The gamma-Pareto distribution was introduced by [1], where its most important statistical parameters were derived. The gamma-Pareto probability density function (PDF) models the long right tail characteristics as well the gamma-type left behaviour. The first application was to the monthly rainfall data from Jatiwangi station, Jakarta [2]. A comparison between the generalized linear model (GLM) and the gamma-Pareto was made by [3]. An application to the length of hospital stays was made by [4]. The inter-aircraft distances between two closest flying passenger aircraft were analysed by [5]. Two extensions are the Gamma-Pareto (IV) [6] and the weighted gamma-Pareto distribution (WGPD) [7]. In astrophysics, many phenomena are modeled by a Pareto or power law distribution in the absence of more accurate models. Two examples are the distribution in energy of the cosmic rays (CR) and the radio-flux versus frequency in extra-galactic radio-sources. The above astrophysical models require a distribution that is more flexible at low values of the random variable and therefore Sections 2 and 3 analyse the gamma-Pareto and the gamma-Pareto II distributions which add to the Pareto distribution the flexibility of the gamma distribution. The effect of right-truncation and bi-truncation on gamma-Pareto and gamma-Pareto II distributions are analysed in Sections 4, 5 and 6. Section 7 deals with the transition from a probability distribution to a function. Section 8 applies the obtained results to a sample of the diameters of the asteroids and to the distribution in energy of CR.

The gamma-Pareto has PDF

f ( x ; α , c , θ ) = ( θ x ) 1 c ln ( x θ ) α − 1 x Γ ( α ) c α , (1)

and is defined for α > 0 , c > 0 , θ > 0 and x > θ , see formula (2.1) in [1]. Its distribution function (DF) is

F ( x ; α , c , θ ) = 1 + − Γ ( α + 1, ln ( x ) − ln ( θ ) c ) + x − 1 c θ 1 c c − α ( ln ( x ) − ln ( θ ) ) α α   Γ ( α ) . (2)

Its average value, or mean, μ , is

μ ( α , c , θ ) = θ ( 1 − c ) − α , (3)

its variance, σ 2 , is

σ 2 ( α , c , θ ) = θ 2 ( ( 1 − 2 c ) − α − ( 1 − c ) − 2 α ) , (4)

its rth moment about the origin, μ ′ r , is

μ ′ r ( α , c , θ ) = ( − c r + 1 ) − α θ r , (5)

its skewness, μ ˜ 3 , is

μ ˜ 3 ( α , c , θ ) = − 3 ( − 2 c + 1 ) − α ( − c + 1 ) − α + 2 ( − c + 1 ) − 3 α + ( − 3 c + 1 ) − α ( ( − 2 c + 1 ) − α − ( − c + 1 ) − 2 α ) 3 2 , (6)

its kurtosis, μ ˜ 4 , is

μ ˜ 4 ( α , c , θ ) = N D , (7)

with

N = − 4 ( − c + 1 ) 3 α ( − 3 c + 1 ) − α ( − 2 c + 1 ) 2 α + ( − c + 1 ) 4 α ( − 4 c + 1 ) − α ( − 2 c + 1 ) 2 α     + 6 ( − c + 1 ) 2 α ( − 2 c + 1 ) α − 3 ( − 2 c + 1 ) 2 α , (8)

D = ( ( − c + 1 ) 2 α − ( − 2 c + 1 ) α ) 2 , (9)

and the mode, M o d e , is at

M o d e ( α , c , θ ) = e ( α − 1 ) c c + 1 θ . (10)

Random generation of the variate X is obtained by solving the nonlinear equation

F ( x ; α , c , θ ) = R , (11)

where R is the unit rectangular variate and F is given by Equation (2). Once the elements, x i , of the experimental sample with i varying between 1 and n are given, the parameter θ can be derived from the minimum of the sample minus a small quantity, see the discussion in [1] [8]. The two remaining parameters, α and c, can be derived by the numerical solution of the two following equations, which arise from the maximum likelihood estimator (MLE),

− n α c − n ln ( θ ) + ( ∑ i = 1 n ln ( x i ) ) = 0, (12a)

− n Ψ ( α ) − n ln ( c ) + ( ∑ i = 1 n ln ( ln ( x i ) − ln ( θ ) ) ) = 0, (12b)

where Ψ ( z ) is the digamma or Psi function defined as

Ψ ( z ) = Γ ′ ( z ) / Γ ( z ) , (13)

where R z > 0 , see [9]. The Pareto PDF [10] as defined in [11] is

f ( x ; a , c ) = a c x − c − 1 c , (14)

with c > 0 and a > 0 . Figure 1 compares the PDFs of the Pareto and the gamma-Pareto distributions.

A second PDF for comparison is the lognormal, which, according to [11], is

f L N ( x ; m , σ ) = e − 1 2 σ 2 ( ln ( x m ) ) 2 x σ 2 π , (15)

where m is the median and σ a shape parameter. Figure 2 compares the gamma-Pareto and the lognormal.

The third distribution which will be used for comparison is the double Pareto lognormal, see formula (22) in [12], which has PDF

f ( x ; α , β , μ , σ ) = 1 2 α β ( e 1 2 α ( α σ 2 + 2 μ − 2 ln ( x ) ) e r f c ( 1 2 ( α σ 2 + μ − ln ( x ) ) 2 σ )         + e 1 2 β ( β σ 2 − 2 μ + 2 ln ( x ) ) e r f c ( 1 2 ( β σ 2 − μ + ln ( x ) ) 2 σ ) ) x − 1 ( α + β ) − 1 , (16)

where α and β are the Pareto coefficients for the upper and the lower tail, respectively, μ and σ are the lognormal body parameters, and erfc is the complementary error function. Figure 3 compares the gamma-Pareto and the double Pareto lognormal.

The translation Y = X + θ of a gamma-Pareto PDF, see Equation (1), in the random variable Y produces the gamma-Pareto II PDF

f ( x ; α , c , θ ) = θ 1 c ( x + θ ) − 1 − 1 c ln ( 1 + x θ ) α − 1 c α Γ ( α ) , (17)

which is defined for α > 0 , c > 0 , θ > 0 and x > 0 , see formula (2.4) in Table 1 of [1]. The DF is

F ( x ; α , c , θ ) = 1 + − Γ ( α + 1, ln ( x + θ ) − ln ( θ ) c ) + ( x + θ ) − 1 c θ 1 c c − α ( ln ( x + θ ) − ln ( θ ) ) α α   Γ ( α ) , (18)

its average value is

μ ( α , c , θ ) = θ ( ( 1 − c ) − α − 1 ) , (19)

and the mode is at

M o d e ( α , c , θ ) = θ ( e c ( α − 1 ) c + 1 − 1 ) . (20)

The three parameters α , c , θ are found in the framework of the MLE method by solving the three non-linear equations.

− n α c − n ln ( θ ) + ( ∑ i = 1 n ln ( x i + θ ) ) c 2 = 0 , (21a)

− n ln ( c ) − n Ψ ( α ) + ( ∑ i = 1 n ln ( ln ( x i + θ ) − ln ( θ ) ) ) = 0 , (21b)

n − ( ∑ i = 1 n − θ ( c + 1 ) ln ( x i + θ ) + θ ( c + 1 ) ln ( θ ) − x i c ( α − 1 ) ( − ln ( x i + θ ) + ln ( θ ) ) ( x i + θ ) ) = 0. (21c)

We now analyse a right truncated gamma-Pareto, see Equation (1), with PDF

f T ( x ; α , c , θ , x u ) = θ 1 c x − 1 − c c ln ( x θ ) α − 1 c − α α α   Γ ( α ) − Γ ( α + 1, ln ( ( x u θ ) 1 c ) ) + c − α ln ( x u θ ) α x u − 1 c θ 1 c , (22)

which is defined for α > 0 , c > 0 , 0 < θ < x < x u and the subscript T means truncation. Its DF is

F T ( x ; α , c , θ , x u ) = x u 1 c ( − Γ ( α + 1, ln ( ( x θ ) 1 c ) ) c α + c α Γ ( α + 1 ) + ln ( x θ ) α ( x θ ) − 1 c ) c α x u 1 c Γ ( α + 1 ) − c α x u 1 c Γ ( α + 1, ln ( ( x u θ ) 1 c ) ) + ln ( x u θ ) α θ 1 c , (23)

and its average value is

μ T ( α , c , θ , x u ) = A c α x u 1 c Γ ( α + 1 ) − c α x u 1 c Γ ( α + 1, ln ( ( x u θ ) 1 c ) ) + ln ( x u θ ) α θ 1 c , (24)

where

A = ln   ( x u θ ) α ( Γ ( α + 1 ) c α x u 1 c ln ( x u ( x u θ ) − c θ ) − α θ     − c α x u 1 c ln ( x u ( x u θ ) − c θ ) − α Γ ( α + 1, ln   ( θ ( x u θ ) 1 c x u ) ) θ + x u θ 1 c ) . (25)

The two parameters θ and x u are the minimum and the maximum of the sample. The two remaining parameters α and c are derived by MLE.

The right- and left-truncated gamma-Pareto, see Equation (1), has PDF

f D T ( x ; α , c , θ , x l , x u ) = θ 1 c x − 1 − c c ln ( x θ ) α − 1 c − α Γ ( α ) K , (26)

where

K = 1 α   Γ ( α ) × ( − Γ ( α + 1, ln ( ( x u θ ) 1 c ) ) + Γ ( α + 1, ln ( ( x l θ ) 1 c ) )                 + c − α x u − 1 c θ 1 c ln ( x u θ ) α − c − α x l − 1 c θ 1 c ln ( x l θ ) α ) , (27)

which is defined for α > 0 , c > 0 , 0 < θ < x l < x < x u and the subscript DT means double truncation. The DF is

F D T ( α , c , θ , x l , x u ) = 1 α K Γ ( α ) × ( ln ( x θ ) α c − α ( x θ ) − 1 c − c − α ln ( x l θ ) α ( x l θ ) − 1 c       + Γ ( α + 1, ln ( ( x l θ ) 1 c ) ) − Γ ( α + 1, ln ( ( x θ ) 1 c ) ) ) , (28)

and its average value is

μ D T ( α , c , θ , x l , x u ) = 1 α K Γ ( α ) × ( − Γ ( α ) c − α ln ( x l θ ) α ln ( ( x l θ ) − c − 1 c ) − α α θ

      + Γ ( α ) ln ( x u θ ) α ln ( ( x u θ ) − c x u θ ) − α α θ       + c − α ln ( x l θ ) α Γ ( α + 1, ln ( ( x l θ ) − c − 1 c ) ) ln ( ( x l θ ) − c − 1 c ) − α θ       + c − α θ 1 c ln ( x u θ ) α x u c − 1 c − c − α x l ( x l θ ) − 1 c ln ( x l θ ) α       − ln ( x u θ ) α Γ ( α + 1, ln ( ( x u θ ) − c − 1 c ) ) ln ( ( x u θ ) − c x u θ ) − α θ ) . (29)

The left and right truncated (bi-truncated) version of the gamma-Pareto II PDF, see Equation (18), is

f D T ( x ; α , c , θ , x l , x u ) = θ 1 c ( x + θ ) − 1 − 1 c ln ( 1 + x θ ) α − 1 c α Γ ( α ) K ′ , (30)

which is defined for α > 0 , c > 0 , x l > 0 , x u > 0 , θ > 0 , x l < x < x u and

K ′ = Γ ( α + 1, ln ( ( x l + θ θ ) 1 c ) ) − Γ ( α + 1, ln ( ( x u + θ θ ) 1 c ) )       + c − α ln ( x u + θ θ ) α ( x u + θ ) − 1 c θ 1 c − c − α ln ( x l + θ θ ) α ( x l + θ ) − 1 c θ 1 c . (31)

Its DF is

F D T ( α , c , θ , x l , x u ) = 1 K ′ α Γ ( α ) × c − α ( ln ( x + θ ) − ln ( θ ) ) α ( x + θ ) − 1 c θ 1 c       − c − α ( ln ( x l + θ ) − ln ( θ ) ) α ( x l + θ ) − 1 c θ 1 c       + Γ ( α + 1, ln ( x l + θ ) − ln ( θ ) c ) − Γ ( α + 1, ln ( x + θ ) − ln ( θ ) c ) . (32)

And its average value is

μ D T ( α , c , θ , x l , x u ) = 1 K ′ α Γ ( α ) × c − α ( − ln ( x l + θ θ ) α ln ( ( x l + θ θ ) 1 − c c ) − α Γ ( α ) α θ

      + ln ( x l + θ θ ) α ln ( ( x l + θ θ ) 1 c ) − α Γ ( α ) α θ       + ln ( x u + θ θ ) α ln ( ( x u + θ θ ) 1 − c c ) − α Γ ( α ) α θ       − ln ( x u + θ θ ) α ln ( ( x u + θ θ ) 1 c ) − α Γ ( α ) α θ

+ ln ( x l + θ θ ) α Γ ( α + 1, ln ( ( x l + θ θ ) 1 − c c ) ) ln ( ( x l + θ θ ) 1 − c c ) − α θ − ln ( x l + θ θ ) α Γ ( α + 1, ln ( ( x l + θ θ ) 1 c ) ) ln ( ( x l + θ θ ) 1 c ) − α θ − ln ( x u + θ θ ) α Γ ( α + 1, ln ( ( x u + θ θ ) 1 − c c ) ) ln ( ( x u + θ θ ) 1 − c c ) − α θ + ln ( x u + θ θ ) α Γ ( α + 1, ln ( ( x u + θ θ ) 1 c ) ) ln ( ( x u + θ θ ) 1 c ) − α θ − ln ( x l + θ θ ) α ( x l + θ θ ) − 1 c x l + ln ( x u + θ θ ) α ( x u + θ θ ) − 1 c x u ) . (33)

When the data are presented as y = f ( x ) , an interpolating function is obtained by multiplying a PDF by a constant of normalization ϕ

y ( x ; ϕ ) = ϕ × P D F ( x ) . (34)

The gamma-Pareto function, see Equation (1), is

Ψ ( x ; α , c , θ , ϕ ) = ϕ ( θ x ) 1 c ln ( x θ ) α − 1 x Γ ( α ) c α . (35)

The gamma-Pareto R-truncated function, see Equation (25), is

Ψ T ( x ; α , c , θ , x u , ϕ ) = ϕ θ 1 c x − 1 − c c ln ( x θ ) α − 1 c − α α α   Γ ( α ) − Γ ( α + 1, ln ( ( x u θ ) 1 c ) ) + c − α ln ( x u θ ) α x u − 1 c θ 1 c . (36)

The right and left truncated gamma-Pareto function, see Equation (29), is

Ψ D T ( x ; α , c , θ , x l , x u , ϕ ) = ϕ θ 1 c x − 1 − c c ln ( x θ ) α − 1 c − α Γ ( α ) K . (37)

The gamma-Pareto II function, see Equation (18), is

Ψ I I ( x ; α , c , θ , ϕ ) = ϕ θ 1 c ( x + θ ) − 1 − 1 c ln ( 1 + x θ ) α − 1 c α Γ ( α ) . (38)

The left and right truncated gamma-Pareto II function, see Equation (33), is

Ψ I I , D T ( x ; α , c , θ , x l , x u , ϕ ) = ϕ θ 1 c ( x + θ ) − 1 − 1 c ln ( 1 + x θ ) α − 1 c α Γ ( α ) K ′ . (39)

The lognormal function, see Equation (16), is

Ψ L N ( x ; m , σ , ϕ ) = ϕ e − 1 2 σ 2 ( ln ( x m ) ) 2 x σ 2 π . (40)

The double Pareto lognormal function, see Equation (17), is

Ψ D P ( x ; α , β , μ , σ , ϕ ) = ϕ 1 2 α β ( e 1 2 α ( α σ 2 + 2 μ − 2 ln ( x ) ) e r f c ( 1 2 ( α σ 2 + μ − ln ( x ) ) 2 σ )       + e 1 2 β ( β σ 2 − 2 μ + 2 ln ( x ) ) e r f c ( 1 2 ( β σ 2 − μ + ln ( x ) ) 2 σ ) ) x − 1 ( α + β ) − 1 . (41)

We present an initial application of the gamma-Pareto PDFs to a sample of asteroids and the second one to the distribution in energy of CR.

We have analysed the effect of truncation on the gamma-Pareto and the gamma-Pareto II distributions, deriving their distribution functions, average values, and variances. We made two applications to phenomena which present a long right tail often modeled with a power law behaviour, such as the Pareto PDF. In the case of the asteroids, the best results were obtained with the bi-truncated gamma-Pareto II, see Table 1, and in the case of cosmic rays (CR), with the bi-truncated gamma-Pareto, see Table 2. These models allow deriving some analytical formulae for the average value of the CR energy spectrum, which is 2.6 GeV for the bi-truncated gamma-Pareto II function.

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

Zaninetti, L. (2022) New Probability Distributions in Astrophysics: VII. The Truncated Gamma-Pareto Distribution Applied to Cosmic Rays. International Journal of Astronomy and Astrophysics, 12, 132-146. https://doi.org/10.4236/ijaa.2022.121008