma {\left(c+\frac{4}{3}\right)}^2\right)}^2.$(51)

An approximate expression for the median can be obtained by an order two Taylor series for the DF about the mean:

$\begin{array}{c}\text{median}=\frac{1}{12{\pi }^{1/3}{c}^{4/3}\Gamma \left(c\right)}\times (-b\Gamma \left(c+\frac{1}{3}\right)(c\Gamma {\left(c\right)}^{3c+1}\Gamma {\left(c+\frac{1}{3}\right)}^{-3c}{\text{e}}^{\frac{\Gamma {\left(c+\frac{1}{3}\right)}^3}{\Gamma {\left(c\right)}^3}}\\ -2\Gamma \left(c+1,\frac{\Gamma \text{}{\left(c+\frac{1}{3}\right)}^3}{\Gamma {\left(c\right)}^3}\right){\text{e}}^{\frac{\Gamma {\left(c+\frac{1}{3}\right)}^3}{\Gamma {\left(c\right)}^3}}\Gamma {\left(c\right)}^{3c}\Gamma {\left(c+\frac{1}{3}\right)}^{-3c}-6c+2)6^{1/3}).\end{array}$(52)

One way to deduce the parameters associates $b$ with the maximum of the sample; $c$ is found by solving the non-linear Equation (37) with the theoretical variance as given by Equation (47). Another way, the MLE method, maximizes

$\begin{array}{c}\Lambda =2nc\ln \left(2\right)-nc\ln \left(3\right)+nc\ln \left(\pi \right)-3nc\ln \left(b\right)+nc\ln \left(c\right)+n\ln \left(3\right)\\ -n\ln \left(\Gamma \left(c\right)\right)+{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(3c\ln \left(x_j\right)-\ln \left(x_j\right)-\frac{4c\pi x_j^3}{3b^3}\right)}\\ =0.\end{array}$(53)

The values of $b$ and $c$ are found by solving the two following non-linear equations:

$\frac{\partial \Lambda }{\partial b}=\frac{c\left(-3n{b}^3+4\pi \left({\displaystyle \underset{j=1}{\overset{n}{\sum }}{x}_j^3}\right)\right)}{b^4}=0,$(54)

and

$\begin{array}{c}\frac{\partial \Lambda }{\partial c}=2n\ln \left(2\right)-n\ln \left(3\right)+n\ln \left(\pi \right)-3n\ln \left(b\right)+n\ln \left(c\right)+n-n\Psi \left(c\right)\\ -\frac{{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(4\pi x_j^3-9\ln \left(x_j\right)b^3\right)}}{3b^3}\\ =0.\end{array}$(55)

The statistics for the Kiang function in radius of the VAST catalog are given in Table 5 .

As a visual example, Figure 6 reports the Planck 2018 and V²/REVOLVER data in the case of the Kiang distribution in radius.

The truncated gamma variate $H_T\left(x;c,x_l,x_u\right)$ is

$H_T\left(x;c,x_l,x_u\right)=\frac{c{\left(cx\right)}^{c-1}{\text{e}}^{-cx}}{-\Gamma \left(c,c{x}_u\right)+\Gamma \left(c,c{x}_l\right)},$(56)

where $x_l\le x<x_u$, $c>0$.

The truncated distribution function is

$F_T\left(x;c,x_l,x_u\right)=\frac{{\text{e}}^{-c{x}_l}{c}^c{x}_l^c-c^c{x}^c{\text{e}}^{-cx}+\Gamma \left(c+1,cx\right)-\Gamma \left(c+1,c{x}_l\right)}{c\left(\Gamma \left(c,c{x}_u\right)-\Gamma \left(c,c{x}_l\right)\right)}.$(57)

The truncated Kiang PDF has a mean, ${\mu }_T$, given by

${\mu }_T\left(c,x_l,x_u\right)=\frac{-\Gamma \left(c+1,c{x}_l\right)+\Gamma \left(c+1,c{x}_u\right)}{c\left(\Gamma \left(c,c{x}_u\right)-\Gamma \left(c,c{x}_l\right)\right)},$(58)

and the $r$th moment about the origin is, ${{\mu }^{\prime }}_r$, is

$\begin{array}{l}{{\mu }^{\prime }}_r\left(c,x_l,x_u\right)\\ =\frac{-x_u^{c+r}{\text{e}}^{-c{x}_u}{c}^c+x_l^{c+r}{\text{e}}^{-c{x}_l}{c}^c-c^{-r}\Gamma \left(1+c+r,c{x}_l\right)+c^{-r}\Gamma \left(1+c+r,c{x}_u\right)}{\left(\Gamma \left(c,c{x}_u\right)-\Gamma \left(c,c{x}_l\right)\right)\left(c+r\right)}.\end{array}$(59)

The variance is

$\begin{array}{c}{\sigma }^2=-\frac{{\left(-c^c{\text{e}}^{-c{x}_u}{x}_u^{c+1}+c^c{\text{e}}^{-c{x}_l}{x}_l^{c+1}-\frac{\Gamma \left(c+2,c{x}_l\right)}{c}+\frac{\Gamma \left(c+2,c{x}_u\right)}{c}\right)}^2}{{\left(\Gamma \left(c,c{x}_u\right)-\Gamma \left(c,c{x}_l\right)\right)}^2{\left(c+1\right)}^2}\\ +\frac{-c^c{\text{e}}^{-c{x}_u}{x}_u^{c+2}+c^c{\text{e}}^{-c{x}_l}{x}_l^{c+2}-\frac{\Gamma \left(3+c,c{x}_l\right)}{c^2}+\frac{\Gamma \left(3+c,c{x}_u\right)}{c^2}}{\left(\Gamma \left(c,c{x}_u\right)-\Gamma \left(c,c{x}_l\right)\right)\left(c+2\right)}.\end{array}$(60)

One way to find $c$ uses Equation (37) in which ${\sigma }^2$ is given by the previous equation. Another way to find $c$ is the MLE method, which maximizes

$\Lambda =nc\ln \left(c\right)+n\ln \left(\frac{1}{-\Gamma \left(c,c{x}_u\right)+\Gamma \left(c,c{x}_l\right)}\right)+{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(\ln \left(x_i\right)c-c{x}_i-\ln \left(x_i\right)\right)}.$(61)

In this case, $\frac{\partial \Lambda }{\partial c}$ has a complicated expression which is not reported. As an application, Figure 7 reports a typical evaluation of the volume distribution of the Voronoi diagrams in the presence of Poissonian seeds.

We now introduce the truncation in the radius distribution with scale, see Equation (44). The truncated PDF is

$f_T\left(R;b,c,R_l,R_u\right)=\frac{4c^2{\left(\frac{4c\pi R^3}{3b^3}\right)}^{c-1}{\text{e}}^{-\frac{4c\pi R^3}{3b^3}}\pi R^2}{k},$(62)

where $R_l\le R<R_u$, $c>0$, $b>0$ and

$\begin{array}{l}k=4^c{3}^{-c}{\pi }^c{c}^c{b}^{-3c}{\left(R_u^c\right)}^3{\text{e}}^{-\frac{4c\pi R_u^3}{3b^3}}{b}^3-4^c{3}^{-c}{\pi }^c{c}^c{b}^{-3c}{\left(R_l^c\right)}^3{\text{e}}^{-\frac{4c\pi R_l^3}{3b^3}}{b}^3\\ +\Gamma \left(c+1,\frac{4c\pi R_l^3}{3b^3}\right)b^3-\Gamma \left(c+1,\frac{4c\pi R_u^3}{3b^3}\right)b^3.\end{array}$(63)

The distribution function is

$\begin{array}{l}{F}_T\left(R;b,c,R_l,R_u\right)\\ =\frac{b^3\left(4^c{\pi }^c{3}^{-c}{\left(R^3\right)}^c{b}^{-3c}{c}^c{\text{e}}^{-\frac{4c\pi R^3}{3b^3}}+\Gamma \left(c+1\right)-\Gamma \left(c+1,\frac{4c\pi R^3}{3b^3}\right)\right)}{k}\\ -\frac{b^3\left(4^c{\pi }^c{3}^{-c}{\left(R_l^3\right)}^c{b}^{-3c}{c}^c{\text{e}}^{-\frac{4c\pi R_l^3}{3b^3}}+\Gamma \left(c+1\right)-\Gamma \left(c+1,\frac{4c\pi R_l^3}{3b^3}\right)\right)}{k},\end{array}$(64)

and the average value

$\begin{array}{c}{\mu }_T\left(b,c,R_l,R_u\right)=\frac{1}{\left(6ck+2k\right){\pi }^{1/3}}\times (3b^4{c}^{2/3}\Gamma \left(\frac{4}{3}+c,\frac{4c\pi R_l^3}{3b^3}\right)6^{1/3}\\ -3b^4{c}^{2/3}\Gamma \left(\frac{4}{3}+c,\frac{4c\pi R_u^3}{3b^3}\right)6^{1/3}\\ -6b^{-3c+3}{c}^{c+1}{\pi }^{c+\frac{1}{3}}{4}^c{3}^{-c}\left(-{\text{e}}^{-\frac{4c\pi R_u^3}{3b^3}}{R}_u^{3c+1}+{\text{e}}^{-\frac{4c\pi R_l^3}{3b^3}}{R}_l^{3c+1}\right)).\end{array}$(65)

The median can be found by numerically solving the equation

$F_T\left(R;b,c,R_l,R_u\right)=\frac{1}{2}.$(66)

The statistics for the truncated Kiang function in radius of the VAST catalog are given in Table 6 , the high values of the probability do not mean overfitting or limitations in the data-model fit. Figure 8 reports the Planck 2018 and V²/REVOLVER data.