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3
A NEW CLASS OF COS-G FAMILY OF DISTRIBUTIONS
WITH APPLICATIONS
Pankaj Kumar1, Laxmi Prasad Sapkota*2 and Vijay Kumar3
•
1,3Department of Mathematics & Statistics, DDU Gorakhpur University, Gorakhpur, India 2Department of Statistics, Tribhuvan University, Tribhuvan Multiple Campus, Palpa, Nepal [email protected], [email protected], [email protected]
Abstract
This paper introduces a novel family of probability distributions, termed the Cos-G family, which is derived from a trigonometric transformation approach. We present the general structural properties of this family and focus on one of its unique members. This newly proposed distribution, formulated from the inverse Weibull distribution, exhibits flexible hazard rate shapes, including reverse-J, increasing, and inverted bathtub forms. We investigate its fundamental statistical properties and employ the maximum likelihood estimation method to estimate its parameters. The performance of the estimation technique is assessed through a Monte Carlo simulation, revealing that biases and mean square errors decrease as sample size increases, ensuring reliable parameter estimation even for small samples. To illustrate its practical applicability, we fit the suggested model to three real-world datasets and compare its performance against existing models using various goodness-of-fit measures and model selection criteria. The results confirm the superiority of the proposed model in capturing complex data structures.
Keywords: Cos-G distribution, Inverse Weibull, Moment, Estimation, Goodness of fit.
1. Introduction
Real-world events are frequently studied using statistical distributions. Both novel developments for their application and the theory of statistical distributions are thoroughly researched. To explain a variety of real-world phenomena, several families of distributions have been developed. In fact, this fresh advancement in distribution theory is an ongoing practice. The majority of probability distributions suggested in the literature have a lot of parameters, which gives the model more adaptability. Some authors claim that it is challenging to acquire these estimates using numerical resources [1]. For modeling actual data, it is better to develop models with a limited number of parameters and a high level of flexibility. A team of scientists made the decision to use trigonometric functions to seek novel distributions in order to achieve this objective. Trigonometric models have gained popularity among scholars in recent years due to their adaptability and ability to be understood mathematically. Souza et al. [2] suggested a new class of trigonometric cosine distribution with a bathtub-shaped or increasing failure rate function called the Cos-G Class of distribution with base parameters (w > 0) among the various trigonometric G-family. The cumulative distribution function (CDF) for the Cos-G class of distribution are
f K(x;w)
f T n
F(x; w) = - sin(t)dt =1 - cos — K(x; w)
L2'
; x e K.
0
Souza et al. [3] utilized a similar methodology to propose the Sin-G family of distributions and include the Sin-Inverse Weibull distribution in the Sin-G class. Similarly, Souza et al. [4]
introduced a new Tan-G class with an increasing failure rate function or bathtub-shaped failure rate function, and focused on examining the Tan-BXII distribution as a member. A CDF exists for both the Sin-G and Tan-G classes of distributions.
F(x; w)
2 K(x;w)
cos(t)dt = sin
L—K(x;w) ;x eS.
4 K(x; w)
F(x; w)
sec2 (t)dt = tan
n K(x;w)
;x .
where K(x; w) is the CDF of any parent distribution and w is the vector of parameters of the parent distribution. The new sin-G family was created by [5], who also studied the sin-inverse Weibull model in specific. The CDF of the novel sin-G family of distribution are
n K(x;w)(K(x;w)+l)
F(x; w)
cos(t)dt = sin
L-K(x; w)(K(x; w)
l)
; x e S.
Also, Chesneau and Jamal [6] have defined the sine Kumaraswamy-G family of distributions as having two extra parameters to this family. Muhammad et al. [7] have defined the exponentiated sine-G family and analyzed the particular distribution as an exponentiated sine-Weibull distribution. Another trigonometric function-related probability model introduced by [8] is called arctan generalized exponential distribution. Using the sine-G family of distribution [9] have developed a new two-parameter model called sine Burr XII distribution. A new family of distributions related to the Sine function was developed by [10] and used with medical data. As a result, we have observed that the simple functions have a trigonometric distribution and are tractable formally see [3]. Additionally, without the use of any extra parameters, the sine transformation can significantly increase G(x) flexibility [6]. We are drawn to the cosine metamorphosis family because of these appealing qualities. In this research, we created a new family of trigonometric models using the cosine function, which we named the new class of cos-G family (NCC-G) of distributions.
This study is divided into several sections. In Section 2, we introduce the methodology of model development and key functions of the family of distributions. Section 3 presents some general properties of the NCC-G family, while Section 4 discusses methods of estimation. In Section 5, we introduce a specific member of the NCC-G family and present a detailed study, and in the application Section6, we provide the application of this model using three real datasets. Finally, Section 7 contains the conclusion.
2. The NCC-G family of Distribution (NCC-G FD)
In this study, a new family of distributions called NCC-G is suggested using the T-X approach as defined by [ll]. Consider a baseline CDF, represented by G(x;£), and a vector of associated parameters, denoted by £ > 0. The ratio of G(x; £) and l + G(x; £) can be treated as a function of the new family of distributions. For further information, refer to [l2]. Mathematically it can be
expressed as 0 as G(x;£) ^ 0; jG0k) ^ 2 as G(x;£) ^ l The CDF F(x;£) of the
NCC-G family of distributions is defined as
r G(x£) 1 l+G(x;£)
F(x; £ ) = —
sin(t) = l — cos
n
G(x; £) l + G(x; £ )J
; x eS.
(l)
Differentiating the Equation (1), the PDF f (x; £) of the family can be written as
f (x; £ )
n sin
n
G(x; £ ) l + G(x; £ )J
g(x; £ )
(l + G(x; £ ))2
;x .
(2)
2.1. Survival Function
The survival function of NCC-G FD is presented as
R(x; £) = 1 - F(x; £) = cos
n
G(x; £ ) 1 + G(x; £ )J
; x G ft.
2.2. Hazard Function
The Hazard function of NCC-G FD can be expressed as
H(x; £)
f (x; £ ) R(x; £ )
n sin
n
G(x; £) 1 + G(x; £ )J
g(x; £ )
(1 + G(x; £ ))2
cos n
G(x; £) 1 + G(x; £ )
2.3. The Quantile Function
; x G ft.
The quantile function is useful in statistical analysis and modeling, as it provides a way to estimate percentiles and other summary statistics of a probability distribution. Suppose Q(p) is the smallest value of X for which the probability that X ^ to that value is at least p. The quantile function Q(p; £) of CDF F(x; £) of NCC-G FD can be obtained as
Q( p; £ ) = G-1
cos X(1 - p) n - cos-1 (1 - p)_
p G (0,1).
(3)
Using equation (3) we can calculate the median, upper and lower quartile, quartile deviation (QD), coefficient of QD, skewness, and kurtosis, which are presented in Table 1.
Table 1: Various measures based on quantités of NCC-G FD
1
Statistics Expressions
Median G-1 cos-1 (0.5) n-cos-1 (0.5)
Lower Quartile G-1 cos-1 (0.75) n-cos-1 (0.75)
Upper Quartile G-1 ' cos-1 (0.25) n-cos-1 (0.25)
QD 1 G-1 ( cos-1 (0.25) \ G-1 ( cos-1 (0.75) \ 2 G \ n-cos-1 (0.25) / G \ n-cos-1 (0.75) /
Coefficient of QD c-1i cos-1(0.25) \ q-1( cos-1 (0.75) \ ^n-cos-1 (0.25) j ^n-cos-1 (0.75) j
c-1f cos-1 (0.25) \ + c-1f cos-1 (0.75) \ 1 n-cos-1 (0.25W ' 1 n-cos-1 (0.75) J
Skewness ([13]) Q( 3;£ )-2Q( z;? )+Q( ) Q( 4;£ )-Q( )
Kurtosis ([14]) Q( 7;£ )-Q( )-Q( 8;£)+Q( U)
Q( 4;S )-Q( b£)
3. Some Properties of NCC-G FD
3.1. Useful Expansion of NCC-G FD
Exponentiated distributions can be used to generate useful linear expansions. The CDF of the exponentiated-G (Exp-G) distribution for more information see [3, 15, 16], exponentiated distributions have well-known properties for a wide range of baseline CDF G(x; f) with parameter z > 0 is given by
Gz (x; f) = [G(x; f)]z ; x G ft, where x G ft. (4)
The PDF corresponding to (4) can be presented as
gz (x; f) = zg(x; f) [G(x; f)]^z-1) , x G K.
We can express the density function of the NCC-G FD in linear form using the series expansions shown below.
~ ^n y2n+1 y3 y5 y7 y9 sinx = nE0 (-1) j2n+iy. = y-3" +1 - 7 +9-•••;-TO < x <
(1+y)p = E (n)yn = 1 +1 y+^y2 +p(p-3(p -2)y3 + •••;y < 1.
n=0 V / ' ' '
The PDF of NCC-G FD is
to n2i+2(_ 1 )i
f (x,Z) = g(x,Z) E (2i + 1)/ (1 + G(x,Z))2i-1 (G(x,Z))2i+1. (5)
Further expanding Equation (5) using generalized binomial series expansion. The expression for f( x; Z) becomes
TO TO
f (x,Z) = g(x,Z) EZAij(G(x,Z)f+j+1, (6)
i=0j=0
u A n2i+2 (-1)' f 2i - 1
where Ai)= (2i+1)l [ j
3.2. Moments
The rth order moment (y!r) about the origin for the NCC-G FD is
TO
Vr = E(Xr) = J xrf (x)dx (7)
-TO TO
TOTO
= EE j xr (G(x, Z ))2i+j+1 g(x, Z)dx. (8)
i=0 j=0 -TO
Further moments can also be calculated using the quantile function for more detail see [17] as Let G(x; Z) = p ^ g(x; Z)dx = dp;0 < p < 1.
TOTO 1
F'r = E(Xr) = E E Aij/ P2i+j+1 QGg(P)dp,0 < p < 1. i=0 j=0 0
where G(x; Z) = p and QG(p) is the function of quantile.
3.3. Moment Generating Function (MGF)
The MGF (Mx(t)) for the NCC-G FD is
, , TO
TO ik TO TO TO ik <•
Mx (t) = E ü Vr = E E E h M xkg(x, Z) (G(x, Z ))2j dx
k=0 k! i=0 j=0 k=0 k! TO
Let G(x; Z) = p ^ g(x; Z)dx = dp;0 ^ p ^ 1.
, TO
TO TO TO ik ¡.
Mx(t) = E E E ÜAj p2i+j+1 QG(p)dp,0 < p < 1,
i=0 j=0 k=0 k' -TO
where G(x; Z) = p and QG (p) is the quantile function of the baseline distribution.
3.4. Incomplete Moments
y
The Incomplete moments of the NCC-G FD can be defined as Mr(y) = J xrf (x)dx. Therefore
0
incomplete moments for NCC-G FD are given by
y
TO TO rt
Mr(y) = EE N'Xrg(x;Z) (G(x,Z))2l+j+1 dx. (9)
l=0 j=0-TO
Alternately, Mr (y) may be defined in terms of quantile function as
G(y)
co co y
Mr(y) = EEj V2l+j+1 QrG(p)dp;0 < p < 1. l=0 j=0 0
3.5. Mean Residual Life
The mean residual life of the NCC-G FD can be defined as M (y) = Fy) Therefore, the mean residual life for NCC-G FD is given by
1
) l=0 j=0
Alternatively, M(y) may be calculated in term of quantile function as
G(y)
y
1 — f xf(x)dx
MM (y)
F(y)
TOTO p
1 — EEN'/ xg(x; £) {G(x; £)}2i+j+1 dx
_n :_n J
1
MM (y)
F(y)
1 — EEAW p2l+j+1 Qg(p)dp
i=0j=o 0
3.6. Inequality Measure
Lorenz and Bonferroni curves are utilized in various fields such as insurance, econometrics, and demography, among others, to analyze measures of inequality such as income and poverty. i) Lorenz Curve
y
Lorenz curve is defined as = 1 f xf(x)dx, where p is the mean of x, hence Lorenz curve
— TO
for NCC-G FD is given by
y
1 TO TO \>
LF(y) = - E E j xg(x; £) (G(x, £ ))2i+j+1 dx. (10)
1 i=0 j=0 -TO
Alternatively, in terms of quantile function as
G(y)
1 TO TO V
1 n
Lp(y) = - EEN/ P2i+j+1 Qg (p)dp.
1 J
V l=0 j=0 -co
ii) the Bonferroni Curve
The Bonferroni curve is given by BFy = Fy). From Equation (10), the Bonferroni curve for the NCC-G FD is obtained as
y
1 TO TO \>
BHy) = ^ E E Ay7 xg(x; Z) (G(x,Z))2l+j+1 dx.
i=0 j=0
3.7. Entropy
Entropy is a concept used to describe the degree of variation or uncertainty associated with a random variable. Its applicability is widespread and can be observed in various disciplines such as probability theory, medicine, insurance, engineering, life sciences, etc. in general. i) Renyi's Entropy
Entropy, which serves as a measure of the amount of variation or uncertainty associated with a random variable, finds its applications across several disciplines, including engineering, econometrics, and financial mathematics. Renyi [18] proposed the concept of entropy as a metric for quantifying
CO
variability and uncertainty, and it can be computed as follows: Rp(X) = j—p log f {f (x)}p dx ; p > 0 and p = 1. Applying Taylor™s series expansion [f (x, £)]p can be obtained in the form
[f (x; £= np (g(x; £))p
sin n
By considering the function of Taylor series
sin n
G(x; £) 1 + G(x; £ )
G(x;£) ' 1+G(x;£)
(1 + G(x; £))-2p .
at the point s = 1/4, we can write
[sin (ns)]p = ttik (T) (-1)k-r(4)k-sr,
k=0 r=0 V Z V4/
where ak = 1 [{sin (rcs)}p](k) | j. We have selected s = 4 because {sin (ns)}p is infinitely
' s=1
differentiable at this point and sin = -j^ = cos (f), which allows to have a more tractable expression for ak at a given p
œ k /k\ /1 \ k-r
[f (x; £ )]p = np (g(x; £ ))p EE aJk) (-1)k-r - (G(x; £ ))r (1 + G(x; £ ))
k=0 r=0 r 4
-(2p+r)
(11)
Further expanding Equation (11) using generalized binomial series expansion. The expression for [f (x; £)]p becomes
M k ~ . /k\ /1 \k-r /(2p+r)+m-1 \
f (x; £)]p = np EEE (-1)m+k-raAkr) 1 (G(x; £))r+m (g(x; £))p (12)
k=0 r=0m=0 V / V / V /
Substituting [f (x, £)]p into the expression for Rp(X), the Renyi™s entropy for NCC-G family of distribution is given by
Rp (X)
1 - p
log
œ k œ œ
EE E ^mkr (g(x; £ ))p (G(x; £ ))r+mdx
k=0 r=0 m=0 J
, /k\ /,\k-r ((2p+r)+m-1
where ^kr = (-1)m+k-r npa^J (i) ( m
ii) q-Entropy
The q-entropy is given by
H(p)
1 - P
log
1 -J {f (x)}p dx
; p > 0 and p = 1.
Substituting [f (x, £)]p from Equation (12) into the expression for H(p), the q-Entropy for NCC-G FD is given by
H(p)
1 - p
log
œ k œ œ
1 - E E E ^mkr (g(x; £))p (G(x,£))r+m dx
k=0 r=0 m=0
; p > 0 and p = 1.
p
p
1
œ
1
1
œ
iii) Shannon's Entropy
The Shannon's entropy for a random variable X with pdf f (x) is a special case of the Renyi's entropy when p 11. Shannon entropies are defined as yX = E(- log f (x)). For the NCC-G family of distribution is given by
nx = E
- logs EEAi;g(x, £ )(G(x, £ ))2i+j+1 ) U=o j=o
4. Estimation Method
4.1. Maximum Likelihood Estimation (MLE)
The parameters of the NCC-G FD are estimated in this section using the method of maximum likelihood. Given random sample x\,..., xn of size n with parameters vector £ from the NCC-G FD, let u = £T be (p x 1) parameter vectors, then the log density and total log-likelihood function respectively, are given by
l(x; £) = log n + log
and
n
l(x, £) = n log n + E log
sin < n
G(x; £ ) 1 + G(x; £ )
i=1
sin < n
G(xi; £) '1 + G(xi; £)
- 2 log (1 + G(x;£))+ logg(x; £)
- 2 E log (1 + G(xi; £)) + E logg(xi; £). (13)
i=1
i=1
Differentiating Equation (13) gives the score function's components of V(u) = as follows,
dl n f G(xi; £) — = n E coM ^ V '
i=1
Gk(xi;£) _2 E Gk(xi;£) + ngk(xi;£)
1 + G(xi ; £ H (1 + G(xi; £ ))2 ¿Î (1 + G(xi ; £ )) + E g(x,; £ ) ,
where g'k(x, £) = ^, g'k (x, £) = ^, G'k(x; £) = and (x; £) = ^^.
d2£
(xi ;£ d£
(xpj d2£
5. Special member of NCC-G FD
Generalization of several distributions can be made using the NCC-G FD. The special distribution, a new class of cosine inverse Weibull distribution, is introduced in this section.
5.1. New class Cos inverse Weibull (NCC-IW) istribution
The CDF and PDF of the Inverse Weibull (IW) distribution are respectively given by
and
G(x) = 1 - exp(-ax ß; x > 0,a,ß > 0
g(x) = aßx ß 1 exp(-ax ß
Hence using the CDF and PDF of IW, the CDF and PDF of the NCC-IW distribution are given by
F(x; a, ß) = 1 - cos
n
exp(-ax ß) 1 + exp(-ax-ß )_
; x > 0
(14)
f (x; a, ß) = naßx-(ß+1) sin
n
exp(-ax ß) 1 + exp(-ax-ß )_
exp(-ax ß) (1 + exp(-ax-ß ))2
f; x > 0
(15)
The reliability and hazard functions, respectively, are given by
exp(-ax-P)
R(x; a, ß) = cos
n
and
H(x; a,ß) = naßx-^D exp(-ax-)
1 + exp(-ax ß)_
exp(-ax-ß)
x > 0.
>2
sin
n
1 + exp(-ax ß)_
(1 + exp(-ax P)) The quantile function for the NCC-IW distribution is presented below
1
-1
cos n
Qx (p)
- W cos-1(1 - p)
a \n - cos-1 (1 - p)
exp(-ax ß) 1 + exp(-ax-ß)
, p e (0,1)
_ a = 0.15, ß = 0.25
a = 2.75, ß = 0.50
_ a = 3.00, ß = 1.25
- a = 2.20, ß = 2.75
- a = 1.75, ß = 3.50
_ a = 0.10, ß = 0.20
a = 2.50, ß = 0.15
_ a = 3.00, ß = 1.25
- a = 2.20, ß = 2.75
- a = 1.75, ß = 3.50
0 1 2 3 4 5 6 7
Figure 1: Shapes of PDF and HRF of NCC-IW distribution
5.2. Linear Expansion
Using Equation (6), Equation (15) can be expressed in linear form as
TOTO ^
f (x; Z ) = ELQijx-(ß+1) exp { -(2i + j + 2)ax-ß\, (16)
i=0j=0
where n- = n2i+2*ß(-iy (2i - 1 where nij = (2i+1)! I j
5.3. Moments
Using the PDF defined in Equation (16), the rth order non-central moment (^) for the NCC-IW distribution can be presented as
rfß-r'
TOTO M [-ß—
EEn-Kß' ; vß > r,
i=0j=0 [a{(2i + j)+ 2}] ß
where n
n a( 1) (2i 1 t and r(.) is the gamma function.
(2i+1)!
4
3
K 2
0
x
x
n
5.4. Skewness and Kurtosis
Using the Equation (17) we can obtain the first four (r = 1,2,3,4) non-central moments as:
. TO TO _p-1 f a 1 \
Mean = pi = EE°*j [a{(2i + j) + 2}]- p r V p > 1,
i=0j=0 VP/
TO TO W )
P2 = EEQ*j-v P J 7-2; v p > 2,
i=0 j=0
00 00 _
p3 = EEQ*j-v p y p-3; Vp >3,
i=0 j=0
^ CO CO
p4 = EEQ p-4' ■ ^ ^
¿=0j=0 [a{(2i + j)+ 2}] p Similarly, we can calculate the central moments using the above non-central moments as
Pi = p1,
' '2 P2 = P2 - Pi2,
[a{(2i + - j) + p-2 2}]
r №: )
[a{(2i + -/)+ p-3 2}]T
r (^: )
and
P3 = P3 3pip2 + 2pi3,
P4 = p4 - 4p3 p2 + 6p2 pi2 - 2pi4
2
Therefore skewness and kurtosis for the NCC-IW distribution are pi = p| and p2 = pf respec-
p2 p2
tively.
5.5. MGF
The MGF (Mx(i)) for the NCC-IW distribution is
TO TO TO ik Q* r (P--r)
Mx(i) = EEE^r-v p; ; vp > r. (is)
i=0j=0k=0 k [a{(2i + j) + 2}] p
5.6. Incomplete moments
The incomplete moments for NCC-IW distribution are given by
y
Mr(y) = EEQij xr-(s+1) exp{ - (2i + j + 2)ax-p} dx i=0j=0 0
1 ^^ „ T^, (2i + j + 2)ay-p) where 7(.) incomplete gamma function.
p E E Qi) -p i=0 j=0 {(2i + j + 2)a} p
5.7. Mean Residual Life
The mean residual life for the NCC-IW distribution is given by
M (y)
1
F(y) 1
W)
y
TO TO ^ .
V - x-p exp j -(2i + j + 2)ax-p \
nii i=0j=0 0
v - p EE°
p i=0 i=\
1 nthr, (2i + j + 2)ay-p
i=0 j=0
{(2i + j + 2)a} p
5.8. Entropy
i) Renyi™s Entropy The Renyi™s entropy for NCC-IW distribution is given by
Rp(X)
1
1 - p 1
1-p
log log
/v n
EE E Vkrm (*P)p x-p(P+1) exp (-(r + m + p)ax-
k=0 r=0 m=0 0
E EE E * r (№+1»
p
(p-m+1)
k=0 r=0 m=0 y {(r + m + p)a} p +\
, /k\ /,\k-r ((2p+r)+m-1
where Vmkr = (-1)m+k-r nakd) (1) ( m ii) q-Entropy
The q-Entropy for NCC-IW distribution is given by
H(p)
1
1 - p 1
1-p
log log
1 - Vkrm (ap)p J x-p(P+1) exp(-(r + m + p)ax-0
(p-m+1) + 1
1 - Vkr,
(pp nt +1D p
(p-1)(p+1)
{(r + m + p)a} p
+1
, /k\ /,\k-r ((2p+r)+m-1
where p > 0, p = 1 and Vkrm = (-1)m+k-r npak (^J (i) ^ m iii) Shannon's Entropy
The Shannon entropy for the NCC-IW distribution is given by
Vx = E
{TO TO { ^
E E Oijx-(P+1) exp { - (2i + j + 2)ax-p \
i=0 j=0
5.9. Inequality Measure
i) Lorentz Curve: The Lorenz curve for NCC-IW distribution is given by
-F(y)
n W p
EE°'M x-p exp(-a(2i + j + 2)x-p )dx
v i=0 j=0 -TO
a E E o 7(P-1, (2i+j+2)ay-^ v E E 0ij - .
v i=0j=0 {(2i + j + 2)a} P
y
ii) Boneferroni Curve
The Bonferroni curve for the NCC-IW distribution is given by
My)
nF(y)
1
CO CO p
EE T'i x—p exp(—a(2i + j + 1)x—p )dx
i—n —n J
i=0 j=0
_ i EEa-, (2i + j + 2)ay-^
5VF(y) i_0j_0 v {(2i + j + 2)a}
5.10. Parameter estimation of NCC-IW distribution
Our current focus is on determining the parameters of the NCC-IW model through the MLE method. The objective is to compute the MLEs for the parameters a and p. To achieve this, we will examine the log-likelihood of a vector X _ (x1,...,xn)T of size n composed of independent random variables from the NCC-IW distribution.
a, p) _ n log(nap) — (p + 1) E log xi + E log sin
i_1
— 2 E log(1+exp(—ax— p ^ — a E xi
i_i i_1
i_1
n
exp(—ax- p) 1 + exp(—ax—p)
(19)
Partially differentiating the Equation (19) with respect to p and a yields the components of the score function V(u) _ ^dp, Ja) as follows
Kdp'da/
dl n n, nx—p log(xi) exp(—ax—p)
- _ p— e log xi + ™ E i, p \i2 cot
dp p i_1 i_1 (1 + exp(—ax—p )J
2a EE ^ log(xi) exp— x—p) + a EE ^ log(xi).
n
exp(—ax— p) 1 + exp(—ax—p)_
+
i_1 (1 + exp(—ax— p )) i_1
dl
and
di n t— _ — n E
A/v /v
da a
x{ pexp(—ax( p) 1 (1 + exp(—ax—p ))
cot
n
exp(—ax— p)
1 + exp(—ax—p)_
+ 2 E
x( pexp(—ax( p)
E
1(1 + exp(—ax— p )) i_1
The MLEs of p and a are obtained by maximizing l(x; a, p) in p and a, which can be done by solving simultaneously the equation: dp _ 0 and da _ 0.
5.11. Simulation Study
We used the maxLik R package developed by [19] to create samples from the quantile function defined in Equation (14) for various parameter combinations of the NCC-IW distribution. The MLEs were calculated for each sample using the maxLik() function with the BFGS algorithm. This allowed us to test parameter estimation problems, such as the sharpness or flatness of the likelihood function and provided estimates for the size and direction (underestimate or overestimate) of the MLEs bias. We repeated the procedure 1000 times, with 25 samples of sizes ranging from 10 to 250. We then calculated the bias and mean square error (MSE) for each simulation. In addition, we provided lower confidence limit (LCL) and upper confidence limit (UCL) estimated values with a 5% level of significance. The results of the experiment are summarized in Tables 2 and 3, which show the bias and MSEs for each parameter, along with the LB and UB for the MLEs. As the table shows, the MLE method consistently estimates the
1
n
n
p
n
parameters of the proposed model. Moreover, as the sample size increases, the MLEs gradually approach the actual values of a and p. In Figures 2 and 3, we have displayed a clear picture of MSEs with 95% confidence bound (dark region) for a and p.
100 150
Sample size
100 150
Sample size
Figure 2: MSE plots of a and p with 95% CI for initial values a = 0.5 and p = 1.25.
-0.25
-0.50
200
250
0
200
250
100 150
Sample size
V
100 150
Sample size
2
-a 1
0
-0.25
-0.50
-1
0
200
250
Figure 3: MSE plots of a and p with 95% CI for initial values a = 0.75 and p = 1.5.
6. Application
Employing three real data sets, we demonstrate the applicability of the NCC-IW distribution in this section. The data sets employed for the application of the suggested distribution are given below:
6.1. Model Analysis
To analyze the data sets under study, we calculate several widely used goodness-of-fit statistics. The fitted models are then compared using various measures, including the log-likelihood value (-2logL), Akaike information criterion (AIC), Hannan-Quinn information criterion (HQIC), Anderson-Darling (AD), Kolmogorov-Smirnov (KS) with p-values, and Cramer-von Mises (CVM). For additional information see [1]. All the essential computations are carried out in R-software. For the comparison of fitting capability we have selected some models such as inverse Weibull (IW), arctan generalized exponential (ArcTGE) [8], arctan Lomax (ArcTLx) [20], arcsine exponential (ASE) [21], Tan Burr XII (TBXII) [4], New Cosine Weibull (NCW) [22], Exponentiated Cos Weibull (EcosW) [7], arcsine exponentiated Weibull (ASEW) [23], Cos Weibull (CosW) [2] and Sine inverse Weibull (Sin-IW) [3]. Data set I:
The dataset from [24] contains information on the relief times of 20 patients who were administered an analgesic. An analgesic is a type of medication that is commonly used to reduce pain, and the relief time refers to the duration for which the patients experience relief from their pain after taking the medication. The data are "1.1,1.4,1.3,1.7,1.9,1.8,1.6, 2.2,1.7, 2.7, 4.1,1.8,1.5,1.2, 1.4, 3, 1.7, 2.3,1.6, and 2.0".
Table 2: Bias, MSEs, and LCL and UCLfor MLEs with initial values a = 0.5 and f = 1.25.
n biasa biasf msex msef LCLa UCLX LCLf UCLf
10 0.0904 0.1500 0.0444 0.4191 0.3381 1.0299 0.7585 2.5927
20 0.0445 0.0633 0.0143 0.0823 0.3761 0.7924 0.8920 1.9588
30 0.0248 0.0283 0.0079 0.0400 0.3893 0.7076 0.9577 1.7161
40 0.0194 0.0249 0.0056 0.0295 0.3918 0.6853 0.9747 1.6231
50 0.0152 0.0182 0.0040 0.0219 0.4117 0.6556 1.0243 1.5883
60 0.0136 0.0174 0.0034 0.0191 0.4155 0.6376 1.0242 1.5585
70 0.0126 0.0100 0.0027 0.0146 0.4284 0.6194 1.0481 1.512
80 0.0083 0.0138 0.0023 0.0140 0.4260 0.6108 1.0596 1.5091
90 0.0069 0.0141 0.0020 0.0119 0.4239 0.6032 1.0692 1.4905
100 0.0070 0.0074 0.0021 0.0105 0.4316 0.6074 1.0601 1.467
110 0.0072 0.0060 0.0017 0.0095 0.4308 0.5913 1.0866 1.4777
120 0.0037 0.0130 0.0014 0.0090 0.4359 0.5831 1.0944 1.4520
130 0.0060 0.0139 0.0014 0.0083 0.4393 0.5823 1.0997 1.4567
140 0.0062 0.0107 0.0013 0.0074 0.4405 0.5821 1.1054 1.4331
150 0.0058 0.006 0.0013 0.0068 0.4424 0.5814 1.1116 1.4269
160 0.0054 0.0102 0.0012 0.0062 0.4428 0.5776 1.1226 1.4222
170 0.0060 0.007 0.0011 0.0061 0.4505 0.5729 1.1146 1.4056
180 0.0032 0.0044 0.0010 0.0054 0.4455 0.5704 1.1236 1.4032
190 0.0041 0.0059 0.0010 0.0056 0.4487 0.5716 1.1212 1.4118
200 0.0031 0.0026 9.00E-04 0.0049 0.4472 0.5687 1.1261 1.3952
210 0.0044 0.0058 9.00E-04 0.0048 0.4493 0.5619 1.1206 1.3927
220 0.0033 0.0076 8.00E-04 0.0047 0.4526 0.5645 1.1295 1.4010
230 0.0046 -6.00E-04 8.00E-04 0.0047 0.4529 0.5606 1.1258 1.3934
240 0.0036 0.0042 7.00E-04 0.0041 0.4531 0.5606 1.1335 1.3816
250 0.0028 0.0014 7.00E-04 0.0040 0.4505 0.5561 1.1315 1.3801
Table 3: Bias, MSEs and LCL and UCLfor MLEs with initial values a = 0.75 and f = 1.5.
n biasa biasf msex msef LCLa UCLa LCLf UCLf
10 0.1424 0.2518 0.0942 0.7713 0.5183 1.5550 0.9717 3.5167
20 0.0692 0.0882 0.0309 0.1166 0.5616 1.1901 1.0949 2.3295
30 0.0363 0.0789 0.0164 0.0774 0.581 1.0422 1.1612 2.1797
40 0.0294 0.0383 0.0121 0.0458 0.6083 1.0183 1.1821 2.0153
50 0.0227 0.0311 0.0088 0.0337 0.6022 0.9684 1.2209 1.9084
60 0.0193 0.0287 0.0071 0.0276 0.6224 0.9468 1.2437 1.8913
70 0.0162 0.025 0.0063 0.0240 0.6247 0.9314 1.2604 1.8680
80 0.0139 0.0238 0.0055 0.0206 0.6397 0.9192 1.2795 1.8384
90 0.0130 0.0131 0.0048 0.0175 0.6507 0.9126 1.2956 1.8022
100 0.0107 0.0176 0.0043 0.0175 0.6418 0.8926 1.2919 1.8124
110 0.0085 0.0112 0.0038 0.0137 0.6507 0.8873 1.3022 1.7490
120 0.0085 0.0098 0.0033 0.0126 0.652 0.8749 1.3022 1.7395
130 0.0071 0.0152 0.0031 0.0116 0.6577 0.8734 1.3151 1.7406
140 0.0091 0.0123 0.0033 0.0115 0.6556 0.8793 1.3232 1.7467
150 0.0065 0.0139 0.0025 0.0101 0.6637 0.8666 1.3344 1.7227
160 0.0075 0.0051 0.0024 0.0090 0.6693 0.8575 1.3315 1.7045
170 0.0074 0.0110 0.0025 0.0083 0.6709 0.8606 1.3356 1.7040
180 0.0080 0.0105 0.0022 0.0090 0.6712 0.8471 1.3412 1.7089
190 0.0053 0.0116 0.0021 0.0078 0.6685 0.8571 1.3481 1.6881
200 0.0052 0.0046 0.0021 0.0068 0.6694 0.8492 1.3471 1.6665
210 0.0040 0.0056 0.0019 0.0070 0.6731 0.8479 1.3615 1.6799
220 0.0055 0.0051 0.0019 0.0075 0.6758 0.8441 1.3438 1.6867
230 0.0055 0.0049 0.0017 0.0062 0.6799 0.8439 1.3583 1.6725
240 0.0049 0.0086 0.0018 0.0057 0.6770 0.8414 1.3603 1.6553
250 0.0055 0.0092 0.0016 0.0060 0.6822 0.8406 1.3621 1.6657
Table 4: MLEs with SE (in parentheses) (dataset I)
Model Parameter(SE)
NCC-IW(a, p) 3.2906(0.5941) 3.9558(1.0140) -
IW(A, 0) 4.0175(0.7060) 6.0224(2.0083) -
ArcTGE(a, A, 0) 0.0000(1.5645) 19.3864(6.0429) 1.8579(0.2245)
ArcTLx(a, p, 0) 147.2664(44.0127) 0.2871(0.2782) 12.3869(9.6739)
ASE(0) 127.8946(4.8432) - -
ASEW(A, 0, u) 1.0488(0.1284) 104.561(19.0921) 3.1656(0.1303)
NCW(A, 0) 0.2505(0.0810) 2.2930(0.3402) -
TBXII(A, v, 0) 1.3946(0.1597) 10.3624(5.0171) 0.3937(0.2735)
ECosW(p, A, 0) 0.2386(0.0486) 0.2789(0.0712) 2.7222(0.1824)
CosW(p, 5) 2.2183(0.3323) 0.5655(0.0471) -
SinIW(5, 0) 5.3385(1.4594) 2.8386(0.4882) -
In Tables 4, 6, and 8, we have presented the estimated values of the parameters and their corresponding standard error (SE in parentheses) of the models under study using the MLE method. Similarly, in Tables 5, 7, and 9, we have presented the model selection and goodness of fit statistics such as log-likelihood, AIC, HQIC, KS, AD, and CVM for all three data sets. It has been observed that the suggested model NCC-IW has the least statistics as compared to IW, ArcTGE, ArcTLx, ASE, ASEW, NCW, TBXII, ECosW, CosW, and Sin-IW. Hence NCC-G is more flexible (even four trigonometric distributions having three parameters) and provides a good fit. Also, we have displayed the graphical illustrations of the fitted models in Figures 5, 7, and 9. These figures also verified that the NCC-G model can perform well as compared to candidate models.
Table 5: Some selection criteria and goodness-of-fit statistics (dataset-I)
Model
NCC-IW
IW
ArcGE
ArcLmx
ASE
ASEW
NCW
Tan-BXII
CosW
NCosW
Sin-IW
-2logL AIC HQIC KS p(KS) CVM p(CVM) AD p(AD)
31.1170 30.8174 33.4131 35.6262 154.7472 31.1885 48.6870 31.0804 40.6035 37.4854 31.1572
35.1170
34.8174
39.4131
41.6262
156.7472
37.1885
52.6870
37.0804
46.6035
41.4854
35.1572
35.5057 35.2062 39.9962 42.2094 156.9416 37.7716 53.0757 37.6636 47.1867 41.8742 35.5460
0.1148 0.1020 0.1516 0.1240 0.8863 0.1170 0.1467 0.0919 0.1922 0.1770 0.1069
0.9548 0.9854 0.7473 0.9182 0.0000 0.9470 0.7829 0.9959 0.4508 0.5576 0.9763
0.0306 0.0266 0.0767 0.0662 5.1247 0.0363 0.1078 0.0231 0.1840 0.1279 0.0292
0.9770 0.9880 0.7169 0.7806 0.0000 0.9551 0.5521 0.9944 0.3022 0.4681 0.9813
0.1772
0.1545
0.4214
0.5268
31.4397
0.2096
0.7800
0.1377
1.0593
0.7563
0.1808
0.9956 0.9984 0.8256 0.7175 0.0000 0.9877 0.4940 0.9994 0.3267 0.5118 0.9949
~i-1-1-1-1-1-r
1.0 1.5 2.0 2.5 3.0 3.5 4.0
~l-1-1-1-1-T
1.0 1.5 2.0 2.5 3.0 3.5 4.0
Empirical quantiles
Figure 4: KS and Q-Q plots (dataset-I).
x
Figure 5: Estimated PDF (left) and empirical vs estimated CDF (right) (dataset-I). Data set-II
The following data set was obtained by [25] consisting of 128 observations on the time intervals, measured in seconds, between the arrivals of vehicles at a specific location on a road. 0.2, 0.5, 0.8, 0.8, 0.8,1.0, 1.1,1.2, 1.2,1.2, 1.2,1.2, 1.3,1.4, 1.5,1.5, 1.6,1.6, 1.6,1.7, 1.8,1.8, 1.8,1.8, 1.8,1.9, 1.9, 1.9, 1.9,1.9, 1.9,1.9, 2.0, 2.1, 2.1, 2.2, 2.3, 2.3, 2.4, 2.4, 2.5, 2.5, 2.5, 2.6, 2.6, 2.7, 2.8, 2.8, 2.9, 3.0, 3.0, 3.1, 3.2, 3.4, 3.7, 3.9, 3.9, 3.9, 4.6, 4.7, 5.0, 5.1, 5.6, 5.7, 6.0, 6.0, 6.1, 6.6, 6.9, 6.9, 7.3, 7.6, 7.9, 8.0, 8.3, 8.8, 8.8, 9.3, 9.4, 9.5,10.1,11.0,11.3, 11.9,11.9, 12.3,12.9, 12.9,13.0,13.8, 14.5,14.9, 15.3,15.4, 15.9, 16.2, 17.6, 20.1, 20.3, 20.6, 21.4, 22.8, 23.7, 24.7, 29.7, 30.6, 31.0, 33.7, 34.1, 34.7, 36.8, 40.1, 40.2, 41.3, 42.0, 44.8, 49.8, 51.7, 55.7, 56.5, 58.1, 70.5, 72.6, 87.1, 88.6, 91.7, 119.8, 125.3'
Table 6: MLEs with SE (in parentheses) (dataset-II)
model Parameter(SE)
NCC-IW(a, ß)
IW( A, d ) ArcTGE( a, A, e ) ArcTLx( a, ß, d ) ASE( d ) ASEW( A, d, v ) NCW( A, d ) tbxii( a, v, e) ECosW( ß, A, e ) CosW( ß, 5 )
SinIW( 5, e)
0.6748(0.0476) 0.8183(0.0540) 1.00E-04(0.3004)
I.00E-04(1.7292)
II.6716(1.5853) 0.2832(0.0181) 0.3294(0.0389) 1.5673(0.3041) 0.1949(0.0172) 0.5566(0.0350) 3.0753(0.2472)
2.0442(0.1524) 2.6651(0.2527) 0.6685(0.0737) 0.0833(0.0310)
36.6683(2.9617) 0.5770(0.0359) 2.7668(0.7182) 0.4319(0.0000) 0.1655(0.0220) 0.5944(0.0385)
0.0475(0.0063) 1.6100(0.3969)
2.8476(0.1367)
0.2870(0.0962) 0.7179(1.00E-04)
Table 7: Some selection criteria and goodness-of-fit statistics (dataset-II)
model -2logL AIC HQIC KS p(KS) CVM p(CVM) AD p(AD)
NCC-IW 924.4949 928.4949 930.8125 0.0616 0.717 0.1390 0.4253 1.0806 0.3175
IW 921.1562 925.1562 927.4738 0.0604 0.7381 0.1323 0.4488 0.974 0.3711
ArcGE 948.3638 954.3638 957.8402 0.1481 0.0073 0.8221 0.0064 4.4936 0.0050
ArcLmx 929.2980 935.2980 938.7744 0.0976 0.1745 0.2488 0.1899 1.7906 0.1202
ASE 956.6371 958.6371 959.7959 0.1501 0.0063 0.6261 0.0191 4.0879 0.0079
ASEW 912.5793 918.5793 922.0557 0.0886 0.2680 0.1858 0.2969 1.1082 0.3051
NCW 998.8923 1002.892 1005.210 0.1207 0.0480 0.3327 0.1096 2.8599 0.0323
Tan-BXII 917.7461 923.7461 927.2225 0.0787 0.4057 0.2086 0.2516 1.2355 0.2544
ECosW 937.7763 943.7763 947.2527 0.1170 0.0603 0.4531 0.0523 2.8386 0.0332
CosW 927.9831 931.9831 934.3006 0.1162 0.0632 0.3467 0.1003 2.1999 0.0716
Sin-IW 916.8895 920.8895 923.2071 0.0868 0.2903 0.1819 0.3056 1.1174 0.3010
1.0 -
0.8 -
0.6 -
0.4 -
0.2 -
Empirical quantiles
Figure 6: KS and Q-Q plots (dataset-ll).
6
5
4
3
2
0
x
NCC-IW IW
ArcTGE
ArcTL
ASE
ASEW
NCW
TBXII
ECosW
CosW
SinIW
I
20
40
60
0.4 0.2
Empirical Fitted
50
100
x
x
Figure 7: Estimated PDF (left) and empirical vs estimated CDF (right) (dataset-ll). Data set-III
We have used the real data reported by [26] and it represents the failure time of 30 items "0.602, 0.603, 0.603, 0.615, 0.652, 0.663, 0.688, 0.705, 0.761, 0.770, 0.868, 0.884, 0.898, 0.901, 0.911, 0.918, 0.935, 0.953, 0.983,1.009,1.040,1.097, 1.097, 1.148,1.296,1.343,1.422, 1.540, 1.555,1.653"
Table 8: MLEs with SE (in parentheses) (dataset-lll).
Model Parameter(SE)
NCC-IW(a, f) 3.178(0.4814) 0.4482(0.1032) -
IW(A, 0) 3.8881(0.555) 0.4275(0.1108) -
ArcTGE( a, A, 0 ) 4.2E-07(1.3892) 31.5836(5.4017) 4.168(0.1055)
ArcTLx( a, f, 0 ) 143.7539(4.3514) 0.3264(0.2077) 18.9209(10.5266)
ASE( 0 ) 194.3346(5.9316) - -
ASEW( A, 0, v ) 1.2673(0.1458) 41.0829(6.893) 5.4445(0.3188)
NCW( A, 0 ) 1.172(0.1733) 2.662(0.3434) -
TBXII(A, v, 0) 0.7768(0.1457) 7.3693(3.3169) 0.6354(0.5186)
ECosW( f, A, 0 ) 0.1726(0.0315) 2.368(0.0046) 3.2934(5.00E-04)
CosW( f, 5 ) 2.5699(0.3352) 1.0907(0.0648) -
SinIW( 5,0 ) 0.8297(0.1375) 2.7635(0.3757) -
Table 9: Some selection criteria and goodness-of-fit statistics (dataset-lll).
Model -2logL AIC HQIC KS p(KS) CVM p(CVM) AD p(AD)
NCC-IW 8.7135 12.7135 13.6100 0.1608 0.4197 0.0992 0.5920 0.6056 0.6413
IW 7.6494 11.6494 12.5459 0.1432 0.5700 0.0793 0.6995 0.5146 0.7306
ArcGE 7.2658 13.2658 14.6105 0.0901 0.9679 0.0479 0.8924 0.3760 0.8712
ArcLmx 12.3954 18.3954 19.7402 0.1262 0.7260 0.0542 0.8542 0.5154 0.7297
ASE 224.6097 226.6097 227.0579 0.9413 0.0000 8.6137 0.0000 62.4475 0.0000
ASEW 6.4640 12.4640 13.8087 0.1141 0.8293 0.0561 0.8419 0.4091 0.8385
NCW 26.5153 30.5153 31.4118 0.1328 0.6649 0.0797 0.6969 0.7164 0.5440
Tan-BXII 8.9087 14.9087 16.2535 0.1074 0.8792 0.0536 0.8575 0.4021 0.8456 •
ECosW 12.0969 18.0969 19.4416 0.1238 0.7477 0.1053 0.5622 0.7074 0.5514
CosW 9.8726 13.8726 14.7692 0.1004 0.9230 0.0741 0.7304 0.5449 0.7001
Sin-IW 7.1457 11.1457 12.0422 0.1111 0.8526 0.0567 0.8383 0.4210 0.8265
1.0 -
0.8 -
0.6 -
0.4 -
0.2 -
0.0 -
Figure 8: KS and Q-Q plots (dataset-lll).
Figure 10: Countour plots for three data sets under study respectively.
7. Conclusion
A new family of distributions, known as the Cos-G family, has been developed by transforming the cosine function based on the ratio of CDF G(x) and 1 + G(x) of a baseline distribution. The general properties of this distribution family have been described. To create a member of this family with a hazard function that is increasing, decreasing, or inverted bathtub-shaped, the Inverse Weibull distribution was used as a baseline distribution. The resulting distribution, called NCC-IW, was analyzed for its statistical properties, and its parameters were estimated using the MLE method. The estimation procedure was evaluated through a Monte Carlo simulation, which showed that the biases and mean square errors decrease as the sample size increases, even for small samples. The NCC-IW distribution was then applied to three real data sets, and using model selection criteria and goodness-of-fit test statistics, it was shown to outperform other existing models with more parameters. This suggests that the Cos-G family and its member distribution have wide applications in fields such as medical science, reliability engineering, and survival analysis, and can lead to the development of new models in the future.
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