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56
The power continuous Bernoulli distribution: Theory and applications
Christophe Chesneau1 Festus C. Opone2
•
1 Department of Mathematics, LMNO, University of Caen, 14032 Caen, France.
2 Department of Statistics, University of Benin, Benin City, Nigeria. [email protected] [email protected]
Abstract
The continuous Bernoulli distribution is a recently introduced one-parameter distribution with support [0,1], finding numerous applications in applied statistics. The idea of this article is to propose a natural extension of this distribution by adding a shape parameter through a power transformation. We introduce the power continuous Bernoulli distribution, aiming to extend the modeling scope of the continuous Bernoulli distribution. Basics of its mathematical properties are derived, such as the shapes of the related functions, the determination of various moment measures, and an evaluation of the overall amount of its randomness via the Renyi entropy. A statistical analysis of the distribution is then performed, showing how it can be applied when dealing with data. Estimates of the parameters are discussed through the maximum likelihood method. A Monte Carlo simulation study investigates the asymptotic behavior of these estimates. The flexibility of the power continuous Bernoulli distribution in real-life data fitting is analyzed using two data sets. Also, fair competitors are considered to highlight the accuracy of this distribution. At all stages, numerous graphics and tables illustrate the findings.
Keywords: Continuous Bernoulli distribution; moments; quantiles; entropy; data fitting.
1. Introduction
In order to understand the mathematical foundation of the study, let us first present the so-called continuous Bernoulli distribution as introduced in [19].
Definition 1. The continuous Bernoulli distribution with parameter A G [0,1], also denoted as CB(A), is defined by the following probability density function (pdf):
1
1, A = 2 and x G [0,1],
f (x; A):
where cA is the following constant:
cAAx(1 - A)1-x, A e (0,1)/{2} and x e [0,1], (1)
0, x e [0,1],
Ca = 2arctanha- 2A). (2)
Here, arctanh(x) denotes the inverse hyperbolic tangent defined by arctanh(x) = (1/2) ln[(1 + x)/(1 — x)] (as a minor remark, the following expressions are equivalent: 2 arctanh(1 — 2A) = ln(1 — A) — ln(A) = ln(1/A — 1)).
Alternatively, the CB(A) distribution can be defined by its cumulative distribution function (cdf), which is given by
F(x; A)
0,
x,
Ax (1 - A) + A - 1
2A — 1
1,
x < 0, 1
A = - and x G [0,1],
A G (0,1)/{!
x > 1.
and x G [0,1],
(3)
Thus, the CB(A) distribution, like the power distribution, is a one-parameter continuous distribution with support of [0,1]. It is useful in a variety of fields, including probability theory, statistics, with an emphasis on machine learning. In particular, it is good at simulating the pixel intensities of natural images in deep learning and computer vision, especially when putting up variational autoencoders. We advise the reader to [19] and [13] for more information on these topics.
More broadly, bounded support distributions have proven useful in modeling real-world data, particularly in scenarios where the data are measured in percentages and proportions. So when the observations take on value within the unit interval [0,1]. In recent decades, the beta and Kumaraswamy distributions have gained more popularity in this regard. However, there are situations where these classical distributions provide poor fit in data analysis. This has become a quest for many researchers to develop alternative bounded distributions with better flexibility in real-life data fitting. With this in mind, [12] introduced the log-Lindley distribution, [21] developed the unit-logistic distribution, [1] created the log-Xgamma distribution, [20] proposed the unit-Gompertz distribution, [23] developed the Kumaraswamy unit-Gompertz distribution, [16] examined the unit-Burr XII distribution, [15] introduced the unit-Chen distribution, [26] developed the transmuted Marshall-Olkin extended Topp-Leone distribution, [6] proposed the log-XLindley distribution, [2] studied the unit-Rayleigh distribution, etc. The CB(A) distribution belongs to the list.
In this paper, by including a shape parameter, we hope to increase the flexibility of the CB(A) distribution for a variety of applications. In other words, for a random variable X following the CB(A) distribution, we consider the distribution of the power random variable Y = , where a > 0. In this way, we introduce the power continuous Bernoulli distribution with parameters a and A, PCB(a, A) distribution for short. The used power scheme is somewhat classic in statistics, and allows to flexibilize various "rigid distributions". We may mention the Weibull distribution, which is the power version of the exponential distribution, the power Lindley distribution by [9], which is the power version of the Lindley distribution (see [18]), etc. Recent examples include the power beta distribution by [5], the power Lomax distribution by [28], the power Ailamujia distribution by [14], etc.
In fact, at the time of writing, no extensions of the CB(A) distribution exist, and the PCB(a, A) distribution is a strong contender for being useful from both theoretical and applied perspectives. After a detailed presentation, we investigate its main features, such as the related probability functions, moments of various kinds, and entropy (Renyi entropy). Then, we examine the practice on the statistical side. We estimate the PCB(a, A) distribution parameters, i.e., a and A, by the maximum likelihood (ML) method. A Monte Carlo simulation study is then conducted to validate the asymptotic behavior of these estimates. We present significant applications of the PCB(a, A) distribution in a data fitting context, with the use of two real-life data sets: one containing trade share data, and the other containing tensile strength of polyester fibers. In addition, several distributions are considered for fair comparison in terms of efficiency in fitting. Illustrations, via tables and graphics, are given to support the findings. We have thus laid the foundation for the use of the PCB(a, A) distribution for statistical purposes.
The organization of the paper is as follows: Section 2 describes the PCB(a, A) distribution, including its underlying functions of interest. A moment analysis is performed in Section 3. The entropy is studied in Section 4. Parameter estimation, simulation study and real-life data fitting
are developed in Section 5. A conclusion is given in Section 6.
2. Power continuous Bernoulli distribution
The PCB(a, A) distribution is defined below through its related probabilistic functions.
Definition 2. Based on its stochastic definition and the functions (1) and (3), the PCB(a, A) distribution with a > 0 and A G [0,1] is defined by the following cdf:
F(x; a, A) = F(xa; A)
0,
xa,
Ax" (1 - A)
x < 0
1—xa
+ A — 1
1
2 and x G [0,1],
2A — 1
1,
A
A G (0,1)/j2 x > 1,
and x G [0,1],
(4)
or, equivalently, by the following pdf:
f (x; a, A) = axa—1 f (xa; A) „a-1
ax
A = 2 and x G [0,1],
cAaxa—1A^(1 — A)1—x", A G (0,1)/{1} and x G [0,1],
(5)
v0, x G [0,1],
where cA is the constant defined in (2).
Basically, by taking a = 1 into (4) and (5), we obtain the cdf and pdf of the CB(A) distribution, as described in (3) and (1), respectively.
It is important to note that the PCB(a, A) distribution has one mode in the case A G (1/2,1), and it is given by the following mathematical formula: x = [(a — 1)/(2a arctanh(1 — 2A))]1/a. In this case, the PCB(a, A) distribution is unimodal, and the mode differs from 0 if, and only if, a G [0,1), i.e., a = 1. So by considering the power version of the CB(A) distribution, we introduce a unimodality property that can be used quite efficiently for statistical aims, including data fitting purposes.
Figure 1 presents the plots for f (x; a, A) in order to illustrate the effect of the parameter a on its possible shapes.
(a)
(b)
Figure 1: Pdf plots of the PCB(a, A) distribution at different choices of the parameter settings: (a) (a, A) G {(3,0.2), (4,0.3), (0.5,0.1), (0.1,0.4), (9,0.6)} and (b) (a, A) G {(5,0.1), (3.5,0.1), (0.2,0.3), (1.5,0.1)}
Clearly, we observe that the pdf of the PCB(a, A) distribution accommodates decreasing (reversed-J) or increasing, left-skewed, right-skewed and symmetric shapes.
As a complementary function to the pdf, the hazard rate function (hrf) of the PCB(a, A) distribution is given as
h(x; a, A)
f (x; a, A) 1 - F(x; a, A)
ax'
a-1
1 - xa'
c*Aaxx-1 Axa (1 - A)1-xa A - Axa (1 - A)1-xa
,0,
A = 2 and x G [0,1], A G (0,1)/{1} and x G [0,1],
x G [0,1],
where c
A
(2A — 1)cA = —2arctanh(1 — 2A). The graphical representation of this function is
displayed in Figure 2.
Figure 2: Hrf plots of the VCB(a, A) distribution at different choices of the parameter settings: (a, A) £ {(5,0.1), (0.1,0.9), (0.6,0.7), (2,0.9)}
Figure 2 indicates that the PCB(a, A) distribution exhibits increasing and bathtub-shaped hazard properties. These are demanded properties for data analysis purposes with values in [0,1].
As the inverse function of the cdf, the quantile function (qf) of the PCB(a, A) distribution is
given as
Q(x; a, A) = F—1 (x; a, A)
1
x1/a, A = ^ and x £ [0,1],
(ln[(2A — 1)x + 1 — A] — ln(1 — A) \1/a , 11 J (6)
{ ln(A) — ln(1 — A) -7 ' A £ (0,1)^2}and x £ [(U].
By inserting x = 1/2 in (6), we obtain the median of the PCB(a, A) distribution, which is given by
2—1/a, A = 2,
( ln(2) + ln(1 — A) \'
iln(2)+ ln(1 - an a G (0 1)^11 \ 2 arctanh(1 - 2A) J , A G (0,1)^2j
Traditionally, the qf and random values from the uniform distribution over [0,1] can be used to generate random values from a random variable Y following the PCB(a, A) distribution. Table 1
shows some quantiles from the A) distribution using the expression in (6) as illustrative
numerical examples.
Table 1: Some values of the qf of the PCB(a, A) distribution.
x a = 0.5, A = 0.3 a = 0.5, A = 0.8 a = 1, A = 0.3 a = 1, A = 0.8
0.05 0.0012 0.0102 0.0342 0.1008
o.1 0.0048 0.0358 0.0694 0.1893
o.2 0.0205 0.1149 0.1432 0.3390
0.3 0.0493 0.2144 0.2219 0.4630
0.4 0.0938 0.3235 0.3063 0.5688
o.5 0.1577 0.4369 0.3971 0.6610
0.6 0.2455 0.5516 0.4955 0.7427
0.7 0.3635 0.6661 0.6029 0.8161
0.8 0.5199 0.7793 0.7210 0.8828
0.9 0.7264 0.8907 0.8523 0.9438
The quantile values of the PCB(a, A) distribution fall into [0,1] for different parameter values. On the other hand, based on the qf, advanced quantile modeling can be performed. For more information, see [10].
3. Moments
The moment measures of the PCB(a, A) distribution are of interest to describe it in an in-depth manner in terms of central, dispersion, and form parameters, and reveal some statistical features.
The following proposition is about the mathematical expressions of the moments of a random variable following the PCB(a, A) distribution.
Proposition 1. Let Y be a random variable following the PCB(a, A) distribution and m be an integer. Then the m-th moment (or raw moment) of Y is given by
M„
E(Ym) a
a + m
(1 - A)c,
Y-
[2 arctanh(1 - 2A)]m/a+1
(1 - A)c,
[—2arctanh(1 - 2A)]m/a+1
m
— + 1,2arctanh(1 - 2A)
a
Y+
m
La + 1, -2 arctanh(1 - 2A)
A = i
a g (o, 2
A G ( 2,1
where
l>u l>u
Y- (x, u)= tx-1 e-tdt, Y+(x, u) = tx-1etdt. Jo Jo
(7)
Proof. For the case A = 1/2, we have
c+œ
Mm
{+ M {1 -1 / xmf (x; a, A)dx = a / xmxa 1 dx J — rn Jo
a
a + m
For the case A € (0,1) / {1/2}, by introducing a random variable X with the CB(A) distribution, we have
/ + TO f 1
xm/af(x; A)dx = cj xm/aAx(1 - A)1-xdx.
-to Jo
Since m/a is not necessarily an integer, let us distinguish the case A € (0,1/2) and the case A € (1/2,1).
In the case A £ (0,1/2), by applying the change of variable y = 2 arctanh(1 — 2A)x > 0, we obtain
r 1 r 1
mm = cj xm/aexln(A)+(1—x)ln(1—A)dx = (1 — A)cJ xm/ae—xI2arctanh(1—2A)]dx
J0 J0
(1 - A)ca
r2arctanh(1-2A)
ym/ae-ydy
[2 arctanh(1 - 2A)]m/a+1 J0
(1 - A)ca
[2 arctanh(1 - 2A)]m/a+1
Y-
m
— + 1,2arctanh(1 - 2A)
La
In the case A £ (1/2,1), we must take into account a sign detail; by applying the change of variable y = —2 arctanh(1 — 2A)x > 0, we obtain
r1 r1
Mm = cj xm/aexln(A)+(1—x)ln(1—A)dx = (1 — A)cJ xm/ae—xI2arctanh(1—2A)]dx 00
(1 - A)ca
-2arctanh(1-2A)
[-2 arctanh(1 - 2A)]m/a+1 J0
ym/aeydy
(1 - A)ca
[-2 arctanh(1 - 2A)]m/a+1
Y+
m
La + 1, -2 arctanh(1 - 2A)
The desired expressions are obtained, ending the proof.
□
It is worth noting that the integral function y— (x, u) corresponds to the lower incomplete gamma function, which is implemented in most of the mathematical software. In the case a = 1, m/a is an integer, and we have
1
m+1
A = ^
Mm = 1 [2arcta^c2A)]m+^ — [m + 1,2a'cta"h(1 — »)], A £ (0,1) / { 2,
giving the m-th moment related to the CB(a, A) distribution, which missing in the list of properties in [19]. In this particular case, by using the expression y— (2, u) = 1 — (1 + u)e—u, we refind the mean of Y as precised in [19, Equation (8)]:
Mi
1 2,
--(1 - A)C\^ 12Y- [2,2arctanh(1 - 2A)]
[2 arctanh(1 - 2A)]2 ' 1 v n
A1
+
A = ^
A G (0,1) /{!}.
" 2A - 1 2 arctanh(1 - 2A)'
More generally, based on the expression of the moments established in Proposition 1, we can easily derive the mean of a random variable Y following the PCB(a, A) distribution; it is given as
M1
a +1'
A = -
(1 - A)ca
1
[2 arctanh(1 - 2A)]1/a+1
(1 - A)ca
[-2 arctanh(1 - 2A)]1/a+1 as well the moment of order 2 of Y:
a
Y-
- + 1,2arctanh(1 - 2A)
Y+
- + 1, -2 arctanh(1 - 2A)
a.
A G \0, 2 A G (2,1
M2
a + 2'
A = x
(1 - A)ca
[2 arctanh(1 - 2A)]2/a+1
(1 - A)ca [-2 arctanh(1 - 2A)]2/a+1
Y-
2
- + 1,2arctanh(1 - 2A) a 2
Y+ —+ 1, -2 arctanh(1 - 2A)
1 :2,
A g (0,2
A G( 2,1
a.
The variance of Y follows from the standard formula: a2 = M2 — M^. The m-th central moment of Y is given by
Mm = E[(Y - M1)m] = jj (m\ (-1)m-kMkM^ k=o V k)
Based on these central moments, the skewness and kurtosis coefficients of Y are, respectively, given by
S _ M3 K _ M4
Numerical computation of the mean, variance, measures of skewness and kurtosis for the PCB(a, A) distribution are shown in Table 2.
Table 2: Theoretical moment measures of the PCB(a, A) distribution
a A M1 S K
0.5 0.1 0.1755 0.0528 1.6646 5.0278
0.4 0.3001 0.0837 0.8087 2.4314
0.9 0.5152 0.0923 -0.1377 1.7674
1.0 0.1 0.3301 0.0665 0.7430 2.5785
0.4 0.4663 0.0827 0.1417 1.8116
0.9 0.6699 0.0664 -0.7388 2.5633
2.0 0.1 0.5234 0.0562 0.0559 2.0882
0.4 0.6394 0.0575 -0.4382 2.2506
0.9 0.7962 0.0360 -1.3390 4.3644
From Table 2, we conclude that the PCB(a, A) distribution can be left- and right-skewed, as negative and positive values for S are observed, and it has all kurtosis states, as K varies around the limit value of 3.
Complement: Alternative measures of skewness and kurtosis are the ones based on the qf of the distribution as proposed by [7] and [22], respectively. The Galton skewness and the Moors kurtosis are, respectively, defined as
S = Q(6/8; a, A) — 2Q(4/8; a, A) + Q(2/8; a, A) G Q(6/8; a, A) — Q(2/8; a, A)
and
K = Q(7/8; a, A) — Q(5/8; a, A) + Q(3/8; a, A) — Q(1/8; a, A) M Q(6/8; a, A) — Q(2/8; a, A) .
In order to complete the previous numerical work, Figure 3 presents the nature of the Galton skewness and Moors kurtosis of the PCB(a, A) distribution.
(a)
(b)
Figure 3: Plots of (a) the Galton skewness and (b) Moors kurtosis for the PCB (a, A) distribution with a G [0,2] and A G [0,1]
From Figure 3, we can observe that the Galton skewness seems monotonic according to a and A, with possible negative and positive values. On the other hand, the Moors kurtosis is more complex, being non-monotonic in a. This illustrates the versatility of the PCB(a, A) distribution on these form aspects.
4. Entropy
The amount of randomness in the PCB(a, A) distribution is now the object of all the attention. In order to accomplish this, we recall that the Rényi entropy of a random variable X with pdf f (x) is given by
Re
1- e
ln
C+œ
f(x)edx
with 0 > 0 and 0 = 1. The following proposition is about the mathematical expression of the Rényi entropy of a random variable Y following the PCB(a, A) distribution.
Proposition 2. Let 0 > 0 with 0 = 1, and Y be a random variable following the PCB(a, A) distribution. Then the Rényi entropy of Y is given by
1
1 ln
- e
1
1 ln
- e
1
1 ln
- e
e(a - 1) + 1
T_(1-A)9tr1)( 1)/ 1 Y- j(e - 1)(a - 1) + 1,2earctanh(1 - 2A)1
[2e arctanh(1 - 2A)](e-1)("-1)/"+1 ' [a +' ( )J
' 'ecLe-1 r (e - 1)(a - 1) + 1, -2earctanh(1 - 2A)1
Re =
1 — e ^ [2earctannu — 2A)|("—vc—v-^1
1 . ( (1 — A)e cjae—1 ^
. v [—2earctanh(1 — 2A)](e—1)(»—1)/»+1
where y— (x, u) and Y+(x, u) are defined as in (7). Proof. For the case A = 1/2, we have
a=2-
A e
' A G(
f(x; a, A)edx = xe(a-1)dx = -œ J 0
(a - 1) + 1'
Hence,
Re =
1
ln
1 - 0 \9(a - 1) + 1)' For the case A G (0,1)/ {1/2}, by applying the change of variable y = xa, we have
/ + œ f 1
f (x; a, A)0dx = ceAae / x0(a-1)A0xa (1 - A)0(1-x")dx
-œ J0
/■ 1
= ceAae-1jo y(0-1)(a-1)/aA0y(1 - A)0(1-y)dy.
Let us now distinguish the case A G (0,1/2) and the case A G (1/2,1).
• In the case A G (0,1/2), by applying the change of variable t = 20 arctanh(1 - 2A)y > 0, we obtain
/+œ c 1
f(x;a,A)0dx = c0a0-1 f y(0-1)(a-1)/ae0yln(A)+0(i-y)ln(i-A)dy
-œ J0
i-1
= (1 - A)0c0a0-1 J y(0-l)(a-l)/ae-y[20arctanh(l-2A)]dy
t(0-1)(a-1)/a e-tdt
(1 - A)ecAae-1 r2earctanh(1-2A)
[2e arctanh(1 - 2A)](e-1)(a-1)/a+1 J0
(1 - A)ecAae-1
[2e arctanh(1 - 2A)](e-1)(a-1)/a+1
Y-
- 1)(a - 1)
+ 1,2e arctanh(1 - 2A)
Hence,
R = -JLlJ (1 - A)ecAae-1
Re 1 - e X [2earctanh(1 - 2A)](e-1)(»-1)/»'
(e - 1)(a - 1) + 1,2earctanh(1 - 2A)U .
1
œ
a
1
2
e
a
e
a
a
Y
In the case A € (1/2,1), by applying the change of variable t = —20 arctanh(1 — 2A)y > 0, we obtain
/ + TO C1
f(x; a,A)0dx = c0a0—1 f y(0—1)(a—1)/ae0yln(A)+0(1—y)ln(1—A)dy -to Jo
f 1
= (1 — A)0c0a0—1 J y(0—1)(a—1)/ae—y[20arctanh(1—2A)]dy
0
(1 - A)9c^-1 f -20arctanh(1-2A) i(fl_1)(._1)/a^
[-20 arctanh(1 - 2A)](0-1)(a-1)/a+1 Jo
_(1 - A)0c0Aa0-1_ [(0 - 1)(a - 1)
[-20 arctanh(1 - 2A)](0-1)(a-1)/a+17+
+ 1,-20 arctanh(1 - 2A)
R = <1 - Y+ rc^-iiia-l) + 1, -20 arctanh(1 - 2A)l } .
Hence,
" \ [—20arctanh(1 — 2A)](0—1)(a—1)/a
We end the proof by compiling the above expressions. □
Table 3 shows some numerical values of the Renyi entropy of the PCB(a, A) distribution.
Table 3: Numerical results of the Renyi entropy of the PCB(a, A) distribution
7 a = 1, A = 0.3 a = 1, A = 0.8 a = 2, A = 0.3 a = 2, A = 0.8
0.01 -0.0003 -0.0008 -0.0019 -0.0061
0.03 -0.0009 -0.0024 -0.0057 -0.0813
0.5 -0.0148 -0.0390 -0.0712 -0.2523
0.7 -0.0207 -0.0542 -0.0901 -0.3290
2.0 -0.0574 -0.1443 -0.1604 -0.6414
4.0 -0.1069 -0.2468 -0.2056 -0.8486
7.0 -0.1624 -0.3379 -0.2360 -0.9827
9.0 -0.1892 -0.3756 -0.2474 -1.0320
Consequently, from Table 3, some useful properties of the Renyi entropy provided in [11] are applicable here. In particular, (i) for any 01 < 02, we have R02 < R01, (ii) the Renyi entropy can be negative.
5. Statistical applications This section is devoted to the applicability of the PCB(a, A) distribution.
5.1. Estimation
In the setting of the PCB(a, A) distribution, we aim to estimate the unknown parameters a and A based on data that can be conceptually fitted with this distribution. To accomplish this, we can use the ML method, described as follows: Let y1,..., yn represent n independent observations from a random variable Y following the PCB(a, A) distribution. Then, based on the pdf indicated
in (5), the likelihood function is specified by
n
L(a, A; yi,..., yn) = n f (Ví; a, A)
i=1
n yi
i=1
a-1
A = i
n yi i=1
a-1 r ^
aE=1 $ (1 - A)n-EU *, A G (0,1)/j1}.
The ML estimates (MLEs) of a and A are given by
(a, A) = argmax(a,A) L(a, A; yx,..., y„). Alternatively, they are defined by
(a, A) = argmax(a,A) ¿(a, A; y1,..., y„),
where l(a, A;yi,...,yn) refers to the log-likelihood function given by
t(a, A; y1,...,yn) = ln [L(a, A;y1,...,yn)] f n
nln(a) + (a — 1) ^ ln(y;),
A =
2'
n ln(cA) + n ln(a) + (a - 1) E ln(y;) + ln(A) E ya + ln(1 - A)
E ya
A G (0,1)/
They can be obtained by solving the following non-linear equations with respect to a and A: d£(a, A; yi,..., yn) = Q d£(a, A; yi,..., Vn) = Q
da
dA
The standard errors of a and A can be approximated, and they are denoted in the next as se(a) and se(A), respectively. The advantage of the ML method is that it guarantees interesting properties for the MLEs, such as asymptotic unbiasedness and normality. More information on these properties can be found in [4]. Based on the MLEs, we can estimate all the underlying functions of the PCB(a, A) distribution. In particular, an estimate of the cdf F(x; a, A) is given by F(x) = F(x; a, A) and an estimate of the pdf f (x; a, A) is given by f(x) = f (x; a, A).
5.2. Simulation study
In this portion, we investigate the asymptotic behavior of the MLEs of a and A using Monte Carlo simulation. Random samples from the PCB(a, A) distribution were generated using (6). The simulation is repeated N = 2000 times for different sample sizes n £ {20,50,100,200,500} and different choices of the parameter values (a = 0.3, A = 0.1), (a = 0.5, A = 0.3) and (a = 0.8, A = 0.6). The performance of the MLEs is examined using various statistical criteria presented below. For $ £ {a, A}, we consider
1
N
1. the average bias (Bias) defined by n ^ — $), where the index i refers to the i-th experiment among the N,
i=1
N
2. the root mean square error (RMSE) defined by ^ ^ ((pi — ^)2,
i=1
3. the coverage probability (CP) of the 95% confidence interval defined by
1
N
N EI - u* se((j>i) < <p < <pi + M* se((pi)),
i=1
n
a
cn a
A
1
i=1
1
n
i=1
i=1
i=1
where I(.) is the indicator function, se(<) is the standard error related to <pi and = 1.959964. Table 4 presents the simulation results based on these criteria.
Table 4: Simulation results for the unknown parameters estimates of PCB(a, A) distribution
Bias RMSE CP
Parameters n a A a A a A
25 0.0189 0.0234 0.0807 0.1314 0.9420 0.8065
a = 0.3 50 0.0056 0.0156 0.0520 0.0934 0.9455 0.8425
A = 0.1 100 0.0042 0.0062 0.0358 0.0565 0.9570 0.9000
200 0.0028 0.0028 0.0255 0.0406 0.9450 0.9060
500 0.0006 0.0014 0.0155 0.0250 0.9520 0.9395
25 0.0528 -0.0061 0.1736 0.2073 0.9440 0.8120
a = 0.5 50 0.0228 -0.0040 0.1139 0.1608 0.9545 0.8675
A = 0.3 100 0.0095 0.0075 0.0819 0.1251 0.9470 0.9150
200 0.0037 0.0040 0.0554 0.0902 0.9515 0.9300
500 0.0029 -0.0002 0.0341 0.0569 0.9575 0.9405
25 0.1847 -0.1243 0.3970 0.2635 0.9485 0.8380
a = 0.8 50 0.0959 -0.0711 0.2636 0.2073 0.9420 0.8835
A = 0.6 100 0.0460 -0.0380 0.1832 0.1628 0.9445 0.9090
200 0.0189 -0.0155 0.1276 0.1221 0.9620 0.9285
500 0.0065 -0.0051 0.0827 0.0818 0.9540 0.9365
From Table 4, we notice that the RMSE of both MLEs decreases as the sample size n increases. While a is a positively biased parameter estimate, A can be both positively and negatively biased. Furthermore, the CP of both MLEs approaches 0.95, and the CP of A increases as the sample size n increases.
5.3. Real-life data fitting
Among other purposes, the PCB(a, A) distribution can be used for fitting data with values into [0,1]. We thus illustrate this application by considering two real-life data sets, and compare their fit with the ones obtained from some existing distributions with support of [0,1]. More specifically, we consider the following recently developed unit distributions, including the beta and Kumaraswamy distributions.
1. Marshall-Olkin extended Kumaraswamy distribution (MOEKD) introduced by [8], and defined with the following pdf: aabxa-1 (1 - xa )b-1
1 - a (1 - xa)b]
f (x; a, a, b) = —-(-, x G [0,1], where a = 1 - a, with a, a, b > 0.
2. Marshall-Olkin extended Topp-Leone distribution (MOETLD) introduced by [25], and
specified by the following pdf:
2aA(1 — x) [1 — (1 — x)2]A-1 f (x; a, A) =-(-^-(-^-2, x G [0,1], with a, A > 0.
1 - a(1 - [1 - (1 - x)2]A)
3. Unit-Gompertz distribution (UGD) introduced by [20], and defined with the following pdf: f (x; a, b) = abx-a-1e-b(x-a-1), x G (0,1], with a, b > 0.
4. Unit-Burr XII distribution (UBXIID) introduced by [16], and defined with the following pdf: f (x; a,ß) = aßx-1(-ln(x))ß-1 (1 + (-ln(x))ß)-a-1, x G (0,1], with a,ß > 0.
5. Kumaraswamy distribution introduced by [17], and characterized by the following pdf: f (x; a, b) = abxa—1(1 — xa)b, x £ (0,1], with a, b > 0.
6. Beta distribution reported in [24], and defined with the following pdf:
f (x; a, fi) = ^ xa—1 (1 — x)fi—1, x £ (0,1), where r(x) refers to the standard gamma function, with a, fi > 0.
7. Continuous Bernoulli distribution (CBD) reported in [30], and defined with the pdf given in (1).
For clarity in exposition, we finally mention that the PCB(a, A) distribution will sometimes be denoted as PCBD in the figures and tables to come.
Data set 1: The first data set consists of trade share data from [3]. The trade share data are as follows: 0.140501976, 0.156622976, 0.157703221, 0.160405084, 0.160815045, 0.22145839, 0.299405932, 0.31307286,0.324612707,0.324745566,0.329479247,0.330021679,0.337879002,0.339706242,0.352317631, 0.358856708,0.393250912,0.41760394,0.425837249,0.43557933,0.442142904,0.444374621,0.450546652, 0.4557693,0.46834656,0.473254889,0.484600782,0.488949597,0.509590268,0.517664552,0.527773321, 0.534684658,0.543337107,0.544243515,0.550812602,0.552722335,0.56064254,0.56074965,0.567130983, 0.575274825,0.582814276,0.603035331,0.605031252,0.613616884,0.626079738,0.639484167,0.646913528, 0.651203632,0.681555152,0.699432909,0.704819918,0.729232311,0.742971599,0.745497823,0.779847085, 0.798375845, 0.814710021, 0.822956383, 0.830238342, 0.834204197, and 0.979355395. The data set is approximately symmetric with a skewness value of 0.0059. Details of this data set can be accessed in [29].
Data set 2: The second data set relates to 30 measurements of the tensile strength of polyester fibers reported in [20]. It was first reported in [27]. The data are as follows: 0.023, 0.032, 0.054, 0.069, 0.081, 0.094, 0.105, 0.127, 0.148, 0.169, 0.188, 0.216, 0.255, 0.277, 0.311, 0.361, 0.376, 0.395, 0.432, 0.463, 0.481, 0.519, 0.529, 0.567, 0.642, 0.674, 0.752, 0.823, 0.887, and 0.926. The data set is right-skewed with a skewness value of 0.5193.
Figure 4 presents the boxplot for the two data sets, showing some of their quantile characteristics.
(a) (b)
Figure 4: Boxplot for (a) data set 1 and (b) data set 2
Figure 4 further supports the claim that data set 1 is approximately symmetric while data set 2 is right-skewed. Observe also that there are no outliers in the two data sets.
The distribution comparison will be based on the distribution parameter estimates, log-likelihood (LogL), Akaike information criterion (AIC), and Kolmogorov-Smirnov test statistic (K-S), along with the corresponding p-value. Tables 5 and 6 present the summary statistics for data sets 1 and 2, respectively.
Table 5: Summary statistics for data set 1
Models Estimates LogL AIC K-S p-value
PCBD a = = 2.8491 15.1002 -26.2005 0.0565 0.9837
A = = 0.0094
a = = 0.3011
MOEKD a = = 3.0586 14.3183 -22.6367 0.0582 0.9783
b = = 1.9513
MOETLD a = 0.6630 14.3606 -24.7211 0.0568 0.9831
A = 3.3521
Beta a = = 2.7940 13.9561 -23.9121 0.1162 0.3546
ß = 2.6038
UGD a = 0.6162 10.8759 -17.7518 0.1098 0.4235
b = 1.0921
UBXIID a = 2.1247 14.1186 -24.2371 0.0578 0.9804
ß = 2.2237
Kumaraswamy a = 2.3297 13.6251 -23.2503 0.0689 0.9142
b = 2.7630
CBD A = 0.5424 0.0734 1.8532 0.1834 0.0287
Table 6: Summary statistics for data set 2
Models Estimates LogL AIC K-S p-value
PCBD a = 1.1240 3.4469 -2.8938 0.0578 0.9998
A = 0.1069
a= 0.4363
MOEKD a= 1.1874 3.6043 -1.2088 0.0627 0.9992
b= 1.2582
MOETLD a= 1.0929 2.9136 -1.8272 0.0672 0.9978
A= 1.0628
Beta a= 0.9666 3.3051 -2.6101 0.1646 0.3515
ß= 1.6203
UGD a= 1.0373 3.9488 -3.8976 0.0734 0.9932
b= 0.4213
UBXIID a= 1.0331 1.0390 1.9220 0.0993 0.9007
ß= 1.8465
Kumaraswamy a= 0.9627 3.3110 -2.6221 0.0649 0.9987
b= 1.6084
CBD A= 0.1565 3.3118 -4.6236 0.0594 0.9997
5.4. Discussion of the results
In the model selection concept, the model that best fits the data set is traceable to the one having the maximized LogL, least value in terms of AIC and K-S with the highest p-value. A close look at Tables 5 and 6 reveals that the PCB(a, A) distribution outperforms the competitors in fitting the two data sets under study. In particular, all the comparison criteria in Table 5 are in favor of the PCB(a, A) distribution. Whereas, in Table 6, we observe that LogL as a criterion supports the Marshall-Olkin extended Kumaraswamy and unit-Gompertz distributions over the PCB(a, A) distribution, while the AIC supports the unit-Gompertz and continuous Bernoulli distributions over the PCB(a, A) distribution. However, the PCB(a, A) distribution outperforms all of the
competitors in terms of the K-S statistic and its corresponding p-value. Figures 5,6,7 and 8 show additional evidence of its flexibility over the competitors.
Especially, Figure 5 displays the estimated pdf and cdf fits of the distributions for data set 1.
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
data data
Figure 5: Estimated pdf and cdf fits of the distributions for data set 1
We observe that the estimated pdf fit of the PCB(a, A) distribution perfectly captures the shape of the unimodal histogram, and the estimated cdf fit approaches well the curvature of the empirical cdf.
In Figure 6 the probability-probability (P-P) plots of the distributions for data set 1 are shown.
Figure 6: P-P plost of the distributions for data set 1
Visually, the P-P line of the PCB(a, A) distribution better adjusts the associated scatter plot than the others.
Figure 7 is analogous to Figure 5 but for data set 2.
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8
data data
Figure 7: Estimated pdfand cdf fits of the distributions for data set 2
In Figure 7, the estimated pdf fit of the PCB(a, A) distribution captures well the decreasing shape of the histogram, and the estimated cdf fit approaches correctly the concave trend of the empirical cdf.
Figure 8 is analogous to Figure 6 but for data set 2.
Figure 8: P-P plots of the distributions for data set 2
In Figure 8, the P-P line of the PCB(a, A) distribution is quite acceptable in terms of fitting, as for some other competitors.
In summary, from these figures, it is clear that the fit accuracy of the PCB(a, A) distribution is excellent, making it a golden distribution to analyze the considered data sets.
6. Conclusion
We proposed a natural extension of the novel continuous Bernoulli distribution by adding a shape parameter through power transformation. The so-called power continuous Bernoulli distribution is aimed at extending the modeling scope of the continuous Bernoulli distribution. Some of its mathematical properties were derived (moments, quantiles, entropy, etc.). A parametric estimation exercise has been given through the maximum likelihood method, and the asymptotic behavior of the parameter estimates was investigated through a Monte Carlo simulation study. Finally, we illustrate the flexibility of the power continuous Bernoulli distribution in real-life data fitting using two real data sets. The potential for probability and statistics applications, such as regression modeling and machine learning applications, is enormous, and this study provides the first steps in that direction.
Conflict of Interests
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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