The experimental sample consists of the data x i with i varying between 1 and n; the sample mean, x ¯ , is

x ¯ = 1 n ∑ i = 1 n     x i , (1)

the unbiased sample variance, s 2 , is

s 2 = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2 , (2)

and the sample rth moment about the origin, x ¯ r , is

x ¯ r = 1 n ∑ i = 1 n ( x i ) r . (3)

The Lindley probability density function (PDF) with one parameter, f ( x ) , is

f ( x ; c ) = c 2 e − c x ( x + 1 ) 1 + c ;     x > 0 , c > 0 (4)

where x > 0 and c > 0 .

The cumulative distribution function (CDF), F ( x ) , is

F ( x ; c ) = 1 − ( 1 + c x 1 + c ) e − c x ;     x > 0 , c > 0. (5)

At x = 0 , f ( 0 ) = c 2 1 + c and is not zero.

The average value or mean, μ , is

μ ( c ) = 2 + c c ( 1 + c ) , (6)

the variance, σ 2 , is

σ 2 ( c ) = c 2 + 4 c + 2 c 2 ( 1 + c ) 2 . (7)

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

μ ′ r = c − r Γ ( r + 2 ) + c 1 − r Γ ( r + 1 ) 1 + c , (8)

where

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

is the gamma function, see [11]. The central moments, μ r , are

μ 3 = 2   c 3 + 12 c 2 + 12 c + 4 c 3 ( 1 + c ) 3 (10a)

μ 4 = 9   c 4 + 72 c 3 + 132 c 2 + 96 c + 24 c 4 ( 1 + c ) 4 (10b)

More details can be found in [2].

The Lindley PDF with two parameters TPLD [3] is

f ( x ; b , c ) = c 2 ( b + x ) e − c x b c + 1 , (11)

where x > 0 , c > 0 and b   c > − 1 . The CDF of the TPLD is

F ( x ; c , b ) = 1 − ( b c + c x + 1 ) e − c x b c + 1 . (12)

The average value or mean of the TPLD is

μ ( b , c ) = b c + 2 c ( b c + 1 ) , (13)

and the variance of the TPLD is

σ 2 ( b , c ) = b 2 c 2 + 4 b c + 2 c 2 ( b c + 1 ) 2 . (14)

The mode of the TPLD is at

M o d e = 1 − b c c , (15)

see Equation (2.3) in [3]. The rth moment about the origin for the TPLD, μ ′ r , is

μ ′ r = c 1 − r b Γ ( r + 1 ) + c − r Γ ( r + 2 ) b c + 1 . (16)

The two parameters b and c can be obtained by the following match

μ = x ¯ (17a)

σ 2 = s 2 , (17b)

which means

b ^ = − ( s 2 + x ¯ 2 ) ( x ¯   − 2 s 2 + 2 x ¯ 2 − 2 s 2 ) ( x ¯   − 2 s 2 + 2 x ¯ 2 + x ¯ 2 − s 2 ) ( 2   x ¯ + − 2 s 2 + 2 x ¯ 2 ) , (18)

and

c ^ = 2   x ¯ + − 2 s 2 + 2 x ¯ 2 s 2 + x ¯ 2 . (19)

The power Lindley PDF with two parameters (PLD) according to [3] is

f ( x ; b , c ) = c b 2 ( 1 + x c ) x c − 1 e − b x c b + 1 , (20)

where b, c and x > 0 . The CDF of the PLD is

F ( x ; c , b ) = ( − b x c − b − 1 ) e − b x c + b + 1 b + 1 . (21)

The average value or mean of the PLD is

μ ( b , c ) = ( b − c − 1 c + b c − 1 c c + b − c − 1 ) Γ ( c + 1 c ) ( b + 1 ) c , (22)

and the variance of the PLD is

σ 2 ( b , c ) = N A D A , (23)

where

N A = − b − 2 c − 1 ( Γ ( c + 1 c ) ) 2 c 2 + b − 2 c − 1 Γ ( c + 2 c ) b c 2 − b 2   c − 2 c ( Γ ( c + 1 c ) ) 2 c 2     − 2 ( Γ ( c + 1 c ) ) 2 b − 2 + c c c 2 + b − 2 + c c Γ ( c + 2 c ) b c 2 − 2 b − 2 c − 1 ( Γ ( c + 1 c ) ) 2 c     + 2 b − 2 c − 1 Γ ( c + 2 c ) b c + b − 2 c − 1 Γ ( c + 2 c ) c 2 − 2 ( Γ ( c + 1 c ) ) 2 b − 2 + c c c     + b − 2 + c c Γ ( c + 2 c ) c 2 − b − 2 c − 1 ( Γ ( c + 1 c ) ) 2 + 2 b − 2 c − 1 Γ ( c + 2 c ) c , (24)

and

D A = ( b + 1 ) 2 c 2 . (25)

The mode of the PLD is at

M o d e = − c b + 1 + ( b 2 + 4 ) c 2 + ( − 2 b − 4 ) c + 2 c − 1 2 c b . (26)

The rth moment about the origin for the PLD is

μ ′ r = b − r + c c Γ ( r + c c ) + b − r c Γ ( r + 2   c c ) b + 1 . (27)

The two parameters b and c of the PLD can be found by numerically solving the nonlinear system given by Equation (17a) and Equation (17b).

The generalized Lindley PDF with three parameters (GLD) according to [12] is

f ( x ; a , b , c ) = b 2 ( b x ) a − 1 ( c x + a ) e − b x ( c + b ) Γ ( a + 1 ) , (28)

where a, b, c and x > 0 . The CDF of the GLD is

F ( x ; a , c , b ) = e − 1 / 2 b x ( x a / 2 ( c b a / 2 + b a / 2 + 1 ) M a / 2 , a / 2 + 1 / 2 ( b x ) + b a + 1 x a e − 1 / 2 b x ( a + 1 ) ) ( c + b ) Γ ( a + 2 ) , (29)

where M μ ,   ν ( z ) is the Whittaker M function, see [11]. The average value or mean of the GLD is

μ ( a , b , c ) = a b + a c + c b ( c + b ) , (30)

and the variance of the GLD is

σ 2 ( a , b , c ) = a b 2 + 2 c b a + c 2 a + 2 c b + c 2 b 2 ( c + b ) 2 . (31)

The hazard rate function, h ( x ; a , b , c ) , of the GLD is

h ( x ; a , b , c ) = − b a + 1 x a − 1 ( c x + a ) e − b x ( a + 1 ) e − 1 / 2 b x x a / 2 ( c b a / 2 + b a / 2 + 1 ) M a / 2 , a / 2 + 1 / 2 ( b x ) + x a b a + 1 ( a + 1 ) e − b x − ( c + b ) Γ ( a + 2 ) , (32)

and Figure 1 reports an example. Here the CDF, Equation (29), and the hazard rate function, Equation (32), are reported in closed form in contrast to what was asserted by [12]. The mode of the GLD is at

M o d e = − a b + a c + a 2 b 2 + 2   a 2 b c + a 2 c 2 − 4   a b c 2 b c . (33)

The rth moment about the origin for the GLD is

μ ′ r = Γ ( r + a ) ( b − r c a + b − r c r + b − r + 1 a ) ( c + b ) Γ ( a + 1 ) , (34)

and in particular the third moment is

μ ′ 3 = Γ ( 3 + a ) ( a b + a c + 3 c ) ( c + b ) Γ ( a + 1 ) b 3 . (35)

The three parameters a, b and c of the GLD can be obtained by numerically solving the following three non-linear equations

μ = x ¯ (36a)

σ 2 = s 2 (36b)

μ ′ 3 = x ¯ 3 . (36c)

The new generalized Lindley PDF with three parameters (NGLD) according to [13] is

f ( x ; a , b , c ) = ( c a + 1 x a − 1 Γ ( b ) + c b x b − 1 Γ ( a ) ) e − c x ( 1 + c ) Γ ( a ) Γ ( b ) , (37)

where a, b, c and x > 0 . The CDF of the NGLD is

F ( x ; a , c , b ) = N B ( 1 + c ) Γ ( b + 2 ) Γ ( a + 2 ) , (38)

where

N B = Γ ( b + 2 ) x a c a + 1 e − c x a + Γ ( a + 2 ) x b c b e − c x b − Γ ( b + 2 ) c Γ ( a + 1, c x ) a     + Γ ( b + 2 ) x a c a + 1 e − c x + Γ ( a + 2 ) x b c b e − c x − Γ ( b + 2 ) c Γ ( a + 1, c x )     + Γ ( b + 2 ) Γ ( a + 2 ) c − Γ ( a + 2 ) Γ ( b + 1, c x ) b + Γ ( b + 2 ) Γ ( a + 2 )     − Γ ( a + 2 ) Γ ( b + 1, c x ) (49)

where Γ ( a , z ) is the incomplete Gamma function, defined by

Γ ​ ( a , z ) = ∫ z ∞   t a − 1 e − t d t , (40)

see [11]. The average value of the NGLD is

μ ( a , b , c ) = a c + b c ( 1 + c ) , (41)

and the variance of the NGLD is

σ 2 ( a , b , c ) = a 2 c − 2   a b c + a c 2 + b 2 c + a c + b c + b c 2 ( 1 + c ) 2 . (42)

The rth moment about the origin for the NGLD is

μ ′ r = c − r + 1 Γ ( r + a ) Γ ( b ) + c − r Γ ( r + b ) Γ ( a ) ( 1 + c ) Γ ( a ) Γ ( b ) , (43)

and the third moment is

μ ′ 3 = Γ ( 3 + a ) Γ ( b ) c + Γ ( 3 + b ) Γ ( a ) c 3 ( 1 + c ) Γ ( a ) Γ ( b ) . (44)

The three parameters a, b and c of the NGLD are obtained by numerically solving the three non-linear Equation (36a), Equation (36b) and Equation (36a).

The new weighted Lindley PDF with two parameters (NWL) according to [14] is

f ( x ; b , c ) = − c 2 ( 1 + b ) 2 ( 1 + x ) ( − 1 + e − c b x ) e − c x b ( c b + b + c + 2 ) , (45)

where b, c and x > 0 . The CDF of the NWL is

F ( x ; c , b ) = N C b ( c b + b + c + 2 ) , (46)

where

N C = − e − c x b 2 c x + e − c ( 1 + b ) x b c x − e − c x b 2 c − 2   e − c x b c x + e − c ( 1 + b ) x b c     + e − c ( 1 + b ) x c x − e − c x b 2 − 2 e − c x b c − e − c x c x + b 2 c + e − c ( 1 + b ) x c     − 2 e − c x b − e − c x c + b 2 + c b + e − c ( 1 + b ) x − e − c x + 2   b . (47)

The average value of the NWL is

μ ( a , b , c ) = b 2 c + 2 b 2 + 3 c b + 6 b + 2 c + 6 ( c b + b + c + 2 ) c ( 1 + b ) , (48)

and the variance of the NWL is

σ 2 ( a , b , c ) = N D c 2 ( b c + b + c + 2 ) 2 ( 1 + b ) 2 , (49)

where

N D = b 4 c 2 + 4 b 4 c + 4 b 3 c 2 + 2 b 4 + 18 b 3 c + 7 b 2 c 2 + 12 b 3 + 32 b 2 c     + 6 b c 2 + 24 b 2 + 30 b c + 2 c 2 + 24 b + 12 c + 12. (50)

The rth moment about the origin for the NWL is

μ ′ r = N E b ( b c + b + c + 2 ) , (51)

where

N E = − ( c 1 − r b 1 − r ( 1 + b b ) − r + b − r ( 1 + b b ) − r c − r r − c − r b 2 r + c 1 − r b − r ( 1 + b b ) − r     − c 1 − r b 2 + b − r ( 1 + b b ) − r c − r − c − r b 2 − 2 c − r b r − 2 c 1 − r b − 2 c − r b − c − r r       − c 1 − r − c − r ) Γ ( 1 + r ) . (52)

The two parameters b and c of the NWL can be found by numerically solving the nonlinear system given by Equation (17a) and Equation (17b).

The merit function χ 2 is computed according to the formula

χ 2 = ∑ i = 1 n ( T i − O i ) 2 T i , (61)

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 ) , (62)

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 evaluated by

χ r e d 2 = χ 2 / N F , (63)

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 [16], which involves the degrees of freedom and χ 2 . According to [16] p. 658, the fit “may be acceptable” if Q > 0.001 .

The Akaike information criterion (AIC), see [17], is defined by

AIC = 2 k − 2 ln ( L ) , (64)

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 from the χ 2 statistic L ∝ e x p ( − χ 2 2 ) where χ 2 has

been computed by Equation (65), see [18] [19]. Now the AIC becomes

AIC = 2 k + χ 2 . (65)

The Kolmogorov-Smirnov test (K-S), see [20] [21] [22], does not require binning the data. The K-S test, as implemented by the FORTRAN subroutine KSONE in [16], finds the maximum distance, D, between the theoretical and the astronomical CDF as well the significance level P K S , see formulas 14.3.5 and 14.3.9 in [16]; if P K S ≥ 0.1 , the goodness of the fit is believable.

The first test is performed on NGC 2362 where the 271 stars have a range 1.47 M ⊙ ≥ M ≥ 0.11 M ⊙ , see [23] and CDS catalog J/MNRAS/384/675/Table 1.

The second test is performed on the low-mass IMF in the young cluster NGC 6611, see [24] and CDS catalog J/MNRAS/392/1034. This massive cluster has an age of 2 - 3 Myr and contains masses from 1.5 M ⊙ ≥ M ≥ 0.02 M ⊙ . Therefore the brown dwarfs (BD) region, ≈ 0.2 M ⊙ is covered.

The third test is performed on γ Velorum cluster where the 237 stars have a range 1.31 M ⊙ ≥ M ≥ 0.15 M ⊙ , see [25] and CDS catalog J/A + A/589/A70/Table 5.

The fourth test is performed on young cluster Berkeley 59 where the 420 stars have a range 2.24 M ⊙ ≥ M ≥ 0.15 M ⊙ , see [26] and CDS catalog J/AJ/155/44/Table 3.

Let X be a random variable defined in [ 0, ∞ ] ; the lognormal PDF, following [27] or formula (14.2) in [28], is

PDF ( x ; m , σ ) = e − 1 2 σ 2 ( l n ( x m ) ) 2 x σ 2 π , (66)

where m is the median and σ the shape parameter. The CDF is

CDF ( x ; m , σ ) = 1 2 + 1 2 erf ( 1 2 2 ( − l n ( m ) + l n ( x ) ) σ ) , (67)

where erf ( x ) is the error function, defined as

erf ( x ) = 2 π ∫ 0 x   e − t 2 d t , (68)

see [11]. The average value or mean, E ( X ) , is

E ( X ; m , σ ) = m e 1 2 σ 2 , (69)

the variance, V a r ( X ) , is

V a r = e σ 2 ( e σ 2 − 1 ) m 2 , (70)

the second moment about the origin, E 2 ( X ) , is

E ( X 2 ; m , σ ) = m 2 e 2   σ 2 . (71)

The statistics for the lognormal distribution for these four astronomical samples of stars are reported in Table 1.

The statistics for the Lindley distribution and its generalizations are reported in the following tables: Table 2 for the Lindley distribution with one parameter, Table 3 for the TPLD, Table 4 for the PLD, Table 5 for the GLD, Table 6 for the NGLD and Table 7 for the NWL. The best fit for NGC 2362 is obtained with the PLD, see Figure 2.

The best fit for NGC 6611 is obtained with the Lindley PDF with one parameter, see Figure 3.

The best fit for γ Velorum is obtained with the lognormal PDF, see Figure 4.

The best fit for the young cluster Berkeley 59 is obtained with the NGLD, see Figure 5.

The statistics for the DTL with three parameters are reported in Table 8. Figure 6 reports the CDF of the DTL for NGC 6611 which is the best fit of the various distributions here analysed for this cluster.