_{1}

^{*}

We demonstrate that certain astrophysical distributions can be modelled with the truncated Weibull distribution, which can lead to some insights: in particular, we report the average value, the
*r*th moment, the variance, the median, the mode, the generation of random numbers, and the evaluation of the two parameters with maximum likelihood estimators. The first application of the Weibull distribution is the initial mass function for stars. The magnitude version of the Weibull distribution is applied to the luminosity function for the Sloan Digital Sky Survey (SDSS) galaxies and to the photometric maximum of the 2MASS Redshift Survey (2MRS) galaxies. The truncated Weibull luminosity function allows us to model the average value of the absolute magnitude as a function of the redshift for the 2MRS galaxies.

The Weibull distribution was originally introduced to model the fracture strength of brittle and quasi-brittle materials, see [

Let X be a random variable defined in [ 0, ∞ ] ; the two/parameter Weibull distribution function (DF), F ( x ) , is

F ( x ; b , c ) = 1 − e − ( x b ) c , (1)

where b and c, both positive, are the scale and the shape parameters, see [

f ( x ; b , c ) = c x c − 1 e − ( x b ) c b c . (2)

We now introduce the function

Γ i = Γ ( 1 + i / c ) , (3)

the average value or mean, μ , is

μ ( b , c ) = b Γ 1 , (4)

the variance, σ 2 , is

σ 2 ( b , c ) = b 2 ( − Γ 1 2 + Γ 2 ) , (5)

the skewness is

skewness ( b , c ) = 2 Γ 1 3 − 3 Γ 2 Γ 1 + Γ 3 ( Γ 1 2 − Γ 2 ) 2 − Γ 1 2 + Γ 2 , (6)

and the kurtosis

kurtosis ( b , c ) = − 3 Γ 1 4 − 6 Γ 2 Γ 1 2 + 4 Γ 1 Γ 3 − Γ 4 ( − Γ 1 2 + Γ 2 ) 2 . (7)

The rth moment about the origin for the Weibull distribution, μ ′ r , is

μ ′ r ( b , c ) = b r Γ ( c + r c ) , (8)

where r is an integer and

Γ ( z ) = ∫ 0 ∞ e − t t z − 1 d t , (9)

is the gamma function, see [

e ln ( ln ( 2 ) ) + c ln ( b ) c , (10)

and the mode is at

c − 1 c c b . (11)

Random generation of the Weibull variate X is given by

X : b , c ≈ − ln ( 1 − R ) c b (12)

where R is the unit rectangular variate. The two parameters b and c can be derived by the numerical solution of the two following equations which arise from the maximum likelihood estimator (MLE)

c b ( ∑ i = 1 n ( x i b ) c − n ) = 0 , (13a)

− n ln ( b ) + n c + ∑ i = 1 n − ( x i b ) c ln ( x i b ) + ln ( x i ) = 0 , (13b)

where x i are the elements of the experimental sample with i varying between 1 and n.

Let X be a random variable defined in [ x l , x u ] ; the truncated two-parameter Weibull DF, F T ( x ) , is

F T ( x ; b , c , x l , x u ) = − e − ( x b ) c + e − ( x l b ) c − e − ( x u b ) c + e − ( x l b ) c , (14)

and the PDF, f T ( x ) , is

f T ( x ; b , c , x l , x u ) = − ( x b ) c c e − ( x b ) c x ( e − ( x u b ) c − e − ( x l b ) c ) , (15)

see Section 2.1 in [

The inequality which fixes the range of existence is ∞ > x u > x > x l > 0 . We report the indefinite integral which characterizes the average value or mean, μ T ,

I ( b , c , x l , x u , x ) = ∫ x f T ( x ; b , c ) d x , (16)

which is

I ( b , c , x l , x u , x ) = I N I D , (17)

where

I N = − 2 c x e − 1 2 b − c x c + c ( − ln ( x ) + ln ( b ) ) b 1 2 − c ( 2 ( 1 2 + c ) 2 b c M 1 2 2 c + 1 c , 1 2 3 c + 1 c ( b − c x c ) + c ( ( 1 2 + c ) b c + 1 2 c x c ) M 1 2 c − 1 , 1 2 3 c + 1 c ( b − c x c ) ) , (18)

and

I D = ( 1 + c ) ( 2 c + 1 ) ( 3 c + 1 ) ( e − x u c b − c − e − x l c b − c ) (19)

where M μ , ν ( z ) is the Whittaker M function, see [

μ ( b , c , x l , x u ) = I ( b , c , x l , x u , x = x u ) − I ( b , c , x l , x u , x = x l ) , (20)

for a comparison, see Equation (5) in [

M ( b , c , x l , x u , x ) = ∫ x r f T ( x ; b , c ) d x , (21)

which is

M ( b , c , x l , x u , x ) = M N ( − e − x u c b − c + e − x l c b − c ) ( c + r ) ( 2 c + r ) ( 3 c + r ) , (22)

where

M N = 2 c e − 1 / 2 b − c x c + c ( − ln ( x ) + ln ( b ) ) ( 2 x r / 2 b r / 2 ( c + r / 2 ) 2 M 1 + 1 / 2 r c , 3 / 2 + 1 / 2 r c ( b − c x c ) + M 1 / 2 r c , 3 / 2 + 1 / 2 r c ( b − c x c ) c ( 1 / 2 c x c + r / 2 b r / 2 − c + b r / 2 x r / 2 ( c + r / 2 ) ) ) . (23)

The rth moment about the origin for the truncated Weibull distribution is therefore

μ ′ r , t = M ( b , c , x l , x u , x = x u ) − M ( b , c , x l , x u , x = x l ) . (24)

The variance, σ T 2 ( b , c , x l , x u ) , of the truncated Weibull distribution is given by

σ T 2 ( b , c , x l , x u ) = μ ′ 2, t − ( μ ′ 1, t ) 2 . (25)

The m e d i a n T in the case x u > m e d i a n T > x l is at

m e d i a n T = ( x u c + x l c ) b − c − ln ( e x u c b − c 2 + e x l c b − c 2 ) c b , (26)

and the m o d e T in the case x u > m o d e T > x l is at

m o d e T = c − 1 c c b , (27)

which is the same value as that for the Weibull pdf. Random generation of the truncated Weibull variate X is given by

X : b , c , x l , x u ≈ ( x u c + x l c ) b − c − ln ( − R e x u c b − c + R e x l c b − c + e x u c b − c ) c b , (28)

where R is the unit rectangular variate. The four parameters x_{l}, x_{u}, b and c can be obtained in the following way. Consider a sample X = x 1 , x 2 , ⋯ , x n and let x ( 1 ) ≥ x ( 2 ) ≥ ⋯ ≥ x ( n ) denote their order statistics, so that x ( 1 ) = max ( x 1 , x 2 , ⋯ , x n ) , x ( n ) = min ( x 1 , x 2 , ⋯ , x n ) . The first two parameters x l and x u are

x l = x ( n ) , x u = x ( 1 ) . (29)

The MLE is obtained by maximizing

Λ = ∑ i n ln ( f T ( x ; b , c , x l , x u ) ) . (30)

The two derivatives ∂ Λ ∂ b = 0 and ∂ Λ ∂ c = 0 generate two non-linear equations in b and c which are

N 1 ( − e − ( x u b ) c + e − ( x l b ) c ) b = 0, (31a)

N 2 ( − e − ( x u b ) c + e − ( x l b ) c ) c = 0, (31b)

where

N 1 = ( ( − e − ( x u b ) c + e − ( x l b ) c ) ∑ i = 1 n ( x i b ) c + ( ( − ( x l b ) c − 1 ) e − ( x l b ) c + e − ( x u b ) c ( ( x u b ) c + 1 ) ) n ) c , (32)

and

N 2 = ( − e − ( x u b ) c + e − ( x l b ) c ) c ∑ i = 1 n − ( x i b ) c ln ( x i b ) + ln ( x i ) + n ( ( ( x l b ) c ln ( x l b ) c − c ln ( b ) + 1 ) e − ( x l b ) c − e − ( x u b ) c ( ( x u b ) c ln ( x u b ) c − c ln ( b ) + 1 ) ) . (33)

This section reports the luminosity functions (LFs) for the Weibull distribution and the truncated Weibull distribution.

The Schechter function, introduced by [

Φ ( L ; α , L * , Φ * ) d L = ( Φ * L * ) ( L L * ) α exp ( − L L * ) d L , (34)

here α sets the slope for low values of L, L * is the characteristic luminosity and Φ * is the normalization. The equivalent distribution in absolute magnitude is

Φ ( M ) d M = 0.921 Φ * 10 0.4 ( α + 1 ) ( M * − M ) exp ( − 10 0.4 ( M * − M ) ) d M , (35)

where M * is the characteristic magnitude as derived from the data. We now introduce the parameter h, which is H_{0}/100, where H_{0} is the Hubble constant. The scaling with h is M * − 5 log 10 h and Φ * h 3 [ Mpc − 3 ] . In order to derive the Weibull LF we start from the PDF as given by Equation (2),

Ψ ( L ; c , L * , Ψ * ) d L = Ψ * ( L L * ) c c e − ( L L * ) c L d L , (36)

where L is the luminosity, L * is the characteristic luminosity and Ψ * is the normalization and the version in absolute magnitude is

Ψ ( M ; c , M * , Ψ * ) d M = 0.4 Ψ * 10 ( − 0.4 M + 0.4 M * ) c c e − 10 ( − 0.4 M + 0.4 M * ) c ln ( 10 ) d M . (37)

We start with the truncated Weibull PDF with scaling as given by Equation (15)

Ψ ( L ; c , L * , Ψ * , L l , L u ) d L = Ψ * − ( L L * ) c c e − ( L L * ) c L ( e − ( L u L * ) c − e − ( L l L * ) c ) d L , (38)

where L is the luminosity, L * is the characteristic luminosity, L l is the lower boundary in luminosity, L u is the upper boundary in luminosity, and Ψ * is the normalization. The magnitude version is

Ψ ( M ; c , M * , Ψ * , M l , M u ) d M = Ψ * − 0.4 ( 10 0.4 M * − 0.4 M ) c c e − ( 10 0.4 M * − 0.4 M ) c ( ln ( 2 ) + ln ( 5 ) ) e − ( 10 − 0.4 M l + 0.4 M * ) c − e − ( 10 0.4 M * − 0.4 M u ) c d M , (39)

where M is the absolute magnitude, M * the characteristic magnitude, M l the lower boundary in magnitude, M u the upper boundary in magnitude and Ψ * is the normalization. The mean theoretical absolute magnitude, 〈 M 〉 , can be evaluated as

〈 M 〉 = ∫ M l M u M × Ψ ( M ; c , M * , Ψ * , M l , M u ) d M ∫ M l M u Ψ ( M ; c , M * , Ψ * , M l , M u ) d M . (40)

This section reviews the adopted statistics, applies the truncated Weibull distribution to the initial mass function (IMF) for stars, models the LF for galaxies and QSOs, explains the photometric maximum in the number of galaxies of the 2MRS, and traces the cosmological evolution of the average absolute magnitude.

The merit function χ 2 is computed according to the formula

χ 2 = ∑ i = 1 n ( T i − O i ) 2 T i , (41)

where n is the number of bins, T i is the theoretical value, and O i is the experimental value represented by the frequencies. The theoretical frequency distribution is given by

T i = N Δ x i p ( x ) , (42)

where N is the number of elements of the sample, Δ x i is the magnitude of the size interval, and p ( x ) is the PDF under examination.

A reduced merit function χ r e d 2 is given by

χ r e d 2 = χ 2 / N F , (43)

where N F = n − k is the number of degrees of freedom, n is the number of bins, 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 [

The Akaike information criterion (AIC), see [

AIC = 2 k − 2 ln ( L ) , (44)

where L is the likelihood function and k 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 χ 2 statistic L ∝ exp ( − χ 2 2 ) where χ 2 has been computed by Equation (41), see [

AIC = 2 k + χ 2 . (45)

The Kolmogorov-Smirnov test (K-S), see [_{KS}, see formulas 14.3.5 and 14.3.9 in [

We tested the truncated Weibull distribution on four samples of stars: NGC 2362 (271 stars), the young cluster NGC 6611 (207 stars), the γ Velorum cluster (237 stars), and the young cluster Berkeley 59 (420 stars), for more details, see Section 5.2 of [

Graphical displays of the empirical PDF visualized through histograms as well as the theoretical PDF for NGC 6611 are reported in

A test has been performed on the u * band of SDSS as in [

A careful examination of

Another case is the LF for QSO in the case 0.3 < z < 0.5 , see [

Cluster | parameters | AIC | χ r e d 2 | Q | D | P_{KS} | LN |
---|---|---|---|---|---|---|---|

NGC 2362 | b = 0.726 , c = 2.2 , x l = 0.12 , x u = 1.47 | 39.5 | 1.96 | 0.011 | 0.011 | 0.576 | N |

NGC 6611 | b = 0.483 , c = 1.011 , x l = 0.019 , x u = 1.46 | 47.77 | 2.48 | 8.4 × 10^{−}^{4} | 0.059 | 0.45 | Y |

γ Velorum | b = 0.153 , c = 0.745 , x l = 0.158 , x u = 1.317 | 31.24 | 1.45 | 0.107 | 0.063 | 0.292 | Y |

Berkeley 59 | b = 0.347 , c = 1.143 , x l = 0.16 , x u = 2.24 | 83.71 | 4.73 | 9.74 × 10^{−}^{10} | 0.122 | 6.35 × 10^{−}^{6} | N |

LF | parameters | χ r e d 2 |
---|---|---|

Schechter | M * = − 17.92 , α = − 0.9 , Φ * = 0.03 / Mpc 3 | 0.689 |

Weibull | M * = − 16.69 , c = 0.728 , Ψ * = 0.0718 / Mpc 3 | 0.650 |

M * | Ψ * | α | χ 2 | χ r e d 2 | Q | AIC |
---|---|---|---|---|---|---|

−23.75 | 8.85 × 10^{−}^{7} | −1.37 | 10.49 | 1.49 | 0.162 | 16.49 |

M * | Ψ * | α | χ 2 | χ r e d 2 | Q | AIC |
---|---|---|---|---|---|---|

−20.566 | 9.26 × 10^{−}^{6} | 0.471 | 10.08 | 1.44 | 0.183 | 16.08 |

has smaller χ r e d 2 compared to the Schechter LF.

In the pseudo-Euclidean universe, the correlation between the expansion velocity and distance is

V = H 0 D = c l z , (46)

where H 0 is the Hubble constant, H 0 = 100 h km ⋅ s − 1 ⋅ Mpc − 1 , with h = 1 when h is not specified, D is the distance in Mpc, c l is the speed of light and z is the redshift. In the pseudo-Euclidean universe, the flux of radiation, f, expressed in units of L ⊙ Mpc 2 , where L ⊙ represents the luminosity of the sun, is

f = L 4 π D 2 , (47)

where D represents the distance of the galaxy expressed in Mpc, and

D = c l z H 0 . (48)

The joint distribution in z and f for a generic LF, Φ ( z 2 z c r i t 2 ) is

d N d Ω d z d f = 4 π ( c l H 0 ) 5 z 4 Φ ( z 2 z c r i t 2 ) , (49)

where d Ω , d z and d f represent the differentials of the solid angle, the redshift, and the flux, respectively, and

z c r i t 2 = H 0 2 L * 4 π f c l 2 (50)

where L * is the characteristic luminosity, for more details, see [

d N ( z ; c , Ψ * , z c r i t ) d Ω d z d f = 4 z 2 c l 5 Ψ * ( z 2 z c r i t 2 ) c c π z c r i t 2 e − ( z 2 z c r i t 2 ) c H 0 5 L * . (51)

The above number of galaxies in z and f has a maximum at z = z max which is the solution of the following non-linear equation

− 8 z c l 5 Ψ ( z 2 z c r i t 2 ) c c π z c r i t 2 e − ( z 2 z c r i t 2 ) c ( ( z 2 z c r i t 2 ) c c − c − 1 ) = 0. (52)

A first numerical evaluation of the position in z of the above equation is reported in units of z c r i t , see the blue dashed line in

c. As an example when c = 1 / 2 , the nonlinear equation for the photometric maximum is

− 2 z c l 5 Ψ * z 2 z c r i t 2 π z c r i t 2 e − z 2 z c r i t 2 ( z 2 z c r i t 2 − 3 ) = 0, (53)

which has a physical solution at

z max = 3 z c r i t . (54)

A third approximate result is obtained using a Taylor expansion of Equation (52) around z = 2 z c r i t of order 3, which gives

z max = z c r i t × 2464 c c 3 − 28 c 3 16 c + 2 2 c + 2 c 3 − 4 c 3 256 c + 464 c c 2 − 12 c 2 16 c + 2 2 c + 2 c 2 + c 16 c − c 4 c − A − 4 c c ( 1264 c c 2 − 2 c 2 256 c − 14 c 2 16 c + 2 c 2 4 c + 3 c 64 c − 9 c 16 c + 3 c 4 c − 16 c + 4 c ) , (55)

where

A = ( − 4 c 4 16 c + 40 × 64 c c 4 + 32 × 1024 c c 4 + 56 × 64 c c 3 − 48 c 3 256 c − 8 c 3 16 c + 14 × 64 c c 2 − 3 c 2 16 c + 2 c 16 c − 3 c 2 256 c − 2 c 64 c + 8 × 1024 c c 3 − 60 × 256 c c 4 − 4 × 4096 c c 4 + 16 c ) 1 / 2 . (56)

A graphical display of the Taylor solution is reported in

The mean redshift for galaxies 〈 z 〉 is

〈 z 〉 = ∫ 0 ∞ z d N d Ω d z d f d z ∫ 0 ∞ d N d Ω d z d f d z . (57)

The mean redshift for the Weibull LF as a function of z c r i t when c = 1 / 2 is

〈 z 〉 ( z c r i t ) = 4 z c r i t when c = 1 / 2 , (58)

or as a function of the flux

〈 z 〉 ( f ) = 2 π f 10 0.4 M ⊙ − 0.4 M * H 0 π f c l when c = 1 / 2 , (59)

where M ⊙ = 3.39 is the reference magnitude of the sun at the considered bandpass, or as a function of the apparent magnitude

〈 z 〉 ( m ) = 4 × 10 − 5 e 0.921 M ⊙ − 0.921 m 10 0.4 M ⊙ − 0.4 M * H 0 e 0.921 M ⊙ − 0.921 m c l when c = 1 / 2 . (60)

The absolute magnitude which can be observed as a function of the limiting apparent magnitude, m L , is

M L = m L − 5 log 10 ( c z H 0 ) − 25 , (61)

where m L = 11.75 for the 2MRS catalog.

The theoretical average absolute magnitude of the truncated Weibull LF, see Equation (40), can be compared with the observed average absolute magnitude of the 2MRS as a function of the redshift. To fit the data, we assumed the following empirical dependence on the redshift for the characteristic magnitude of the truncated Weibull LF.

M * = − 25.14 + 4 ( 1 − ( z − z min z max − z min ) 0.7 ) . (62)

This relationship models the decrease of the characteristic absolute magnitude as a function of the redshift and allows us to match the observational and theoretical data. The lower bound in absolute magnitude is given by the minimum magnitude of the selected bin, the upper bound is given by Equation (61), the characteristic magnitude varies according to Equation (62) and

Truncated Weibull distribution

We derived the PDF, the DF, the average value, the rth moment, the variance, the median, the mode, an expression to generate random numbers and the way to obtain the two parameters, b and c, by the MLE for the truncated Weibull distribution.

Weibull luminosity function

We derived the Weibull LF in the standard and the truncated case: the application to both the SDSS Galaxies and to the QSOs in the range of redshift [ 0.3,0.5 ] yields a lower reduced merit function compared to Schechter LF, see

Cosmological applications

The number of galaxies as functions of the redshift, the flux and the solid angle for the Weibull LF in the pseudo-Euclidean universe presents a maximum which can be compared with the observed one for the 2MRS, see

The author declares no conflicts of interest regarding the publication of this paper.

Zaninetti, L. (2021) New Probability Distributions in Astrophysics: V. The Truncated Weibull Distribution. International Journal of Astronomy and Astrophysics, 11, 133-149. https://doi.org/10.4236/ijaa.2021.111008