




































©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 7doi: 10.28924/ada/ma.5.7
A Proposal of New Extended Symmetric Cosine Distribution

Christophe Chesneau
Department of Mathematics, LMNO, University of Caen-Normandie, 14032 Caen, France

christophe.chesneau@gmail.com

Abstract. This article presents an extended symmetric version of the cosine distribution. The cor-responding probability density function is constructed by a special linear combination of cosine andsine functions. These trigonometric functions are activated by two adjustable parameters with the aimof generating modulable oscillatory shapes. This gives the new distribution greater flexibility andapplicability than the cosine distribution. Its main characteristics are then examined, focusing on itsfunctional properties, the key moment measures and the generation of distributions with different sup-port. A new skewed version of the standard normal distribution is also derived. Potential applicationsin various fields are discussed. Two simulated data examples are presented and analyzed, showingthe superior performance of the new distribution compared to another two-parameter extended versionof the cosine distribution.

1. Introduction
Symmetric distributions model variables whose values are equally likely to deviate from a centralvalue in both directions. Well-known examples include the normal (Gaussian), logistic, Student(often symbolized by T), Cauchy and Laplace distributions, all of which are defined over the entirereal line, i.e., R. However, many practical situations require distributions defined over a boundedinterval. For example, physical measurements, such as lengths, weights and concentrations cannotbe negative and often have upper limits. In addition to measurements with positive values, otherexamples include proportions, cosine or sine values of an angle, correlations, normalized test scoresor normalized risk metrics, which are naturally bounded over an interval. In such cases, dependingon the exact context, it may be a good idea to retain the symmetry property while ensuring thatthe support of the distribution is bounded.Several symmetric distributions over bounded intervals have been extensively studied in theliterature. These include the uniform distribution with support of the form [−υ, υ] with υ > 0,which assigns equal probability to all points within [−υ, υ]; it is the simplest form of boundedsymmetric distribution. We can also mention the truncated normal distribution, which attempts to
Received: 12 Aug 2024.
Key words and phrases. mathematical analysis; symmetric distribution; cosine distribution; moment measures; skewednormal distribution; statistical modeling. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.7
https://orcid.org/0000-0002-1522-9292


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 2apply the properties of the normal distribution, including its symmetry, to a given interval. This isdone by truncating the tails beyond certain limits and renormalizing the probability density function(pdf) accordingly. See [8], [10] and [4]. Another notable example is the cosine (C) distribution, whosepdf has a symmetric bell-shaped curve similar to that of the normal distribution, but is defined overa finite interval. Let us develop it for the purposes of this article. The (standard) C distributionwith support [−1, 1] is defined by the following pdf:
f◦(x) =

1

2
[1 + cos(πx)], x ∈ [−1, 1], (1)

and f◦(x) = 0 for x 6∈ [−1, 1]. It is clear that f◦(−x) = f◦(x) for any x ∈ [−1, 1], justifyingthe symmetry of this pdf around x = 0. Furthermore, since cos(−π) = −1, the curve of this pdfstarts at 0, increases smoothly for x ∈ [−1, 0), reaches its maximum value 1 at x = 0 and thendecreases at the same rate thanks to the symmetry. It thus resembles a "single cosine wave",which is very close to the symmetric bell-shaped curve of the pdf of the normal distribution. The Cdistribution is the best known trigonometric distribution with bounded support. We also note thata scaled version of this distribution exists in the literature, with support [−π, π]. For more details,see [18], [12], [13], [16], and [20].The C distribution has been the subject of recent developments in distribution theory and practice.In particular, various characterisations of the C distribution were examined in [1], an asymmetricsystem based on it to produce a new skewed standard normal distribution was considered in [19], anatural two-parameter symmetric version of the C distribution was proposed in [2], original distri-butions based on the deformation of the cumulative distribution function (cdf) of the C distributionwere constructed in [7], and two different two-parameter asymmetric versions of the C distributionwere proposed in [5] and [6].Despite the variety of existing symmetric distributions over bounded intervals, there is a contin-uous need for new candidates that offer greater flexibility and innovative modeling capabilities. Inthis article, we propose a new extended symmetric C (ESC) distribution. It is designed to retain thesymmetric properties of the C distribution, while introducing two additional parameters to enhanceits adaptability. In particular, one of the parameters activates an additional trigonometric termcapable of introducing oscillatory shapes in the curves of the corresponding pdf. From a statisti-cal point of view, this feature allows the pdf to accommodate nuanced patterns for a normalizedhistogram of the data, and thus may be preferable in some situations. We illustrate this claimusing the maximum likelihood (ML) estimation for the two parameters and the means of two sim-ulated data sets. We also show that it can be more accurate in the fitting exercise than anothertwo-parameter modified C distribution introduced in [5]. Complementing this practical aspect, weexamine some understandable properties of the ESC distribution that can be used beyond thepurposes of the article. These include its functional properties, the key moment measures, and

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 3the generation of distributions with different supports. A new skewed normal distribution is alsoderived and discussed.The rest of this article is structured as follows: Section 2 presents a general distribution resultfrom which we derive the mathematical formulation of the ESC distribution. Section 3 derives itsmain properties and additional results. Section 4 discusses possible applications and illustratesthe flexibility of the ESC distribution with examples. Finally, Section 5 concludes the article andsuggests directions for future research.
2. The ESC distribution

2.1. A general distribution result. The result below is theoretical; it shows how to choose someparameters of a special linear combination of trigonometric functions in order to satisfy the condi-tions of a valid pdf. The function considered is inspired by the pdf of the C distribution as definedin Equation (1), with the aim of making it more flexible in a functional sense.
Theorem 2.1. For any a ∈ R, b ∈ R, and c ∈ R, let us set

f (x ; a, b, c) = c
[
1 + a cos(πx){1− b[sin(πx)]2}

]
, x ∈ [−1, 1],

and f (x ; a, b, c) = 0 for any x 6∈ [−1, 1]. If the following conditions on a, b, and c are satisfied,
then f (x ; a, b, c) is a valid pdf for a random variable with support [−1, 1]:

c =
1

2
, a ∈ [−1, 1], b ∈ [0, 1].

Proof. To define a valid pdf, we need to check that the following three conditions are met:
• (condI): f (x ; a, b, c) is continuous on R, except possibly, for a finite number of values for x .
• (condII): f (x ; a, b, c) ≥ 0 for any x ∈ R.
• (condIII): ∫ +∞−∞ f (x ; a, b, c)dx = 1.The condition (condI) is immediate; f (x ; a, b, c) is a linear combination of continuous trigonometricfunctions, it is continuous on R, except possibly, at the extremes, i.e., x = −1 and x = 1.Let us now investigate the condition (condII). This condition is obvious for any x 6∈ [−1, 1], since

f (x ; a, b, c) = 0. Let us concentrate on the case x ∈ [−1, 1]. Using c > 0, and the standardtriangle inequality, i.e., |u − v | ≥ |u| − |v | for any u ∈ R and v ∈ R, we have
f (x ; a, b, c) ≥ c

{
1− |a|| cos(πx)||1− b[sin(πx)]2|

}
.

Since b ∈ [0, 1] and sin(πx) ∈ [0, 1], we have |1 − b[sin(πx)]2| = 1 − b[sin(πx)]2 ∈ [0, 1]. Thiscombined with | cos(πx)| ≤ 1 and a ∈ [−1, 1] gives
f (x ; a, b, c) ≥ c [1− |a|| cos(πx)|] ≥ c(1− |a|) ≥ 0.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 4The condition (condIII) is now examined. Since c = 1/2, using standard integral rules and
sin(−π) = sin(π) = 0, we have∫ +∞
−∞

f (x ; a, b, c)dx =

∫ 1
−1
f (x ; a, b, c)dx = c

∫ 1
−1

[
1 + a cos(πx){1− b[sin(πx)]2}

]
dx

= c

∫ 1
−1

{
1 + a cos(πx)− ab cos(πx)[sin(πx)]2

}
dx

= c

{
x +

a

π
sin(πx)−

ab

3π
[sin(πx)]3

}∣∣∣∣x=+1
x=−1

= c

{
1 +

a

π
sin(π)−

ab

3π
[sin(π)]3 − (−1)−

a

π
sin(−π) +

ab

3π
[sin(−π)]3

}
= c

{
1 +

a

π
× 0−

ab

2π
× 0 + 1−

a

π
× 0 +

ab

2π
× 0
}
= 2c = 1.

This ends the proof. �

This theorem is the key theoretical result for deriving the ESC distribution. It is described inthe next subsection.
2.2. Definition of the ESC distribution. Based on Theorem 2.1, we define the ESC distribution bythe following pdf:

f∗(x ; a, b) =
1

2

[
1 + a cos(πx){1− b[sin(πx)]2}

]
, x ∈ [−1, 1],

and f∗(x ; a, b) = 0 for any x 6∈ [−1, 1], where a ∈ [−1, 1] and b ∈ [0, 1]. We say "symmetric"because, for any x ∈ R, f∗(x ; a, b) satisfies f∗(−x ; a, b) = f∗(x ; a, b); it is immediate for any x 6∈
[−1, 1] since f∗(x ; a, b) = 0, and, for any x ∈ [−1, 1], this follows from the facts that cos(−πx) =
cos(πx) and [sin(−πx)]2 = [− sin(πx)]2 = [sin(πx)]2.Obviously, for a = 1 and b = 0, we have f∗(x ; a, b) = f◦(x) as given in Equation (1); the ESCdistribution is thus reduced to the C distribution. More generally, note that, for any x ∈ [−1, 1],we can write

f∗(x ; a, b) = f◦(x) + g(x ; a, b),

where
g(x ; a, b) =

1

2
cos(πx)

{
a − 1− ab[sin(πx)]2

}
,

which can be thought of as a two-parameter perturbative trigonometric function of f◦(x) in thiscontext.Figure 1 shows the curves of the pdf of the ESC distribution under different but complementaryconfigurations: when a is fixed at a positive value and b varies, when a is fixed at a negative valueand b varies, when b is fixed and a varies, and a summary of the different plots.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 5

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.2

0
.4

0
.6

0
.8

1
.0

x
p
d
f

a = 1  b = 0.001
a = 1  b = 0.2
a = 1  b = 0.4
a = 1  b = 0.7
a = 1  b = 1

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.2

0
.4

0
.6

0
.8

1
.0

x

p
d
f

a = − 1  b = 0.001
a = − 1  b = 0.2
a = − 1  b = 0.4
a = − 1  b = 0.7
a = − 1  b = 1

(I) (II)

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.5

1
.0

1
.5

x

p
d
f

a = 1  b = 0.2
a = 0.5  b = 0.2
a = 0.1  b = 0.2
a = − 0.5  b = 0.2
a = − 1  b = 0.2

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.5

1
.0

1
.5

x

p
d
f

a = 1  b = 0.001
a = 0  b = 0.6
a = − 1  b = 0.001
a = 0.6  b = 0.2
a = − 1  b = 1

(III) (IV)
Figure 1. Curves of the pdf of the ESC distribution (I) when a = 1 and b varies,(II) when a = −1 and b varies, (III) when b = 0.2 and a varies, and (IV) a summaryof the different plots

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 6In this figure, we can see that all the curves are symmetrical with respect to the axis x = 0, witha general bell or inverted bell shape. We distinguish typical smooth curves and curves with smalloscillatory shapes, mainly caused by the additional sine term in the pdf. Compared to the curves ofthe pdf of the C distribution, these nuanced shapes can be a plus for the fitting perspective whenthe histogram shows some complex symmetric features. This claim will be made more concrete inthe application section, i.e., Section 5.
3. Functions and properties

The main functions of the ESC distribution, moment measures and some transformed ESC dis-tributions are determined in this section.
3.1. The cdf. An important function, the cdf of the ESC distribution, is shown in the result below.
Proposition 3.1. The cdf of the ESC distribution is given as

F∗(x ; a, b) =
1

2

[
1 + x +

a

π
sin(πx)

{
1−

b

3
[sin(πx)]2

}]
, x ∈ [−1, 1],

F∗(x ; a, b) = 0 for any x < −1 and F∗(x ; a, b) = 1 for any x > 1.

Proof. Since the support of the ESC distribution is [−1, 1], we immediately know that F∗(x ; a, b) =
0 for any x < −1 and F∗(x ; a, b) = 1 for any x > 1. For any x ∈ [−1, 1], we have
F∗(x ; a, b) =

∫ x

−∞
f∗(t; a, b)dt =

∫ x

−1
f∗(t; a, b)dt =

1

2

∫ x

−1

[
1 + a cos(πt){1− b[sin(πt)]2}

]
dt

=
1

2

∫ x

−1

{
1 + a cos(πt)− ab cos(πt)[sin(πt)]2

}
dt

=
1

2

{
t +

a

π
sin(πt)−

ab

3π
[sin(πt)]3

}∣∣∣∣t=x
t=−1

=
1

2

{
x +

a

π
sin(πx)−

ab

3π
[sin(πx)]3 − (−1)−

a

π
sin(−π) +

ab

3π
[sin(−π)]3

}
=
1

2

{
1 + x +

a

π
sin(πx)−

ab

3π
[sin(πx)]3

}
=
1

2

[
1 + x +

a

π
sin(πx)

{
1−

b

3
[sin(πx)]2

}]
.

The specified expression is found. �

Clearly, we have
F∗(0; a, b) =

1

2

[
1 + 0 +

a

π
sin(π × 0)

{
1−

b

3
[sin(π × 0)]2

}]
=
1

2
,

which means that the median of the ESC distribution is 0 as expected, since the symmetry pointof the corresponding pdf is x = 0.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 7Figure 2 represents the curves of the obtained cdf for different values of a and b (those consideredin the subfigure (IV) of Figure 1).

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.2

0
.4

0
.6

0
.8

1
.0

x

c
d

f

a = 1  b = 0.001

a = 0  b = 0.6

a = − 1  b = 0.001

a = 0.6  b = 0.2

a = − 1  b = 1

Figure 2. Curves of the cdf of the ESC distribution for different values of a and b
This figure confirms the flexibility of the ESC distribution, as various concave and convex in-creasing curves are observed. However, the "degree of distortion" is limited; in a sense, some areasof the rectangle [−1, 1]× [0, 1] cannot be reached.

3.2. Quantile function. The quantile function (qf) of the ESC distribution is defined as the inversefunction of F∗(x ; a, b), i.e., F−1∗ (x ; a, b). Let us denote it by Q∗(p; a, b), for p ∈ [0, 1]. Due to thetrigonometric complexity of F∗(x ; a, b), it has no closed form expression. However, we can determineit numerically, for fixed values of a and b. Table 1 illustrates this claim, with also different valuesof p.
Table 1. Some values of Q∗(p; a, b) for different values of a, b, and p.
p → 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a = 1, b = 0.001 -1.00 -0.48 -0.33 -0.21 -0.10 0.00 0.10 0.21 0.33 0.48 1.00
a = 0, b = 0.6 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

a = −1, b = 0.001 -1.00 -0.90 -0.79 -0.67 -0.52 0.00 0.52 0.67 0.79 0.90 1.00
a = 0.6, b = 0.2 -1.00 -0.64 -0.43 -0.26 -0.13 0.00 0.13 0.26 0.43 0.64 1.00
a = −1, b = 1 -1.00 -0.90 -0.78 -0.61 -0.41 0.00 0.41 0.61 0.78 0.90 1.00

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 8For example, for a = 0.6, b = 0.2 and p = 0.6 we have Q∗(p; a, b) = 0.13. For p = 0.5,we refind that the median of the ESC distribution is 0. This table shows that a computationalquantile analysis of the ESC distribution is possible, including the determination of various quantilemeasures (see [11] and [14]).
3.3. Hazard rate function. Complementary to the cdf, we can determine the survival function of theESC distribution. It is obtained as S∗(x ; a, b) = 1− F∗(x ; a, b), so that

S∗(x ; a, b) =
1

2

[
1− x −

a

π
sin(πx)

{
1−

b

3
[sin(πx)]2

}]
, x ∈ [−1, 1],

S∗(x ; a, b) = 1 for any x < −1 and S∗(x ; a, b) = 0 for any x > 1.The hazard rate function (hrf) of the ESC distribution is obtained by the following ratio formula:
h∗(x ; a, b) = f∗(x ; a, b)/S∗(x ; a, b), which can be expressed as

h∗(x ; a, b) =
1 + a cos(πx){1− b[sin(πx)]2}

1− x − (a/π) sin(πx) {1− (b/3)[sin(πx)]2} , x ∈ [−1, 1],

and h∗(x ; a, b) = 0 for any x 6∈ [−1, 1]. The shape behavior of this function is informative aboutthe flexibility of the ESC distribution. With this in mind, Figure 3 shows the curves of this hrf fordifferent values of a and b.

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.5

1
.0

1
.5

2
.0

2
.5

3
.0

x

c
d
f

a = 1  b = 0.001

a = 0  b = 0.6

a = − 1  b = 0.001

a = 0.6  b = 0.2

a = − 1  b = 1

Figure 3. Curves of the hrf of the ESC distribution for different values of a and b
In this figure, we distinguish smooth increasing curves as well as oscillating curves, which arecharacteristics of the trigonometric type distributions. This shows that the ESC distribution is ableto model both simple and complex statistical situations involving data in [−1, 1].

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 93.4. Moment analysis. Moment analysis is particularly interesting for distributions with boundedsupports, as most of the derived moment measures and functions are automatically well defined.
3.4.1. Main moment results. We start this analysis with the mean and variance associated withthe ESC distribution in the result below.
Proposition 3.2. Let X be a random variable with the ESC distribution. Then we have

E(X) = 0

and
V(X) =

1

3
+
2a

π2

(
2b

9
− 1
)
,

where E and V denote the mean and variance operators, respectively.

Proof. The symmetry of the ESC distribution (around 0) implies that X and −X follow the ESCdistribution (with the same parameters). We thus have E(X) = E(−X) = −E(X), so that E(X) =
0. On the other hand, we have V(X) = E(X2)− [E(X)]2 = E(X2), where

E(X2) =
∫ +∞
−∞

x2f∗(x ; a, b)dx =

∫ 1
−1
x2f∗(x ; a, b)dx

=
1

2

∫ 1
−1
x2
[
1 + a cos(πx){1− b[sin(πx)]2}

]
dx

=
1

2

∫ 1
−1
x2dx +

a

2

∫ 1
−1
x2 cos(πx){1− b[sin(πx)]2}dx

=
1

3
+
a

2

∫ 1
−1
x2 cos(πx){1− b[sin(πx)]2}dx.

For the remaining integral term, using two integrations by parts in a row and [sin(πx)]2 = 1 −
[cos(πx)]2, we get∫ 1

−1
x2 cos(πx){1− b[sin(πx)]2}dx

=

[
x2
1

π
sin(πx)

{
1−

b

3
[sin(πx)]2

}]∣∣∣∣x=1
x=−1

−
2

π

∫ 1
−1
x sin(πx)

{
1−

b

3
[sin(πx)]2

}
dx

= 0−
2

π

∫ 1
−1
x sin(πx)

{
1−

b

3
+
b

3
[cos(πx)]2

}
dx

= −
2

π

{ [
−x
1

π
cos(πx)

{
1−

b

3
+
b

9
[cos(πx)]2

}]∣∣∣∣x=1
x=−1

+
1

π

∫ 1
−1
cos(πx)

{
1−

b

3
+
b

9
[cos(πx)]2

}
dx

}
= −

2

π

{
2

π

(
1−
2b

9

)
+

(
1−

b

3

)
1

π

∫ 1
−1
cos(πx)dx +

b

9π

∫ 1
−1
[cos(πx)]3dx

}
= −

2

π

{
2

π

(
1−
2b

9

)
+ 0 +

b

9π

∫ 1
−1
cos(πx){1− [sin(πx)]2}dx

}

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 10

= −
2

π

{
2

π

(
1−
2b

9

)
+

b

9π

[
1

π
sin(πx)

{
1−
1

3
[sin(πx)]2

}]∣∣∣∣x=1
x=−1

}
= −

2

π

{
2

π

(
1−
2b

9

)
+ 0

}
=
8b

9π2
−
4

π2
=
4

π2

(
2b

9
− 1
)
.

Hence, we have
V(X) =

1

3
+
a

2
×
4

π2

(
2b

9
− 1
)
=
1

3
+
2a

π2

(
2b

9
− 1
)
.

This concludes the proof. �

Now let us complete this result. Let X be a random variable with the ESC distribution. Then,for any positive integer n, since the ESC distribution is symmetric (around 0), we have
E(X2n+1) = 0.

More generally, for any odd function p(x), we have E[p(X)] = 0. In particular, we rediscover
E(X) = 0, and we can mention that E(X3) = 0. On the other hand, using several integraltechniques and trigonometric formulas, we find that

E(X4) =
1

5
+
4a

π4

[
2(3π2 − 20)b

27
− π2 + 6

]
.

Since E(X) = 0, the skewness coefficient of X is equal to
γ(X) =

1

[E(X2)]3/2
E(X3) = 0.

The ESC distribution therefore has a neutral skewness. Furthermore, the kurtosis coefficient of Xcan be calculated as
β(X) =

1

[E(X2)]2
E(X4) =

81π4 + 60a
[
(6π2 − 40)b − 27(π2 − 6)

]
5 [2a(2b − 9) + 3π2]2

.

Since the above moment measures are not easy to handle, especially for β(X), we propose anumerical work. Table 2 gives some values of the moment measures for different values of a and b.
Table 2. Some values of moment measures of a random variable X with the ESCdistribution for different values of a and b.

E(X) E(X2) E(X3) E(X4) γ(X) β(X)

a = 1, b = 0.001 0.000 0.131 0.000 0.041 0.000 2.406
a = 0, b = 0.6 0.000 0.333 0.000 0.200 0.000 1.800

a = −1, b = 0.001 0.000 0.536 0.000 0.359 0.000 1.249
a = 0.6, b = 0.2 0.000 0.217 0.000 0.108 0.000 2.294
a = −1, b = 1 0.000 0.491 0.000 0.330 0.000 1.368

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 11For the values considered, we have β(X) < 3, indicating that the ESC distribution is mainlyplatykurtic. Thus, for the same variance, the distribution is relatively "flattened".We now determine the incomplete mean associated with the ESC distribution.
Proposition 3.3. Let X be a random variable with the ESC distribution. Then, for any x ∈ [−1, 1],
we have

E(X1{X≤x}) =
1

4
(x2 − 1) +

a

2π2

[
πx sin(πx)

{
1−

b

3
[sin(πx)]2

}
+ cos(πx)

{
1−

b

3
+
b

9
[cos(πx)]2

}
+ 1−

2b

9

]
,

where 1{.} denotes the indicator operator.

Proof. For any x ∈ [−1, 1], we have
E(X1{X≤x}) =

∫ x

−∞
tf∗(t; a, b)dt =

∫ x

−1
tf∗(t; a, b)dt

=
1

2

∫ x

−1
t
[
1 + a cos(πt){1− b[sin(πt)]2}

]
dt

=
1

2

∫ x

−1
tdt +

a

2

∫ x

−1
t cos(πt){1− b[sin(πt)]2}dt

=
1

4
(x2 − 1) +

a

2

∫ x

−1
t cos(πt){1− b[sin(πt)]2}dt.

For the remaining integral term, using an integration by parts and [sin(πx)]2 = 1− [cos(πx)]2, weget ∫ x

−1
t cos(πt){1− b[sin(πt)]2}dt

=

[
t
1

π
sin(πt)

{
1−

b

3
[sin(πt)]2

}]∣∣∣∣t=x
t=−1

−
1

π

∫ x

−1
sin(πt)

{
1−

b

3
[sin(πt)]2

}
dt

=
1

π
x sin(πx)

{
1−

b

3
[sin(πx)]2

}
−
1

π

∫ x

−1
sin(πt)

{
1−

b

3
+
b

3
[cos(πt)]2

}
dt

=
1

π
x sin(πx)

{
1−

b

3
[sin(πx)]2

}
−
1

π

[
−
1

π
cos(πt)

{
1−

b

3
+
b

9
[cos(πt)]2

}]∣∣∣∣t=x
t=−1

=
1

π
x sin(πx)

{
1−

b

3
[sin(πx)]2

}
+
1

π2
cos(πx)

{
1−

b

3
+
b

9
[cos(πx)]2

}
+
1

π2

(
1−
2b

9

)
.

Hence, we have
E(X1{X≤x}) =

1

4
(x2 − 1) +

a

2π2

[
πx sin(πx)

{
1−

b

3
[sin(πx)]2

}
+ cos(πx)

{
1−

b

3
+
b

9
[cos(πx)]2

}
+ 1−

2b

9

]
.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 12This ends the proof. �

Clearly, if we take x = 1, we refind that E(X1{X≤1}) = E(X) = 0. More generally, theexpression of this incomplete mean can be useful in assessing the expected value of X within acertain range, which is crucial in areas such as reliability. It also helps in the calculation andinterpretation of other related functions, such as the mean residual life function. The incompletemean also plays a theoretical role in some characterization results (see [17]).We now determine the mean deviation associated with the ESC distribution.
Proposition 3.4. Let X be a random variable with the ESC distribution. Then we have

E(|X|) =
1

2
+
2a

π2

(
2b

9
− 1
)
.

Proof. Since the pdf f∗(x ; a, b) is symmetric (around 0), we have
E(|X|) =

∫ +∞
−∞

|x |f∗(x ; a, b)dx =
∫ 1
−1
|x |f∗(x ; a, b)dx

= 2

∫ 1
0

xf∗(x ; a, b)dx =

∫ 1
0

x
[
1 + a cos(πx){1− b[sin(πx)]2}

]
dx

=

∫ 1
0

xdx + a

∫ 1
0

x cos(πx){1− b[sin(πx)]2}dx

=
1

2
+ a

∫ 1
0

x cos(πx){1− b[sin(πx)]2}dx.

For the remaining integral term, using an integration by parts and [sin(πx)]2 = 1− [cos(πx)]2, weget ∫ 1
0

x cos(πx){1− b[sin(πx)]2}dx

=

[
x
1

π
sin(πx)

{
1−

b

3
[sin(πx)]2

}]∣∣∣∣x=1
x=0

−
1

π

∫ 1
0

sin(πx)

{
1−

b

3
[sin(πx)]2

}
dx

= 0−
1

π

∫ 1
0

sin(πx)

{
1−

b

3
+
b

3
[cos(πx)]2

}
dx

= −
1

π

[
−
1

π
cos(πx)

{
1−

b

3
+
b

9
[cos(πx)]2

}]∣∣∣∣x=1
x=0

=
1

π2

(
−1 +

2b

9
− 1 +

2b

9

)
=
2

π2

(
2b

9
− 1
)
.

We therefore have
E(|X|) =

1

2
+
2a

π2

(
2b

9
− 1
)
.

This concludes the proof. �

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 13As the ESC distribution is symmetric, the associated mean deviation of X can be considered asa measure of dispersion of the same unit as the modeled variable. It is comparable to the standarddeviation of X , which is defined as the square of the corresponding variance. We notice also thefollowing formula:
V(X)− E(|X|) =

1

3
−
1

2
= −
1

6
,

which has the property of being independent of a and b.The moment generating function of the ESC distribution can be determined with mathematicalefforts. This is developed in the result below.
Proposition 3.5. Let X be a random variable with the ESC distribution. Then the moment gener-
ating function of X is given as ϕ(t; a, b) = E

[
etX
]
, and can be expressed as

ϕ(t; a, b) = sinh(t)

{
1

t
−
at[π2(9− 2b) + t2]
(t2 + π2)(t2 + 9π2)

}
, t ∈ R,

where sinh(t) = (et − e−t)/2.

Proof. We have
ϕ(t; a, b) = E

[
etX
]
=

∫ +∞
−∞

etx f∗(x ; a, b)dx =

∫ 1
−1
etx f∗(x ; a, b)dx

=
1

2

∫ 1
−1
etx
[
1 + a cos(πx){1− b[sin(πx)]2}

]
dx

=
1

2

∫ 1
−1
etxdx +

a

2

∫ 1
−1
cos(πx)etxdx −

ab

2

∫ 1
−1
cos(πx)[sin(πx)]2etxdx.

The last two integral terms require special treatment. It follows from [9, Formula number 2.663.3]that ∫ 1
−1
cos(πx)etxdx = −

2t sinh(t)

t2 + π2
.

On the other hand, after a work on the formula in [9, Formula number 2.664.2], we obtain∫ 1
−1
cos(πx)[sin(πx)]2etxdx = −

4π2t sinh(t)

(t2 + 9π2)(t2 + π2)
.

We therefore have
ϕ(t; a, b) =

1

2

[
1

t
etx
]∣∣∣∣x=1
x=−1

−
at sinh(t)

t2 + π2
+

2abπ2t sinh(t)

(t2 + 9π2)(t2 + π2)

=
sinh(t)

t
−
at sinh(t)

t2 + π2
+

2abπ2t sinh(t)

(t2 + 9π2)(t2 + π2)

= sinh(t)

{
1

t
−
at[π2(9− 2b) + t2]
(t2 + π2)(t2 + 9π2)

}
.

The stated formula is established, concluding the proof. �

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 14We rediscover the moment generating function associated with the C distribution by taking a = 1and b = 0, i.e,
ϕ◦(t) =

π2 sinh(t)

t(t2 + π2)
, t ∈ R.

See [1] (with the following configuration: µ = 0 and σ = 1). The characteristic function associatedwith the ESC distribution follows from the same mathematical development. By introducing i , theimaginary unit solution of the equation x2 + 1 = 0, since sinh(i t) = i sin(t), we have
φ(t; a, b) = ”ϕ(i t; a, b)” = sinh(i t)

{
1

i t
−

ait[π2(9− 2b) + (i t)2]
((i t)2 + π2)((i t)2 + 9π2)

}
= sin(t)

{
1

t
−
at(2π2b + t2 − 9π2)
(t2 − 9π2)(t2 − π2)

}
, t ∈ R.

This function also fully defines the ESC distribution. In fact, we have the following formulalinking the pdf and the characteristic function:
f∗(x ; a, b) =

1

2π

∫ +∞
−∞

φ(t; a, b)e−itxdt, x ∈ [−1, 1],

which can be written as, for any x ∈ [−1, 1],
1

2

[
1 + a cos(πx){1− b[sin(πx)]2}

]
=
1

2π

∫ +∞
−∞

sin(t)

{
1

t
−
at(2π2b + t2 − 9π2)
(t2 − 9π2)(t2 − π2)

}
e−itxdt

or, equivalently,
1 + a cos(πx){1− b[sin(πx)]2} =

1

π

∫ +∞
−∞

sin(t)

{
1

t
−
at(2π2b + t2 − 9π2)
(t2 − 9π2)(t2 − π2)

}
e−itxdt.

This two-parameter formula can be of some mathematical interest, especially in harmonic analysisdealing with the Fourier transform of various trigonometric functions.We can also note that, since X is symmetric (around 0), we have ϕ(t; a, b) = ϕ(−t; a, b), aproperty we can also check using the expression we found. As a direct consequence, for any t ∈ R,we have
E[cosh(tX)] =

1

2
[ϕ(t; a, b) + ϕ(−t; a, b)] = ϕ(t; a, b)

= sinh(t)

{
1

t
−
at[π2(9− 2b) + t2]
(t2 + π2)(t2 + 9π2)

}
,

recalling that cosh(x) = (ex + e−x)/2.We end this part by investigating the moments of the power of the sine transformed random vari-able with the ESC distribution. These moments have the property of being very simple, dependenton an adjustable positive integer, and independent of a and b.
Proposition 3.6. Let n be a positive integer, and X be a random variable with the ESC distribution.
Then we have

E
{
[sin(πX)]2n

}
=
(2n)!

22n(n!)2

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 15

and

E
{
[sin(πX)]2n+1

}
= 0.

Proof. We have
E
{
[sin(πX)]2n

}
=

∫ +∞
−∞
[sin(πx)]2nf∗(x ; a, b)dx =

∫ 1
−1
[sin(πx)]2nf∗(x ; a, b)dx

=
1

2

∫ 1
−1
[sin(πx)]2n

[
1 + a cos(πx){1− b[sin(πx)]2}

]
dx

=
1

2

∫ 1
−1
[sin(πx)]2ndx +

a

2

∫ 1
−1
cos(πx)[sin(πx)]2n{1− b[sin(πx)]2}dx.

After a work on the formula in [9, Formula number 2.513.1], we find that∫ 1
−1
[sin(πx)]2ndx =

(2n)!

22n−1(n!)2
. (2)

We therefore have
E
{
[sin(πX)]2n

}
=
(2n)!

22n(n!)2
+
a

2

[
1

π
[sin(πx)]2n+1

{
1

2n + 1
−

b

2n + 3
[sin(πx)]2

}]∣∣∣∣x=1
x=−1

=
(2n)!

22n(n!)2
+ 0 =

(2n)!

22n(n!)2
.

On the other hand, since p(x) = [sin(πx)]2n+1 is an odd function, we have already discussed that
E
{
[sin(πX)]2n+1

}
= E [p(X)] = 0. The desired results are obtained. �

3.4.2. Secondary moment results. During our investigations of the ESC distribution, we found othermoment results of potential interest. We present them below, considering a random variable X withthe ESC distribution. The mathematical details are omitted for reasons of space.
• A special moment formula is as follows:

E
[

1√
X + 1

]
=

1

12
√
2

[
3a(b − 4)C(2)− ab

√
3C[2

√
3] + 24

]
,

where C(x) denotes the Fresnel C integral defined by C(x) = ∫ x0 cos(πt2/2)dt (see [9,Section 8.25]). This integral is implemented in most mathematical software, such as R withthe package entitled pracma.
• Another special moment formula is as follows:
E
[
e−|X|

]
=
e[aπ2(9− 2b) + 1 + a + 10π2 + 9π4]− 1 + a + π2(9a − 10− 2ab)− 9π4

e(1 + π2)(1 + 9π2)
.

More succinct moment formulas are given below, some of which may be useful for varioustheoretical or practical applications involving the ESC distribution.
• We have

E
{

1

1 + [cos(πX)]2

}
=
1√
2
.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 16

• We have
E
{

1

1 + [sin(πX)]2

}
=
1√
2
.

The two results for the moments above are therefore the same.
• We have

E
{

1

1 + [tan(πX)]2

}
=
1

2
.

• We have
E {cos[π|X|]} =

a

8
(4− b).

• We have
E {sin[π|X|]} =

2

π
.

• We have
E [| cos(πX)|] =

2

π
.

• We have
E [| sin(πX)|] =

2

π
.The three results for the moments above are therefore the same.

• We have
E
[√
1 + cos(πX)

]
= −

2
√
2

105π
[a(8b − 35)− 105].

• We have
E
{

cos(πX)

1 + [sin(πX)]2

}
= a

{√
2− 1 + b

[√
2−
3

2

]}
.

3.5. Some derived distributions.

3.5.1. Transformed ESC distributions. Since the ESC distribution is new in the literature, it maybe interesting to derive transformed ESC distributions with different properties, such as differentsupports. For this purpose, we consider a random variable X with the ESC distribution andinvestigate the following transformed random variable:
Y = m(X),

where m(x) is a well-defined function on [−1, 1] or (−1, 1).
• For m(x) = πx , we get Y = πX , which is of support [−π, π]. Based on Proposition 3.1, Yhas the following cdf:

F∧(x ; a, b) = F∗

( x
π
; a, b

)
=
1

2π

[
π + x + a sin(x)

{
1−

b

3
[sin(x)]2

}]
, x ∈ [−π, π],

F∧(x ; a, b) = 0 for any x < −π and F∧(x ; a, b) = 1 for any x > π (still with a ∈ [−1, 1]and b ∈ [0, 1]).

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 17

• For m(x) = |x |, we get Y = |X|, which is of support [0, 1]. Based on Proposition 3.1, Yhas the following cdf:
F�(x ; a, b) = 2F∗(x ; a, b)− 1

= x +
a

π
sin(πx)

{
1−

b

3
[sin(πx)]2

}
, x ∈ [0, 1],

F�(x ; a, b) = 0 for any x < 0 and F�(x ; a, b) = 1 for any x > 1.
• For m(x) = (x + 1)/2, we get Y = (X + 1)/2, which is of support [0, 1]. Based onProposition 3.1, Y has the following cdf:

F?(x ; a, b) = F∗(2x − 1; a, b)

= x −
a

2π
sin(πx)

{
1−

b

3
[sin(πx)]2

}
, x ∈ [0, 1],

F?(x ; a, b) = 0 for any x < 0 and F?(x ; a, b) = 1 for any x > 1. This and the previous cdfcan be seen as modified versions of the cdf of the uniform distribution over [0, 1].
• For m(x) = tan[(π/2)x ], we get Y = tan[(π/2)X], which is of support R. Based onProposition 3.1, Y has the following cdf:
F�(x ; a, b) = F∗

[
2

π
arctan(x); a, b

]
=
1

2

{
1 +
2

π
arctan(x) +

a

π
sin[2 arctan(x)]

[
1−

b

3
{sin[2 arctan(x)]}2

]}
=
1

2

{
1 +
2

π
arctan(x) +

2ax

π(1 + x2)

[
1−

4bx2

3(1 + x2)2

]}
, x ∈ R.

This cdf can be seen as a modified cdf of the standard Cauchy distribution.
• For m(x) = sin[(π/2)x ], we get Y = sin[(π/2)X], which is of support [−1, 1]. Based onProposition 3.1, Y has the following cdf:
F∨(x ; a, b) = F∗

[
2

π
arcsin(x); a, b

]
=
1

2

{
1 +
2

π
arcsin(x) +

a

π
sin[2 arcsin(x)]

[
1−

b

3
{sin[2 arcsin(x)]}2

]}
=
1

2

{
1 +
2

π
arcsin(x) +

2a

π
x
√
1− x2

[
1−
4b

3
x2(1− x2)

]}
, x ∈ [−1, 1],

F∨(x ; a, b) = 0 for any x < −1 and F∨(x ; a, b) = 1 for any x > 1.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 18

• For m(x) = (1 − x)/(1 + x), we get Y = (1 − X)/(1 + X), which is of support (0,+∞).Based on Proposition 3.1, Y has the following cdf:
F/(x ; a, b) = 1− F∗

(
1− x
1 + x

; a, b

)
=
1

2

{
1−
1− x
1 + x

−
a

π
sin

[
π(1− x)
1 + x

][
1−

b

3

{
sin

[
π(1− x)
1 + x

]}2]}

=
x

x + 1
−

a

2π
sin

(
2πx

1 + x

){
1−

b

3

[
sin

(
2πx

1 + x

)]2}
, x > 0

and F/(x ; a, b) = 0 for any x ≤ 0. This cdf can be seen as a modified cdf of the standardLomax distribution, i.e., defined by q(x) = x/(x + 1) = 1− (x + 1)−1.
These are only examples of new distributions derived from the ESC distribution. Others may beconsidered depending on the objectives.
3.5.2. A new skewed normal distribution. In [19], the C distribution is used to create a trigonometricskewed version of the standard normal distribution by applying the Azzalini scheme introduced in [3].We can follow the same idea but using the ESC distribution, also for more flexibility thanks to thetwo additional parameters. To be precise, let us consider the pdf of the standard normal distributiondefined by

g(x) =
1√
2π
e−x

2/2, x ∈ R.

We then construct a new trigonometric skewed version of this pdf by considering the followingfunction:
f4(x ; a, b, λ) = 2g(x)F∗(λx ; a, b), x ∈ R,

where λ > 0 in an additional skew parameter, and F∗(x ; a, b) is still the cdf of the ESC distributiongiven in Proposition 3.1. Thus defined, f4(x ; a, b, λ) is a valid pdf. It can be expressed as
f4(x ; a, b, λ) =

1√
2π
e−x

2/2

[
1 + λx +

a

π
sin(λπx)

{
1−

b

3
[sin(λπx)]2

}]
, x ∈

(
−
1

λ
,
1

λ

)
,

completed by
f4(x ; a, b, λ) =

√
2

π
e−x

2/2, x ≥
1

λ

and f4(x ; a, b, λ) = 0 for any x ≤ −1/λ.Figure 4 shows the curves of this pdf for different values of a, b and λ.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 19

−2 −1 0 1 2 3

0
.0

0
.2

0
.4

0
.6

0
.8

x

p
d

f

a = 1  b = 0.001  λ = 1
a = − 1  b = 1  λ = 1.5
a = − 0.5  b = 0.4  λ = 0.5
a = 0.4  b = 0.7  λ = 12
a = − 1  b = 0.1  λ = 1

Figure 4. Curves of the pdf of the derived skewed normal distribution for differentvalues of a, b and λ
This figure illustrates the complexity of the deformations applied to the pdf of the standardnormal distribution based on our trigonometric skew scheme. In particular, we see some abruptchanges and pronounced peaks in the mode, which are quite rare properties in the family of skewednormal distributions.The derived skewed normal distribution may be of interest for the analysis of versatile skeweddata. It is more adaptable than the one developed in [19], mainly due to the presence of theadditional parameter b. We leave this practical aspect to future work.Of course, the same methodology can be applied to other basic symmetric distributions around

0 than the standard normal. For example, we can consider the pdf of the logistic, Student, Cauchyand Laplace distributions for g(x), which opens up new research directions.
4. Applications

This section looks at some statistical aspects of the ESC distribution, mainly when it is goingto be used to analyze data.
4.1. Estimation method. In practice, a variable with values in [−1, 1] can potentially be modeledby a random variable X with the ESC distribution. To get a precise idea of this probabilisticassumption, we assume that the parameters a and b as unknown, and try to estimate them efficientlybased on the available information of X that make up the data. A wide range of estimation methods

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 20can be used to do this. One of the most common and efficient is the ML estimation method. Let usformalize this in the context of the ESC distribution below.Suppose we have n data, observations of X , classically denoted as x1, . . . , xn. Then the MLestimates of a and b are given as
(â, b̂) = argmax(a,b)∈[−1,1]×[0,1] `(a, b),

where
`(a, b) =

n∑
i=1

ln [f∗(xi ; a, b)]

= −n ln(2) +
n∑
i=1

ln
[
1 + a cos(πxi){1− b[sin(πxi)]2}

]
.

This procedure of maximization can be carried out with any mathematical software, such as R withthe function nlminb. Once â and b̂ are obtained, the corresponding estimated pdf is given by
f̂∗(x) = f∗(x ; â, b̂) =

1

2

[
1 + â cos(πx){1− b̂[sin(πx)]2}

]
, x ∈ [−1, 1]. (3)

Using this function, we can visually assess the fit of the ESC distribution to the data. Morespecifically, we have an acceptable result if the corresponding normalized histogram of the datahas a shape that is well fitted by the curve of f̂∗(x) for x ∈ [−1, 1].The goodness of fit of different distributions can be compared using well-established mathemat-ical tools, such as the Akaike information criterion (AIC) and the Bayesian information criterion(BIC). In the context of the ESC distribution, these criteria are defined as
AIC = 2[k − `(â, b̂)], BIC = k ln(n)− 2`(â, b̂),

respectively, where k is the number of unknown parameters, i.e., k = 2.As the main competitor of the ESC distribution, we consider another variant of the C distribution:the AC distribution introduced in [5]. It is defined by the following two-parameter pdf:
f†(x ;α, β) =

β(π2 + β2)

2[π2 + (1− α)β2] sinh(β) [1 + α cos(πx)]e
βx , x ∈ [−1, 1], (4)

and f†(x ;α, β) = 0 for all x 6∈ [−1, 1], where α ∈ [−1, 1] and β ∈ R. Unlike the ESC distribution,the AC distribution is not designed to be exclusively symmetric. This makes it a challenger ofthe ESC distribution when dealing with "possibly symmetric" data, especially when the data arenot "perfectly symmetric". For this distribution, we can also use the ML estimation method toestimate the parameters α and β, say α̂ and β̂, respectively, determine the estimated pdf as
f̂†(x) = f†(x ; α̂, β̂), and compute the AIC and BIC for comparison purposes.For more general information on the ML estimation method, see [4].In the remaining parts of this section, for illustrative purposes, we test the ML estimation methodon the ESC distribution using two different examples of simulated data, i.e., data that can be

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 21obtained in credible scenarios (not real data). In particular, we compare the fit obtained with thatof the AC distribution.
4.2. Simulated example 1. In this simulated example, we look at the normalized daily air qualityindex (AQI) in a particular urban area over a period of 57 days. Each measurement of the AQIis contained in the range [−1, 1], where a value close to 0 indicates moderate air quality, whilevalues close to −1 represent poor air quality and values close to 1 represent good air quality.Conceptually, the data capture the natural variability of air quality due to various environmentalfactors, such as traffic patterns, weather conditions and industrial activities.The data are as follows: 0.877, -0.803, 0.623, -0.132, 0.063, 0.282, -0.725, 0.452, -0.290, 0.114,-0.029, -0.520, 0.711, -0.775, 0.906, 0.106, -0.506, 0.194, -0.744, 0.549, -0.339, 0.186, -0.263,0.109, -0.888, 0.739, -0.591, 0.731, -0.253, 0.533, -0.037, 0.307, -0.205, 0.122, -0.106, 0.858,-0.760, 0.586, -0.723, 0.280, -0.496, 0.037, -0.236, 0.159, -0.253, 0.105, -0.901, 0.709, -0.590,0.725, -0.298, 0.432, -0.052, 0.347, -0.194, 0.297, -0.108.To summarize these data, the minimum is −0.926, the first quartile is −0.194, the median is
0.002, the mean is −0.005122, the third quartile is 0.199, and the maximum is 0.920.This information indicates some symmetry in the distribution of the data. The main quantilefeatures of the data are illustrated by a boxplot in Figure 5.

−0.5 0.0 0.5

 

 

D
a

ta

Figure 5. Boxplot of the data of the simulated example 1
This boxplot confirms a certain symmetry of the data, centered around 0, with no outliers. TheESC distribution may therefore be appropriate for their analysis.Using the ML estimation method, we obtain â = 0.5688681 and b̂ = 1. From these estimates,we derive the estimated pdf f̂∗(x). Note that the high value of b̂ makes the estimated pdf of the

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 22ESC distribution really different from that of the C distribution. We also find AIC = 77.49385 and
BIC = 81.57996.The AC distribution and the ML estimation method are used for comparison. The estimates ofthe parameters involved in the AC distribution are α̂ = 0.36297403 and β̂ = 0.02174637, andwe obtain AIC = 79.11784 and BIC = 83.20395. Since the AIC and BIC associated with theESC distribution are lower than those associated with the AC distribution, the ESC distributionprovides the best fit to the data. As a visual check, in Figure 6, we plot the histogram of the dataand overlay the estimated pdfs of the ESC and AC distributions.

 

x

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.2

0
.4

0
.6

0
.8

1
.0

estimated pdf of the ESC distribution
estimated pdf of the AC distribution

Figure 6. Histogram of the data of the simulated example 1 and estimated pdfs ofthe ESC and AC distributions
The normalized histogram is well fitted by the two estimated pdfs, but that of the ESC distributionseems to be closer to the overall shape thanks to oscillatory features. This follows the resultsinterpreted from the values of the AIC and BIC.

4.3. Simulated example 2. This example simulates a different scenario. We consider a series ofmeasurements taken over time from a sensor array monitoring water quality parameters in a river.There are 49 values, ranging from −1 to 1, reflecting the variation in a particular water qualityparameter. Most values are centered around 0, due to the overall stability of the condition of theriver, but with a complex symmetry feature, including a slight asymmetry caused by occasionalfluctuations due to external factors such as run-off, pollution or weather changes.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 23The data are as follows: 0.096, -0.590, 0.811, -0.194, -0.309, 0.593, -0.926, 0.378, -0.635, 0.285,0.116, -0.382, 0.439, 0.020, -0.682, 0.724, -0.771, 0.920, -0.048, -0.121, 0.199, -0.251, 0.445, -0.599, 0.536, -0.807, 0.899, -0.797, -0.044, 0.030, 0.055, -0.035, -0.001, -0.064, 0.072, -0.073,0.131, 0.002, 0.150, -0.140, 0.217, -0.128, 0.192, -0.167, 0.186, -0.136, 0.264, -0.274, 0.163.To summarize these data, the minimum is −0.901, the first quartile is −0.298, the median is
0.037, the mean is 0.005649, the third quartile is 0.347, and the maximum is 0.906. This informationindicates some symmetry in the distribution of the data. As for the previous example, the mainquantile features of the data are illustrated by a boxplot in Figure 7.

−0.5 0.0 0.5

 

 

D
a

ta

Figure 7. Boxplot of the data of the simulated example 2
There are some outliers in this boxplot, but also some symmetry. Again, the ESC distributionmay be able to analyze these data.Using the ML estimation method for the ESC distribution, we obtain â = 0.7993845 and b̂ = 1.From these estimates, we derive the estimated pdf f̂∗(x). Note that the high value of b̂ makes theestimated pdf of the ESC distribution really different from that of the C distribution. We also find

AIC = 57.03759 and BIC = 60.82123.For comparison, we look at the AC distribution. We also use the ML estimation method. Thecorresponding estimates of the parameters are α̂ = 0.65030333 and β̂ = −0.02541488, andwe get AIC = 58.46521 and BIC = 62.24885. Since the AIC and BIC associated with the ESCdistribution are lower than those associated with the AC distribution, the ESC distribution providesthe best fit for the data. To support this claim, in Figure 8, we show the histogram of the data andoverlay the estimated pdfs of the ESC and AC distributions.

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 24

 

x

−1.0 −0.5 0.0 0.5 1.0

0
.0

0
.5

1
.0

1
.5

estimated pdf of the ESC distribution
estimated pdf of the AC distribution

Figure 8. Histogram of the data of the simulated example 2 and estimated pdfs ofthe ESC and AC distributions
We see that both estimated pdf fit the general shape of the normalized histogram. The advantageof the pdf of the ESC distribution is that it better captures the bars around x = 0, and also theoriginal variability of the data represented by the other bars.It is worth noting that during our statistical investigations, other simulated data examples wereproduced and it results that the ESC distribution was not uniformly the best compared to the ACdistribution. This is mainly due to the fact that the ESC distribution is only designed to analyzedata with a symmetric distribution, whereas the AC distribution offers more possibilities to capturedifferent skewnesses and is clearly more appropriate in some scenarios. If the normalized histogramof the data is symmetric in shape, the ESC distribution is more recommended. If there is skewness,the AC distribution is preferable.

5. Conclusion
In this article, we have developed a new symmetric extension of the famous C distribution,called the ESC distribution. In addition to being symmetric, it has the property of having thesupport [−1, 1], while maintaining a high degree of flexibility thanks to two adjustable parameters.In particular, one of them activates an original sine term in the main functions. This results insome oscillations in the pdf and hrf, which can be useful in different data fitting scenarios. Wehave investigated some of the main properties of the ESC distributions. In particular, a momentanalysis expresses several key measures in closed form, including the mean, variance, skewness,

https://doi.org/10.28924/ada/ma.5.7


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 25kurtosis, incomplete mean and mean deviation. A number of transformed ESC distributions withdifferent supports were also defined, together with a new skewed version of the standard normaldistribution. The statistical aspects of the ESC distribution were then examined using the MLestimation of the two parameters and two examples of simulated data. For these examples, thefit of the new distribution is satisfactory and outperforms that of another two-parameter extended(mainly asymmetric) version of the C distribution.Thus, this article provides a valuable extension of the C distribution and opens some perspectivesfor new statistical models with trigonometric properties. The ESC distribution can also be used asit is in different regression contexts, work that deserves a complete study, which we leave to futureresearch.
References

[1] M. Ahsanullah, M. Shakil, Some characterizations of raised cosine distribution, Int. J. Adv. Stat. Probab. 6 (2018)42-49. https://doi.org/10.14419/ijasp.v6i2.14988[2] M. Ahsanullah, M. Shakil, B.M.G. Kibria, On a generalized raised cosine distribution: some properties, characteri-zations and applications, Moroc. J. Pure Appl. 5 (2019) 63-85. https://doi.org/10.2478/mjpaa-2019-0006[3] A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Stat. 12 (1985) 171-178. https:
//www.jstor.org/stable/4615982[4] G. Casella, R.L. Berger, Statistical Inference. 2nd Edition, Duxbury Press, Pacific Grove, (2002).[5] C. Chesneau, The asymmetric cosine distribution, Asymmetry 1 (2024) 1-28. https://doi.org/10.55092/

asymmetry20240004[6] C. Chesneau, Investigating an asymmetric ratio cosine distribution, Pan-Am. J. Math. 3 (2024) 1-22. https://doi.
org/10.28919/cpr-pajm/3-22[7] S.A. El-Shehawy, M.M. Rizk, Study on effect of the deformation technique on the RC-distribution and its properties,Sci. J. Fac. Sci. Menoufia Univ. 28 (2024) 53-62. https://doi.org/10.21608/sjfsmu.2024.357491[8] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2. John Wiley & Sons, New York, (1971).[9] I.S. Gradshteyn, E.M. Ryzhik, Table of Integrals, Series, and Products. Edited by Jeffrey, A. and Zwillinger, D.Academic Press, NY, USA, 7th edition, (2007).[10] N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, vol. 2, 2nd edn, Wiley, New York, USA,(1995).[11] J.F. Kenney, E.S. Keeping, Mathematics of Statistics, Part 1, 3rd ed., Princeton, New Jersey, (1962).[12] M. King, Statistics for Process Control Engineers: A Practical Approach, First Edition, Wiley, New York, USA,(2017).[13] V. Kyurkchiev, N. Kyurkchiev, On the approximation of the step function by raised-cosine and laplace cumulativedistribution functions, Eur. Int. J. Sci. Tech., 4 (2016) 75-84.[14] J.J. Moors, A quantile alternative for kurtosis, J. R. Stat. Soc. D. 37 (1998) 25-32. https://doi.org/10.2307/
2348376[15] R Core Team. R: A Language and Environment for Statistical Computing [Internet], Vienna, Austria, (2016). https:
//www.R-project.org/[16] H. Rinne, Location-Scale Distribution: Linear Estimation and Probability Plotting Using MATLAB, Copyright:Prof. em. Dr Hors Rinne, Department of Economics and Management Science, Justus-Lieblig-University, Giessen,Germany, (2010). http://dx.doi.org/10.22029/jlupub-17571

https://doi.org/10.28924/ada/ma.5.7
https://doi.org/10.14419/ijasp.v6i2.14988
https://doi.org/10.2478/mjpaa-2019-0006
https://www.jstor.org/stable/4615982
https://www.jstor.org/stable/4615982
https://doi.org/10.55092/asymmetry20240004
https://doi.org/10.55092/asymmetry20240004
https://doi.org/10.28919/cpr-pajm/3-22
https://doi.org/10.28919/cpr-pajm/3-22
https://doi.org/10.21608/sjfsmu.2024.357491
https://doi.org/10.2307/2348376
https://doi.org/10.2307/2348376
https://www.R-project.org/
https://www.R-project.org/
http://dx.doi.org/10.22029/jlupub-17571


Eur. J. Math. Anal. 10.28924/ada/ma.5.7 26

[17] M. Shaked, J.G. Shanthikumar, Stochastic Orders and their Applications, Academic Press, New York, (1994).[18] R.K. Sinha, A thought on exotic statistical distributions, J. Math. Comput. Sci. 6 (2012) 366-369. doi.org/10.
5281/zenodo.1334219[19] L. Watagoda, H. Rupasinghe Arachchige Don, J.A. Sanqui, A cosine approximation to the skew normal distribution,Int. Math. Forum 14 (2019) 253-261.[20] R. Willink, Measurement Uncertainty and Probability, Cambridge University Press, First Edition, New York, USA,(2013).

https://doi.org/10.28924/ada/ma.5.7
doi.org/10.5281/zenodo.1334219
doi.org/10.5281/zenodo.1334219

	1. Introduction
	2. The ESC distribution
	2.1. A general distribution result
	2.2. Definition of the ESC distribution

	3. Functions and properties
	3.1. The cdf
	3.2. Quantile function
	3.3. Hazard rate function
	3.4. Moment analysis
	3.5. Some derived distributions

	4. Applications
	4.1. Estimation method
	4.2. Simulated example 1
	4.3. Simulated example 2

	5. Conclusion
	References

