CC BY
18
A NOVEL ASYMMETRIC COMPOUND CLASS OF DISTRIBUTIONS WITH ESTIMATION AND
APPLICATION
A.G. Al-Kilany1, Amal S. Hassan- , L.S. Diab3, and E.S. El-Atfy1
1
Faculty of Science for (girls), Al-Azhar University, Nasr City, 11884, Egypt,
2*Faculty of Graduate Studies for Statistical Research, Cairo University, 12613, Giza, Egypt
amal52_soliman@cu.edu.eg
3Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic
University (IMSIU), Riyadh 11432, Saudi Arabia Correspondence: Email amal52_soliman@cu.edu.eg
Abstract
This paper introduces and discusses the novel asymmetric class of distributions that have the name inverse power Lomax power series (IPLPS). This class of distributions is produced by combining the inverse power Lomax with the power series distributions. This combined approach provides an opportunity for the creation of flexible distributions with significant physical implications in many fields, like biology and engineering. The IPLPS distributions encompass several new compound distributions as sub-models along with a new class of compound distributions. Many statistical features, including moments, quantile function, conditional moments, inverse moments, uncertainty measures, and probability-weighted moments, are obtained. As a special model of the generated class, the parameters of the inverse power Lomax Poisson distribution are estimated by different methods, including least squares, Cramer von Mises, maximum likelihood, and weighted least squares. Through an extensive simulation analysis, the execution of different parameter estimation techniques for the inverse power Lomax Poisson model is performed to show its validity based on its mean squared error and absolute bias. Two real datasets are utilized to show the practicality of the newly generated model. Results show that the inverse power Lomax Poisson distribution provides the most fitted model for these datasets in comparison to other distributions such as power Lomax, Marshall-Olkin power Lomax, power Lomax Poisson, and Topp-Leone Lomax distributions.
Keywords: Power series distributions, inverse power Lomax distribution, moments, compounding, Havrda and Charvat measure, Cramer von Mises.
1. Introduction
Recent academic focus has shifted towards the creation of new univariate distributions. Univariate distributions, whether for theoretical, practical, or combined purposes, hold significant importance in statistical and related fields. Analyzing the reliability of experimental failure components is a primary objective. It's often assumed that these failures occur due to certain processes, yet a thorough investigation into the causes of component failure seems lacking; see Barreto-Souza et al. [1]. Consider a system's lifetime composed of N components, and N is the discrete random variable that follows geometric, Poisson, logarithmic, or binomial distributions.
Power series (PS) is the general form of these chosen distributions. For further information on the PS class of distributions, refer to Noack [2]. Suppose that X denotes the continuous random variable for each component. Consequently, the random variable X= Min(Xi,X2,...,XN) or X= Max(Xi,X2,...,XN) signifies any component lifetimes depending on whether they are arranged in a series or in parallel structure, respectively.
Suppose that the random variable N associated with the PS class of distributions, characterized by a probability mass function, is given by:
P( N = n) = , n = 1,2,.......
C(6)
TO
where an > 0 only dependent on n, and C(d) = ^ andn is finite.
n=1
Several compound lifetime models have been created by combining several lifetime distributions with the PS class of distributions. For instance, the exponential PS [3], the Weibull-PS [4], Lindley-PS [5], exponential Pareto-PS [6], Burr XII-PS [7], exponentiated power Lindley-PS [8], generalized Burr XII-PS [9], odd log-logistic-PS [10], Topp-Leone generalized exponential-PS [11], power function-PS [12], inverse gamma-PS [13], inverse exponentiated Lomax-PS [14], beta exponential-PS [15], unit exponentiated half logistic-PS [16], inverted Nadarajah-Haghighi-PS [17], power quasi Lindley-PS [18], unit Burr XII-PS [19], unit Gompertz-PS [20], log-logistic modified Weibull-PS [21], power inverted Topp-Leone- PS [22] distributions, among others.
Numerous writers have highlighted the significance and usefulness of inverted distributions in many fields, including engineering, economics, and medicine. In this work, the inverse power Lomax (IPL) distribution with three parameters, which was recently presented by Hassan and AbdAllah [23], attracts our attention. The probability density function (PDF) and the cumulative distribution function (CDF) of the IPL distribution, having a, ft > 0, as shape parameters, and its scale parameter X > 0, is defined, respectively, as follows:
g (x)=al x -(ft+i) xP]
X
and,
(
x
1 + X
X
G(x) =
( -ft
1 + -
X
x > 0, (1)
x > 0. (2)
Due to the IPL distribution's non-monotonic failure rate, it offers greater flexibility, making it more appropriate for various practical data modeling and analytic applications. Hassan and Abd-Allah [23] looked at a few statistical characteristics and provided estimators of the parameters in censored samples. Shi and Shi [24] studied how to statistically estimate parameters of the IPL distribution when employing progressive first-failure censoring. The inference of the IEL distribution based on generalized order statistics was discussed by Nassr et al. [25].
This paper's primary objective is to create a novel asymmetric compound class of distributions that is produced by combining the IPL and PS distributions to analyze a system with parallel components; this system is known as the inverse power Lomax power series (IPLPS). We are introducing this class due to the following:
■ To design several distinct models with different symmetric and asymmetric density and hazard rate functions (HRFs) shapes.
■ To go over a few of its statistical characteristics, including moments, quantile function (QF), conditional moments, uncertainty measures, inverse moments, and probability-weighted moments (PWMs).
a
■ To estimate the IPLPS class of distribution parameters, some estimation techniques are taken into consideration, such as weighted least squares (WLS), maximum likelihood (ML), Cramer von Mises (CM), and least squares (LS).
■ To evaluate the effectiveness of various estimates using specific metrics, a dedicated simulation study is conducted for one special model, namely the IPL Poisson (IPLP) distribution.
■ The IPLP distribution, as a sub-model within this class, demonstrates superiority over certain other distributions, as revealed through an analysis of two real-data applications.
This paper's contents are arranged as follows. The IPLPS distributions are introduced in Section 2. Many structural properties of the class are provided in Section 3. Section 4 provides certain examples of the suggested distributions. Parameter estimators for the IPLPS class using different classical methods are shown in Section 5, while Section 6 provides simulation studies. Section 7 presents the application of the suggested distribution's particular case, whereas Section 8 offers concluding findings.
2. Construction of the IPLPS Class
The IPLPS class is introduced in this section. This class of distributions is motivated by a key assumption that renders it suitable for application in each survival and reliability study. Specifically, it assumes that a device's failure arises from the presence of an unspecified number of initial faults, denoted as N, of the same type. These faults remain undetected until they lead to failure and are subsequently fixed.
If we consider Xi, i=1,...,N to represent the time until device failure caused by the ¿th defect supposing that these X/s are independent and identically distributed (iid) IPL random variables, independent of N, then a truncated PS random variable, a distribution within the IPLPS class, can be utilized to model the time until the last failure. This proposed class of distributions can effectively model systems with parallel components, as many biological and industrial applications frequently do. Currently, let us explore a parallel of N iid random variables from the IPL distribution, denoted as Xi, where i=1,...,N.
Assuming that X=max {Xi}N be iid breakdown times of N items connected at a parallel structure, then the conditional CDF of X |N is introduced as:
fx\n=n (x) = [G( x)]n = x
V /
where G(.) is the CDF (2) of the IPL distribution. The joint CDF is given as follows:
1+x
P(X < x, N = n) = P(N = n)Fx\N=n (x) = ■
C(0)
1 + -
A
Hence, the IPLPS class is represented by the marginal CDF of X, which takes the form:
X n fin F (x -y,) = V ^n^-^ C (0)
n=l
-—-
C (0)
-C
-—-
A
(3)
where, y = (a,p,A,9) denotes the set of parameters, A0> 0, represent scale parameters and, P,a> 0 indicate shape parameters. Another simplified form for (3) is as follows:
F (x,y)=c^)C [k (xy^]; x >0,
(4)
x
where, k(r;y) = 0 introduced by:
( r -ß\~a 1 + r
1
Also, the PDF of the IPLPS class of distributions can be
f ( r,w) =
aß0 r -ß-1
1C (0)
1 +
r -^~a-1
C'[k(r;^)]; r > 0.
(5)
The survival function and HRF associated to the IPLPS distribution are expressed as follows, respectively:
- C [k (x;^)] F (x,w) = 1--' ^ ,
C(0)
and,
h(r,w) =
aß0r -ß-1C[k (r;y)]
l[C(0) - C [k ( r;^)]]
1 + -
ß
-a-1
1
Proposition: When 6 approaches zero, the IPL distribution appears as a limiting special case of
the IPLPS distributions
Proof: If 9 approaches zero, then
n-1
lim F (r;^) = 0^0 +
1 +'
1
+ a- 1 lim ^ nan0n 1
0^0 + n=2
1+
1
( -ß\~a 1+i
1 + a, 1 lim ^ nan0n 1 0^0 + n=2
which represents the CDF (2) of the IPL distribution.
Lemma 1: For the IPLPS class of distributions, the density function can be expressed as an infinite mixture of IPL distributions with parameters (na,ß,Z),
to
f (x; ¥) = X P(N = n) g (x; na, ß, Ä). n=1
Proof: The following is an alternative form for the PDF given in Equation (5):
f ( rr,w) =X
n=1
naß0nanr-(ß+1) ( r-ß"(an+1)
1C (0)
1 +
1
(6)
= £ P(N = n)g (r; na, ß,1), n=1
where, g (x; na, ft,X) refers to the IPL distribution's density function (1) with parameters (na, ft,X).
TO
TO
TO
3. Some Statistical Properties
Here, several distinct statistical features of the IPLPS distributions are derived, which may include the quantile function, rth moment and inverse moment, PWMs, conditional moments, and entropy measures.
3.1 Quantile Function The QF of the IPLPS class of X, denoted by xu = Q(u) = F -1 (u), is represented as follows:
—
2
( C-1 [uC(0)]
x-V «
-1
-1 ß
(7)
Particularly, the median, denoted by m, of the IPLPS distribution, is derived by letting u = 0.5 in Equation (7).
3.2 Moments and Inverse Moments
Most important properties for any distribution are concluded using ordinary moments. The rth moment of X can be introduced by using Equation (6) as follows:
^ — X P(N — n) J «xr-ß1) n—1 0
n ( -ß\~(an+1) naß ^r^ , x ß
2
1 + -
2 ,
v
dx.
After simplification, the rth moment of the IPLPS distribution, can be written as:
— (
r r
v
¡îr — X na2ß P(N — n)B 1 - — ,an + ß> r, r —1,2,... (8)
n—1 n n
P Pj
where, B(.,.) denotes the beta function. For r=1 in Equation (8), the mean of the IPLPS distribution is given. Also, we can get the IPLPS moment generating function from the moments by the following equation:
-r
MX(t) = f ^ = f f P(N = n)Bi 1 -r,an + r)
n r! n i r! \ P P J
r=0 r=0 n=1 v r r J
Furthermore, the rth inverse moment for the IPLPS distribution is derived using Equation (6) which leads to:
r
Sr = f nalpP(N = n)B^1 + r ,an - r = 1,2,...
3.3 Conditional Moments
X
œ
X
Studying the conditional moments is very important in lifetime models. The conditional moments of the IPLPS distribution, defined by E (Xr \ X > t), can be introduced by the following lemma.
Lemma 3.1: Supposing that X has the IPLPS (x;^), the rth conditional moment of X, is obtained such that:
r Xx2rßF(tW)
r r 1--,an +-
ß ß
f tß^
2 ,
v y
where B(.,.) refers to the incomplete beta function. Proof: Since
Mr — E
œ
( Xr\X > t ) — =-J xrf ( x; w)dx.
y \ ' F(t;w) J f (
Hence, by inserting the PDF (4) in M r then
M rraPPW = n)7-p-1 ii XF(^) ;
( -(«n+i) i+—
dx.
By simplifying, then the rth conditional moment of the IPLPS class of distributions can be rewritten as,
Mr = X
(1+t P/x)'1
™P(N = ^ i' (1 - z )=rzan+p-1dz =1
naP( N = n)
B
r r 1--.an + -
P
P
1 + -
X
-1 ^
1PF (t;¥) ^ ■ ~ nriP F (t;W) ^
where B (.,., x) is the incomplete beta function and F(x,y) represents the IPLPS survival function.
3.4 Probability-Weighted Moments
Greenwood et al. [26] were the first to propose the PWM approach, with the main goal being the derivation of quantiles and parameter estimators for several generalized distributions that are only analytically represented in reverse form. Eventually, for a random variable X, the PWM is expressed by the following equation:
és,r = E[XsF(x)r] = i xsf(x)(F(x))rdx.
Substituting Equations (4) and (5) in Equation (9) we get:
ape
= 7xs Oilx-P-1 r- x-P
X
( x-P ^a 1 C'\k(r ixr x , \ ( :;+? (C\k(x;^)])rdx.
1 + -
X
{C (e)}
An expansion for ( C\k ( x; iy)\)r can be written as follows:
(9)
(10)
[C(k(x;v))] r =|i an \k(x;^)]n | =(a1 )r (k(x;y))'
1 + a2\k (x;^)] + ^\k (x;^)]2 +... a1 a1
a 3 I
(11)
am+1
m = 1,2,3,
a1
= [a\k(x;¥)]r j Y, cm [k(x;^)] m\ , c
U=0 J
After that, using the Gradshteyn and Ryzhik [27] relation, which states that; for any positive integer m, the following expansion, for a positive integer r, is used:
i
V m=0
cmw'
= i dr,mw' m=0
(12)
where dr o = 1, t > 1 and the coefficients dr, = t-1 i (m(r +1) - t)cmdr t-m. Then, using expansion
m=1
(12) in (11) provides the following:
7
[C(k(x;y))]r = a1 i dr,m \k(x;^:>] m=0
m+r
In addition,
C '\k (x;^)] = i nan \k ( x;^)] n=1
n-1
(13)
X
t
7
Assuming that z = n — 1, then Equation (14) is rewritten in this form:
az+1
C' [k(x; w)] = a1 f bz (z +1) [k(x; w)]z, bz =
z=0 a1
(15)
Hence, the PWM of the IPLPS class of distributions is represented by placing Equations (13) and (15) into (10) and after some simplification,
ana r+m+z+1 TO
ts,r =—-— f a1 r+1dr,mbz(z +1)Jxs-p—1
A{C (0)} m,z=0 0
/ -a(r+m+z+1)-1
x p
1 + -
dx.
Hence,
$s,r = f A *B 1 - — ,a(r + m + z +1) + —\, s < p. m,z=0 \ p p j
where, A =
a0r+m+z+1a1r+1drmbz(z +1)2 p
{C (0)}
r+1
and B (.,.) is the beta function.
3.6 Entropy Measures
Entropy serves as a metric for quantifying the uncertainty within data and finds applications across diverse fields such as science, physics, and engineering. Essentially, higher entropy values indicate greater uncertainty within the data. In this sub-section, expressions for certain entropy measures within the IPLPS class are derived. Let X refers to random variable drawn from IPLPS distributions, so the Renyi entropy (RE) can be represented by the following equation:
1
IR =
8-1
log
J f 8 (x;y)dx
0
,8* 1,8 > 0.
(16)
Suppose IP = J (f (x; dx, then by using PDF (5) and expansion (14) in integral IP, we have
But
0
IP ==fj x -8(P+1) {AC(0)}8 0
8
n-1 '
f -P\ -8(a+1) 1 x P
v
A
f nan [k(x;^)]
n=1
n-1
dx,
f nan [k(x;^)] n =1
= af
f c m [k(x;^)]m
m=0
c• = am+1(m +1), m = 1,2,..
m
a1
According to Ref. [27], the previous equation can be expressed as
-|8
in-1
f nan [k(x;^)] ] n =1
= a f f d 8,m [k (x;^)]m . m = 0
By using Equation (18) in the last term in (17), then
,D ^ . Di 8(P + l)-I s s 8(P + 1)-1
IP = f AmB\ _ -,8a + am +8-
m=0
P
P
8-1
A m =
(a0ai)8 P8-1d8,0mA P ( C (0) )8
(17)
(18)
(19)
Hence, by substituting (19) in Equation (16), the RE of the IPLPS class of distributions takes the following form:
<x>
s
8
Ir (S) = log
I — o
^ . D, S(ß +1) — I _ _ S(ß +1) —I
^A ---,Sa+am + S —
m=0
ß
ß
Tsallis entropy (TE), introduced by Tsallis [28] as a thermodynamic measure, has a wide application across various real-world domains. Generally, TE offers intriguing explanations in physical, chemical, and biological phenomena. The TE measure is represented as:
IfE =
I
S —I
I -J f S( x)dx 0
S I,S> 0.
Using the similar procedure discussed above, the TE is given by 1
ITE =
S — I
I —
'va JS(ß +1) — Ix/U S(ß +1) — ^A mB | J— ,S(a +1) +am —
Vm=0
ß
ß
4. Special Sub-Models
Here, certain special cases of this class are introduced. Graphs depicting the PDF and HRF are presented to showcase the IPLP distribution' flexibility for some chosen values for the parameters. If P = 1, the IPLPS class offers the inverse Lomax PS class of distributions (new-class). Letting C (6) = e 6 -1, the IPLPS distribution turns to the IPLP distribution. Supposing that P = 1, C(6) = e 6 -1 and the IPLPS distribution provides the IL Poisson (ILP) distribution (new).
Setting C(0) = - log(1 -6), the IPLPS class becomes the IPL logarithmic (IPLL) distribution (new).
By putting P = 1, and C(0) = - log(1 -6), the IPLPS distribution provides the IL logarithmic (ILL) distribution (Buzaridah et al. [29]).
Considering that C (6) = 6(1 -6)-1, the IPLPS distribution introduces the IPL geometric (IPLG) distribution (new).
By letting P = 1, and C(6) = 6(1 -6)-1, the IPLPS distribution presents the IL geometric (ILG) distribution (new).
Substituting C(6) = (1 - 6)m -1, that yields the IPL binomial (IPLB) distribution (new). Taking P = 1, besides C(6) = (1 -6)m -1, it gives the IL binomial (ILB) distribution.
The IPLP Distribution
By setting C(6) = e 6 -1, and C'(6) = e 6 in (4) and (5), the PDF and CDF of the IPLP distribution is obtained by:
f\(x;w) = -jßr\ x —ß—I
e° — I)
I+x ß
,k(xw). x > 0,
ek(x;iy) -1
F1 (x; y) =--; x > 0,
e6 -1
where, cc, P denote the shape parameters and 6, X refer to the scale parameters. The HRF of the IPLP distribution is given as follows:
H i(x;w) =
■ —ß—Iek (x;w)
e9 — ek (xW) J
ß
I + -
-a—I
v
; x > 0.
The PDF and HRF plots for the IPLP distribution are given in Figure 1.
Figure 1: PDF and HRF plots for specific parameter values of the IPLP distribution.
Figure 1 indicates that the IPLP distribution's density may exhibit reversed-J, skewed to the right, or unimodal shapes. Moreover, the HRF can take on increasing, decreasing, upside down, or reversed J-shaped forms at different parameter values. This suggests that the IPLP distribution is versatile for fitting datasets with diverse shapes.
5. Parameter Estimation
Here, the parameter estimation for the IPLPS distributions is discussed by applying the ML, LS, WLS, and CM methods.
5.1 Maximum Likelihood Estimators
Let X1, X2, .Xn be a simple random sample from the IPLPS class of distributions with a set of
T
parameters \ = (a,P,A,0) .The likelihood function of this sample, denoted by Ln based on the observed random sample of size n from density (5) is given by:
^aP0lKA„-P-1(. x P"a-1
Ln =
AC(0)
n x'i i=1
1 + -
A
C( k (xi\)).
The log-likelihood, say log Ln,, can be expressed as:
n
n (
log Ln = n log (aP0) - n log (AC(0))-f (P +1) log xi - (a + 1)f log
i=1 i=1
1+
P
A
(20)
+ f log (C ' (k (xi ;\))), i=1
Hence, by differentiating (20) with respect to a, P, A and 0, respectively, yields
5log Ln = n-f^log
da a
i=1
1' xi
P n 0C (k(xi ;\))
A
= C (k (xi ;\))
■
1 + -*-
PYa (
log
V
J
x
1+ i
P
A
A
J
-SI T n n
^ZL = n.-f log xi-f
dP P f f
i=1 A + xi
-P
-P
(a +1) x- P log xi n a0x■ P log x-C" (k (xi ;\)) | x- P - + f--—- - ,.-^-- 1'
i=1
AC'( k ( xi ;\))
, -a-1
A
^^ =-n + (a + 1)f_
dA A f + Ax-P fA2C'{k (x- ;\)
| ^ 0aC"(k(x- ;\))
( -P} 1 + 1
-a-!
A
J
and,
n
dlogLn _ n nC'(0) nC"(k(xf ;v))
-p
X
de 0 c (0) f^c'(k (xf ;v))
Then the ML estimates (MLEs) for the parameters a,p,X and0, denoted by a, p, X and 0), can be derived by setting (dLn /da),(dLn/dp),(dLn/dX) and (dL nfd0) to be zero and solving these equations numerically.
5.2 Least Squares and Weighted Least Squares Estimators
Consider x (i), %(2), ..., xn refers to an observed ordered sample and xi, X2, ..., xn represents n random samples from the IPLPS distribution. Johnson et al. [30] claimed that the distribution's expectation and variance are determined independently of the unknown parameter by
E (F (X (i))) = —, and, Var (F (X {i)) )= f(n -* +1) , v n +1 v ()/ (n + 1)2(n + 2)
where F(X (,■)) indicates the CDF of any given distribution and X(i) denotes the statistic of order i.
So, the LS estimates (LSEs) and WLS estimates (WLSEs) can be given by the minimization of the sum of all squared errors
i2
H(v) = £Vf [F(x(f);^) -E (F(x(i);^))] .
i=1
The LSEs and WLSEs of a, p, X and 0, are produced by the minimization of the preceding function
H(v) = £ Vi i=1
C [k (x (i);^)]
C(0)
n +1
(21)
Based on Equation (21), the LSEs a1,p1,X\ and 01 are provided by using Vj = 1, while the WLSE
a2, p2, X2 and 02 are obtained by putting Vi =
(n +1)2 (n + 2)
i (n - i + 1)
These estimates can be given by solving each of the following equations numerically.
C (k (x (i)V)) i
dH<v) ^
da
i=1
dHjw) _ yv.
dp £ 1
dH(v)
i=1 n
dX
dH(v)
d0
= £ Vi i=1 n
=&
i=1
C(0) n +1
C (k (x (i);v)) i
C(0) n + 1
C (k (x (i);v)) i
C(0) n + 1
C (k (x (i);v)) i
C(0)
n + 1
91 (x(i)v) =
92 (x(i)v) =
93 (x(i)v) =
94 (x(i)v) =
where,
91(x (i)V) = -
92(x (i)W) =
C (0)
a0
-p\~a (
1 + ■
\i)
log
-p
1 + ■
-p
\i)
X
C'( x (i^vX
XC (0)
1+■
\ -a-1
x(J) lnx(i)C '(x(i),V),
n
0
X
V3(x (i),\) =
a0x-
-P( vP
-a-1
(i)
A2C (0)
1 +
x(i)
A
V J
C'( x (i),\),
and,
x (i),\) =
C'(k (x;\))
f -P\
x
C (0)
P
(2L
A
V
1 + -
C (k (x (i),\)) C (0) (C' (0))2 .
5.3 Cramer -von-Mises Estimators
This method can be defined as a type of estimator that relies on minimal distance principles since it relies on the disparity between the empirical distribution function and the CDF estimate. According to Macdonald [31], in this method, the CM estimators are presented as the minimization of the given equation with respect to a,P,A, and 0, respectively,
n2
1
H \ = 12n+f
i=1
C(k(x(i),\)) 2i -1
C(0)
2n
The CM estimates (CMEs) a3,P3,A3, and 03 can be obtained by differentiating the previous equation with respect to a,P,A, 0, respectively, and equating it to zero.
6. Simulation Study
For each estimation problem, the investigation of the estimator's properties is very important. Analytical study of the obtained expressions for the estimators can't be effective due to their complexity. As a result, a numerical study will be established, handling the estimates' sampling distribution independently. This estimation is conducted in order to assess the estimators presented at the preceding section. All calculations are produced by using the Mathematica11.3 program. The performances of the different estimates will be compared according to their absolute bias (AB) and mean squared error (MSE). These numerical procedures will be shown by steps below:
Step 1: 1000 random samples given the sizes of 50, 100, 150, and 200 are conducted from the inverse power Lomax Poisson distribution.
Step 2: Four cases of parameter values have been selected such that: Case 1 = (a = 0.2, P = 0.5, A = 0.5,0 = 0.5), Case 2 = (a = 0.1, P = 0.7,.A = 0.5,0 = 0.5) , Case 3 =(a = 0.35, P = 0.75, A = 0.5,0 = 0.5), Case 4 = (a = 0.7, P = 0.25, A = 0.5,0 = 0.5). Step 3: The MLEs, LSEs, WLSEs, and CMEs are derived for each unknown parameter. Step 4: The ABs and MSEs of different estimates of unknown parameters are calculated. The results are written down in Tables A.1 to A.4 (Appendix A). By the help of these tables, the following conclusions can be concluded to predict the performance for all these different estimates
• For fixed value of A = 0.5 and 0 = 0.5, ABs and MSEs for each a estimates and P estimate values in the MLEs decrease while sample size increases (see Table A.1).
• For fixed values of A and0, the MSEs of CMEs for a and P are decreasing and the sample size will be increasing in the same time (see Table A.3).
• For P = 0.75, and for fixed values of A and 0, the MSEs of the WLSEs increase as the sample size increases (see Table A.2).
• By increasing the sample size, the ABs of MLEs ata = 0.35 and P = 0.75 decrease
consistently, for fixed values for A and 0 as shown in Table A.3.
a
• At n = 50 and k = 0.5, the MSEs have the smallest values for all different sets of parameters at « = 0.35 and P = 0.75, as indicated in Table A.4.
• As the sample size increases, it is evident through all estimation methods that both MSEs and ABs decrease, as demonstrated in Table A.1 for instance.
• Almost in all cases, the estimated MSEs of the MLEs are the smallest compared to other estimation methods across all parameter values.
7. Data Analysis
This section presents the application of the IPLP model on two real data sets, illustrating its practical adaptability and utility. The IPLP distribution is contrasted with alternative models including the power Lomax (PL) [32], PL Poisson (PLP) [33], Topp-Leone Lomax (TLLO) [34], and Marshall Olkins PL (MOPL) [35] distributions for two real datasets.
The first dataset has been introduced by Murthy et al. [36], represents 84 observations recording the failure time for specific aircraft windshield model. The dataset is as follows:
0.04 1.866 2.385 3.443 0.301 1.876 2.481 3.467 0.309 1.899
2.61 3.478 0.557 1.911 2.625 4.57 1.652 2.3 3.344 4.602
1.757 3.578 0.943 1.912 2.632 3.595 1.07 1.914 2.646 3.699
1.124 1.981 2.661 3.779 1.248 2.01 2.224 3.117 4.485 1.652
2.229 3.166 2.688 3.924 1.281 2.038 2.823 4.035 1.281 2.085
2.89 4.121 1.303 2.089 2.902 4.167 1.432 4.376 1.615 2.223
3.114 4.449 1.619 2.097 2.934 4.24 1.48 2.135 2.962 4.255
1.505 2.154 2.964 4.278 1.506 2.19 3 4.305 1.568 2.194
3.103 2.324 3.376 4.663
To examine the utility of the proposed models, various criteria measures, including -2Log-likelihood (L*), Akaike information criterion (A*), Bayesian information criterion (B*), consistent Akaike information criterion (C*), the Kolmogorov-Smirnov distance (K*) and its p-value (K*-PV), and CM statistics (W*) are evaluated. In general, the smaller the value of these statistics, a better fit model for the data will be found. Table 1 offers MLEs for all models that are suggested, and Table 2 lists several goodness of fitting metrics.
Table 1: MLEs for all parameters of the models fitted to first dataset
Model ct P i ê
IPLP 0.1987 4.4769 0.011 3.9019
PL 22.4127 2.3992 270.085
PLP 121.997 1.6083 332.485 3.1412
TLLO 3.7045 4.1343 0.1044
MOPL 7.5277 2.417 7.5784 18.0908
Table 2: Statistical metrics for all models according to the first dataset
Model L* A* B* C* K* W* K*-PV
IPLP 310.726 318.726 319.232 328.449 0.06679 0.05164 0.823436
PL 524.280 530.279 530.579 537.572 0.0717773 0.06906 0.752356
PLP 544.968 552.968 553.475 562.692 0.0713355 0.05521 0.758912
TLLO 464.11 470.11 470.41 477.403 0.132497 0.38564 0.095490
MOPL 616.978 624.978 625.484 634.701 0.0706109 0.05595 0.769575
Table 2 clearly indicates that among all the models fitted, the IPLP model exhibits the lowest values for statistical measures. Hence, it could be regarded as the best model. Figure 2 illustrates non-parametric plots for the first dataset, encompassing total time on test (TTT), box plot, and percentile-percentile (PP) plots. Furthermore, Figure 3 presents the estimated cumulative and density functions
for the fitted models.
Figure 2: The TTT plot, Box plot; and PP-plot for first data
fitted PDFs
X p - ; ■-,
Figure 3: Estimated CDF and PDF for the models fitted to the first dataset.
Depending on Figure 3, the IPLP distribution provides the closest fit to the provided data, and then it is the best model among the other models to analyze these data.
Data 2: This dataset represents 63 aircraft windshield service times, presented by Murthy et al. [36]. The data can be shown as follows:
0.046 1.436 2.592 0.14 1.492 2.6 0.15 1.58 2.67 0.248
1.719 2.717 0.28 1.794 2.819 0.313 1.915 2.82 0.389 1.92
2.878 0.487 1.963 2.95 0.622 1.978 3.003 0.9 2.053 3.102
0.952 2.065 3.304 0.996 2.117 3.483 1.003 2.137 3.5 1.01
2.141 3.622 1.085 2.163 3.665 1.092 2.183 3.695 1.152 2.24
4.015 1.183 2.341 4.628 1.244 2.435 4.806 1.249 2.464 4.881
1.262 2.543 5.14
Table 3 lists the MLEs for all models that are suggested, while Table 4 gives the numerical values of the statistical metrics.
Table 3: MLEs for the unknown parameters of the models fitted to the second dataset
Model a p X 00
IPLP 0.2238 3.8211 0.0233 2.0577
PL 108.647 1.6327 422.985
PLP 131.468 1.3335 256.861 1.8047
TLLO 1.9449 4.5615 0.0834
MOPL 0.7228 3.1157 0.0161 69.0443
Table 4: Statistical metrics for the proposed models according to the second dataset
Model L* A* B* C* K* W* K*-PV
IPLP 536.838 544.839 545.529 553.411 0.0684147 0.056686 0.909991
PL 633.596 639.596 640.002 646.025 0.109307 0.0945244 0.409648
PLP 545.928 553.928 554.617 562.5 0.0898046 0.0571536 0.656499
TLLO 569.62 575.62 576.026 582.049 0.145844 0.273631 0.123926
MOPL 542.74 550.74 551.429 559.312 0.137781 0.204885 0.166429
Results in Table 4 show the utility of the IPLP model as it has the lowest L*, A*, B*, C*, K*, and W* values and has the greatest K*-PV compared to the others, which indicates that the IPLP distribution is the best model. In addition, Figures 4 and 5 give TTT Plot, box plot, and PP-plot, along with the estimated cumulative and densities of the fitted models plot as well, respectively, for the data.
Figure 4: The TTT plot, box plot and PP-plot for second data
fitted PDFs
0 1 2 3 4 5 6
Figure 5: The estimated CDF and PDF for the models fitted to the second dataset
Figure 5 demonstrates that the IPLP distribution closely aligns with the histogram, indicating its superiority over other models for analyzing this data.
8. Concluding Remarks
A novel asymmetric four-parameter IPLPS class of distributions formed by combining the inverse power Lomax and power series distributions is introduced in this paper. This blending technique enables the creation of adaptable distributions with significant implications across diverse fields such as engineering and biology. The IPLPS class includes a new compound class and many novel
compound distributions, which come as new sub-models. Expressions for the QF, conditional moments, inverse moments, PWMs, and uncertainty measures are constructed. Estimation of model parameters is carried out using WLS, ML, CVM, and LS techniques. We assess and compare several parameter estimators for the IPLP distribution using an in-depth simulation study. Additionally, we demonstrate the efficacy of the proposed model using two real datasets, where it exhibits superior fit compared to alternative models.
Appendix A: Tables
Case 1 (a = 0.2, p = 0.5,X = 0.5,0 = 0.5)
n Method Measure a p X 0
50 ML AB 0.006745 0.077041 0.029852 0.115995
MSE 0.007617 0.041746 0.120200 0.197901
LS AB 0.430544 0.487070 0.144346 0.113605
MSE 0.269951 0.238003 0.140357 0.146791
WLS AB 0.288091 0.441809 0.096579 0.384180
MSE 0.149819 0.215351 0.123369 0.789747
CM AB 0.339837 0.487085 0.131450 0.304000
MSE 0.174023 0.238043 0.114371 0.620196
100 ML AB 0.000455 0.057409 0.001818 0.076642
MSE 0.004357 0.023858 0.083599 0.185695
SE 0.000066 0.000143 0.000289 0.000424
LS AB 0.447667 0.491493 0.156331 0.123996
MSE 0.281292 0.241848 0.137407 0.143057
SE 0.000284 0.000017 0.000336 0.000357
WLS AB 0.279684 0.461081 0.073950 0.399205
MSE 0.145609 0.221363 0.119117 0.831226
SE 0.000259 0.000094 0.000337 0.000819
CM AB 0.334056 0.493331 0.132073 0.361127
MSE 0.162428 0.243538 0.102497 0.692470
150 ML AB 0.000429 0.034402 0.004477 0.047673
MSE 0.002593 0.013023 0.065901 0.178032
SE 0.000051 0.000109 0.000257 0.000419
LS AB 0.444868 0.494006 0.155821 0.116442
MSE 0.270110 0.244186 0.128611 0.130407
WLS AB 0.296139 0.462616 0.085070 0.348997
MSE 0.152255 0.222201 0.115882 0.733505
CM AB 0.343748 0.493717 0.146006 0.373102
MSE 0.166483 0.244084 0.097878 0.750284
200 ML AB 0.001551 0.023436 0.012966 0.031401
MSE 0.002002 0.009062 0.056420 0.170785
LS AB 0.452990 0.494879 0.168071 0.114122
MSE 0.275720 0.244991 0.131350 0.134244
WLS AB 0.300224 0.468995 0.087634 0.360930
MSE 0.155801 0.226458 0.117244 0.801838
CM AB 0.338092 0.495897 0.128463 0.359770
MSE 0.163050 0.245969 0.094788 0.693901
Table A.2: Results of simulation study of different estimates for the IPLP distribution: Case 2
Case 2 (a = 0.1, P = 0.7,2 = 0.5,0 = 0.5)
n Method Measure a P 2 0
50 ML AB 0.002242 0.091436 0.052315 0.088154
MSE 0.001661 0.047267 0.112424 0.190250
LS AB 0.514374 0.688831 0.134479 0.098571
MSE 0.348814 0.475298 0.140524 0.136474
WLS AB 0.382270 0.650093 0.080623 0.348311
MSE 0.215249 0.443329 0.119549 0.738434
CM AB 0.408666 0.690469 0.107712 0.382826
MSE 0.221501 0.477186 0.104083 0.756821
100 ML AB 0.000734 0.064783 0.027290 0.064352
MSE 0.000925 0.032769 0.087944 0.176691
LS AB 0.549471 0.692815 0.168821 0.119962
MSE 0.383591 0.480338 0.142214 0.142375
WLS AB 0.356645 0.660133 0.060534 0.444925
MSE 0.191185 0.447833 0.117684 0.891369
CM AB 0.425101 0.694151 0.110837 0.335976
MSE 0.234942 0.481978 0.100780 0.717249
150 ML AB 0.001048 0.047594 0.030172 0.060538
MSE 0.000739 0.026150 0.076517 0.168294
LS AB 0.548601 0.694918 0.164571 0.123721
MSE 0.378798 0.483023 0.138253 0.138424
WLS AB 0.363182 0.662171 0.067058 0.387271
MSE 0.191053 0.449580 0.114219 0.801316
CM AB 0.433434 0.695742 0.138198 0.388423
MSE 0.236302 0.484116 0.098432 0.783373
200 ML AB 0.000355 0.035961 0.024822 0.018055
MSE 0.000581 0.021184 0.065165 0.162195
LS AB 0.557505 0.695863 0.168104 0.133845
MSE 0.390652 0.484309 0.138774 0.144132
WLS AB 0.358781 0.670564 0.063388 0.438549
MSE 0.190053 0.454457 0.111519 0.828456
CM AB 0.421589 0.696359 0.125404 0.424671
MSE 0.228737 0.484958 0.102214 0.859722
Table A.3: Results of simulation study of different estimates for the IPLP distribution: Case 3
Case 3 (a = 0.35, p = 0.75, X = 0.5,0 = 0.5)
n Method Measure a p X 0
ML AB 0.028882 0.038793 0.041519 0.097016
MSE 0.021385 0.024007 0.093766 0.205833
LS AB 0.330981 0.720645 0.172635 0.214191
50 MSE 0.200888 0.522177 0.156952 0.153552
WLS AB 0.171788 0.664002 0.111915 0.315342
MSE 0.094188 0.468671 0.125394 0.676175
CM AB 0.200807 0.723881 0.139624 0.297306
MSE 0.094502 0.526193 0.108873 0.592616
ML AB 0.011510 0.032625 0.034154 0.045939
MSE 0.010988 0.017202 0.073926 0.191679
LS AB 0.343760 0.730998 0.176993 0.239639
100 MSE 0.204578 0.535951 0.150556 0.161608
WLS AB 0.183232 0.681750 0.121003 0.315958
MSE 0.096159 0.485995 0.114875 0.638105
CM AB 0.196654 0.733830 0.131675 0.316076
MSE 0.088577 0.539333 0.100398 0.589016
ML AB 0.009429 0.021253 0.021613 0.045372
MSE 0.008488 0.012985 0.061332 0.185365
LS AB 0.347205 0.736845 0.183141 0.228068
150 MSE 0.198331 0.543845 0.142716 0.156601
WLS AB 0.158364 0.678758 0.094665 0.387196
MSE 0.091379 0.481589 0.118032 0.745790
CM AB 0.197847 0.737241 0.133215 0.322106
MSE 0.088204 0.544097 0.096084 0.576261
ML AB 0.005265 0.018039 0.032937 0.005198
MSE 0.005849 0.010298 0.052733 0.183501
LS AB 0.330284 0.738295 0.161452 0.216034
200 MSE 0.183964 0.545553 0.130130 0.155453
WLS AB 0.181529 0.692042 0.112438 0.315812
MSE 0.096914 0.492804 0.114267 0.617612
CM AB 0.185704 0.739790 0.125346 0.377463
MSE 0.083501 0.541593 0.094968 0.691230
Table A.4: Results of simulation study of different estimates for the IPLP distribution: Case 4
Case 4 (a = 0.7, P = 0.25,2 = 0.5,0 = 0.5)
n Method Measure a P 2 0
50 ML AB 0.002412 0.026550 0.017636 0.062220
MSE 0.056895 0.005150 0.097176 0.176375
LS AB 0.004458 0.236684 0.175355 0.196397
MSE 0.090972 0.056577 0.154987 0.154997
WLS AB 0.118599 0.213013 0.132290 0.293232
MSE 0.095884 0.049828 0.129123 0.667014
CM AB 0.100852 0.237882 0.168351 0.251028
MSE 0.074475 0.057127 0.116832 0.562473
100 ML AB 0.011686 0.012692 0.031510 0.009029
MSE 0.396155 0.002067 0.076041 0.170751
LS AB 0.019738 0.241765 0.204124 0.214818
MSE 0.087939 0.058649 0.158975 0.161972
WLS AB 0.115818 0.222881 0.142248 0.273083
MSE 0.085862 0.051960 0.121264 0.574088
CM AB 0.131302 0.241384 0.149405 0.309364
MSE 0.070562 0.058503 0.105365 0.588599
150 ML AB 0.019717 0.005788 0.035701 0.010170
MSE 0.030271 0.001102 0.058667 0.153535
LS AB 0.012704 0.244189 0.198267 0.207426
MSE 0.082305 0.059729 0.150544 0.160624
WLS AB 0.114129 0.224038 0.144723 0.293595
MSE 0.086913 0.052262 0.124999 0.661949
CM AB 0.133418 0.243985 0.142315 0.290336
MSE 0.070266 0.059684 0.103061 0.557025
200 ML AB 0.012894 0.005315 0.039573 0.030536
MSE 0.025921 0.000908 0.053184 0.153984
LS AB 0.027965 0.245094 0.218293 0.217001
MSE 0.078031 0.060166 0.152014 0.158960
WLS AB 0.103315 0.226621 0.161816 0.278613
MSE 0.085451 0.052928 0.130257 0.629098
CM AB 0.117752 0.244956 0.169220 0.286765
MSE 0.064152 0.060075 0.104847 0.525627
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