The moment generating function (denoted M V ( t ) ) of a continuous random variable V with Iwueze’s distribution structure can be calculated as follows:

M V ( t ) = θ 5 θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ∫ 0 ∞ [ 1 + v + v 2 ] 2 e − ( θ − t ) v d v = θ 5 θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 [ ( θ − t ) − 1 + 2 ( θ − t ) − 2 + 6 ( θ − t ) − 3     + 12 ( θ − t ) − 4 + 24 ( θ − t ) − 5 ] = 1 θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 [ θ 4 ( 1 − t θ ) − 1 + 2 θ 3 ( 1 − t θ ) − 2     + 6 θ 2 ( 1 − t θ ) − 3 + 12 θ ( 1 − t θ ) − 4 + 24 ( 1 − t θ ) − 5 ] (2.10)

Noting the famous binomial series expansion [23] that

( 1 − ∂ ) − l = ∑ π = 0 ∞ ( l + π − 1 π ) ∂ π ; l > 0     and     | ∂ | < 1 , (2.11)

Equation (2.10) reduces to

M V ( t ) = ∑ π = 0 ∞ [ θ 4 + 2 ( π + 1 ) θ 3 + 3 ( π + 1 ) ( π + 2 ) θ 2   + 2 ( π + 1 ) ( π + 2 ) ( π + 3 ) θ + ( π + 1 ) ( π + 2 ) ( π + 3 ) ( π + 4 ) ] ( t θ ) π θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 (2.12)

From Equation (2.12), the ωth generalized derivation of the moments about zero or origin) is

μ ω / = ω ! [ θ 4 + 2 ( ω + 1 ) θ 3 + 3 ( ω + 1 ) ( ω + 2 ) θ 2   + 2 ( ω + 1 ) ( ω + 2 ) ( ω + 3 ) θ + ( ω + 1 ) ( ω + 2 ) ( ω + 3 ) ( ω + 4 ) ] θ ω [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] ; ω = 1 , 2 , ⋯ (2.13)

As a result, precise instances about ωth specific moments about the origin for ω = 1 , 2 , 3 , 4 of Iwueze’s distribution are defined as follows:

μ 1 / = θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] = θ 4 + 2 [ 2 θ 3 + 3 [ 3 θ 2 + 4 [ 2 θ + 5 ] ] ] θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.14)

μ 2 / = 2 θ 4 + 12 θ 3 + 72 θ 2 + 240 θ + 720 θ 2 [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] = 2 θ 4 + 3 [ 4 θ 3 + 4 [ 6 θ 2 + 10 [ 2 θ + 6 ] ] ] θ 2 [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.15)

μ 3 / = 6 θ 4 + 4 [ 12 θ 3 + 5 [ 18 θ 2 + 36 [ 2 θ + 7 ] ] ] θ 3 [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.16)

μ 4 / = 24 θ 4 + 5 [ 48 θ 3 + 6 [ 72 θ 2 + 168 [ 2 θ + 8 ] ] ] θ 4 [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.17)

Its essential moments about the mean, on the other hand, are

μ 2 = σ 2 = θ 8 + 8 θ 7 + 56 θ 6 + 240 θ 5 + 876 θ 4 + 1344 θ 3 + 2304 θ 2 + 2880 θ + 2880 θ 2 [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] 2 (2.18)

μ 3 = 2 [ θ 12 + 12 θ 11 + 114 θ 10 + 664 θ 9 + 3132 θ 8 + 7776 θ 7 + 17064 θ 6   + 30240 θ 5 + 49248 θ 4 + 77760 θ 3 + 103680 θ 2 + 103680 θ + 69120 ] θ 3 [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] 3 (2.19)

μ 4 = 3 [ 3 θ 16 + 48 θ 15 + 560 θ 16 + 4192 θ 13 + 24920 θ 12 + 96576 θ 11   + 319872 θ 10 + 900096 θ 9 + 2257776 θ 8 + 4962816 θ 7   + 9787392 θ 6 + 16657920 θ 5 + 24288768 θ 4 + 28753920 θ 3   + 29859840 θ 2 + 23224320 θ + 11612160 ] θ 4 [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] 4 (2.20)

The factors for evaluating the Iwueze’s distribution variation (VF), dispersion (FD), skewness ( β 1 ) and kurtorsis ( β 2 ) are as follows:

V F = θ 8 + 8 θ 7 + 56 θ 6 + 240 θ 5 + 876 θ 4 + 1344 θ 3 + 2304 θ 2 + 2880 θ + 2880 θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 (2.21)

F D = θ 8 + 8 θ 7 + 56 θ 6 + 240 θ 5 + 876 θ 4 + 1344 θ 3 + 2304 θ 2 + 2880 θ + 2880 θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] (2.22)

β 1 = 2 [ θ 12 + 12 θ 11 + 114 θ 10 + 664 θ 9 + 3132 θ 8 + 7776 θ 7 + 17064 θ 6   + 30240 θ 5 + 49248 θ 4 + 77760 θ 3 + 103680 θ 2 + 103680 θ + 69120 ] [ θ 8 + 8 θ 7 + 56 θ 6 + 240 θ 5 + 876 θ 4   + 1344 θ 3 + 2304 θ 2 + 2880 θ + 2880 ] 3 / 2 (2.23)

β 2 = 3 [ 3 θ 16 + 48 θ 15 + 560 θ 16 + 4192 θ 13 + 24920 θ 12 + 96576 θ 11 + 319872 θ 10 + 900096 θ 9 + 2257776 θ 8 + 4962816 θ 7 + 9787392 θ 6 + 16657920 θ 5 + 24288768 θ 4 + 28753920 θ 3 + 29859840 θ 2 + 23224320 θ + 11612160 ] [ θ 8 + 8 θ 7 + 56 θ 6 + 240 θ 5 + 876 θ 4 + 1344 θ 3 + 2304 θ 2 + 2880 θ + 2880 ] 2 (2.24)

Figure 3 depicts a graphical depiction of the values of β 2 at various positions of θ . Figure 3 shows that β 2 for all varying points of θ , the values of β 2 are greater than three, indicating that the Iwueze’s distribution is leptokurtic in nature. However, the values of β 2 for 0 < θ < 1 .179 drop, while the values of β 2 remain constant for 1 .179 ≤ θ < 1 .459 and increase exponentially for θ ≥ 1.459 . As a result, Table 1 shows varying values of θ and associated dispersion structures of Iwueze’s and some single parameter distributions.

Some statistical properties of Iwueze’s distribution are derived, including the survivorship or existence measurement function, risk measurement function, and average residual measurement life-time function, stochastic ordering of random variate, absolute deviations from average and mid points, bonferroni curve, lorenz curve, bonferroni and lorenz indices, entropy, and stress-strength reliability.

The survivorship or existence measurement function (known as survival rate function [20] [21] [22]) is specified as ( s ( v , θ ) ) , risk measurement function (known as hazard or failure rate function [8] [20] [21] [22] [24]) is specified as ( h ( v , θ ) ) and average residual measurement life-time function (also known as mean residual life function [8] [21] [24]) is specified as ( a r ( v , θ ) ) for a random variable V , V > 0 with theoretical density function f ( v ) and distribution function G ( v ) are specified as

s ( v , θ ) = 1 − G ( v ) (2.25)

h ( v , θ ) = lim Δ v → 0 ( P ( failure   in   the   interval ( v ,   v + Δ v ) ) / Δ v P ( surviving   up   till   time   v ) ) = 1 s ( v , θ ) lim Δ v → 0 ( G ( v + Δ v ) − G ( v ) Δ v ) = G / ( v ) s ( v ) = f ( v , θ ) s ( v , θ ) (2.26)

and

a r ( v , θ ) = 1 s ( v , θ ) ∫ x ∞ [ 1 − G ( t ) ] d t (2.27)

s ( v , θ ) , h ( v , θ ) and a r ( v , θ ) will be simply written as s ( v ) , h ( v ) and a r ( v ) respectively. Therefore, the mathematical expressions for evaluating the Iwueze’s distribution s ( v ) , h ( v ) and a r ( v ) are as follows:

s ( v ) = [ 1 + [ θ 4 v 4 + 2 θ 3 ( θ + 2 ) v 3 + 3 θ 2 [ θ 2 + 2 ( θ + 2 ) ] v 2   + 2 θ [ θ 3 + 3 [ θ 2 + 2 [ θ + 2 ] ] ] v ] θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] e − θ v (2.28)

h ( v ) = θ [ θ 2 + θ 2 v + θ 2 v 2 ] 2 [ θ 4 x 4 + 2 θ 3 ( θ + 2 ) x 3 + 3 θ 2 [ θ 2 + 2 ( θ + 2 ) ] x 2   + 2 θ [ θ 3 + 3 [ θ 2 + 2 [ θ + 2 ] ] ] x + ( θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ) ] (2.29)

a r ( v ) = [ θ 4 v 4 + 2 θ 3 ( θ + 4 ) v 3 + 3 θ 2 [ θ 2 + 4 ( θ + 3 ) ] v 2   + 2 θ [ θ 3 + 6 [ θ 2 + 2 θ + 8 ] ] v + ( θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ) ] θ [ θ 4 x 4 + 2 θ 3 ( θ + 2 ) x 3 + 3 θ 2 [ θ 2 + 2 ( θ + 2 ) ] x 2   + 2 θ [ θ 3 + 3 [ θ 2 + 2 [ θ + 2 ] ] ] x + ( θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ) ] (2.30)

Observe that s ( 0 ) = 1 , h ( 0 ) = θ 5 θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 = f ( 0 ) and

a r ( 0 ) = θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 θ ( θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ) = μ 1 / . Figure 4 and Figure 5 show graphs

of h ( v ) and a r ( v ) of Iwueze’s distribution for some selected but different real points or values of θ respectively. The graphical plot of h ( v ) clearly shows that h ( v ) is monotone non-decreasing function in v and θ , while a r ( v ) is monotone non-increasing function in v and θ .

The comparative behaviour of continuous variables from Iwueze’s distribution can be appropriately examined and\or determined using the stochastic ordering of any two selected random variables, say T , V > 0 . Then, T < V in

1) Stochastic ordering; that is

T ≤ s t o r V if G T ( ς ) ≥ G V ( ς ) for all ς (2.31)

2) Risk Measurement ordering; that is

T ≤ R o r V if h T ( ς ) ≥ h V ( ς ) for all ς (2.32)

3) Average residual Measurement life ordering; that is

T ≤ a r l V if a r T ( ς ) ≥ a r V ( ς ) for all ς (2.33)

4) Likelihood Ratio ordering; that is

T ≤ l o r V if f T ( ς ) f V ( ς ) for all ς (2.34)

These results had been established in the literature for stochastic ordering of distributions [9] [25];

T ≤ l o r V ⇒ T ≤ R o r V ⇒ { T ≤ s t o r V T ≤ a r l V (2.35)

Theorem 1.0

Let T ~ I W ( θ 1 ) and V ~ I W ( θ 2 ) . If θ 1 > θ 2 then T ≤ l o r V and by implication T ≤ R o r V , T ≤ a r l V and T ≤ s t o r V .

Proof: Let T ~ I W ( θ 1 ) and V ~ I W ( θ 2 ) . We obtain that

f T ( ς ) f V ( ς ) = θ 1 5 ( θ 2 4 + 2 θ 2 3 + 6 θ 2 2 + 12 θ 2 + 24 ) θ 2 5 ( θ 1 4 + 2 θ 1 3 + 6 θ 1 2 + 12 θ 1 + 24 ) e − ( θ 1 − θ 2 ) ς ; ς > 0

and

log e f T ( ς ) f V ( ς ) = log e ( θ 1 5 ( θ 2 4 + 2 θ 2 3 + 6 θ 2 2 + 12 θ 2 + 24 ) θ 2 5 ( θ 1 4 + 2 θ 1 3 + 6 θ 1 2 + 12 θ 1 + 24 ) ) − ( θ 1 − θ 2 ) ς

Hence,

d d ς log e f T ( ς ) f V ( ς ) = − ( θ 1 − θ 2 )

By implication, d d ς log e f T ( ς ) f V ( ς ) < 0 if θ 1 > θ 2 . This result implies that

T ≤ l o r V ⇒ T ≤ R o r V ⇒ { T ≤ s t o r V T ≤ a r l V . The result indicated clearly that Iwueze

distribution is ordered in the likelihood ratio and consequently has risk measurement, average residual measurement life and stochastic orderings.

The absolute deviations about the average and mid-points respectively can be used to assess the attitude of variation inherent in a group of observations. The absolute deviations about the average point (denoted ψ 1 ( v ) ) and that about the mid-point (denoted ψ 2 ( v ) ) is defined by

ψ 1 ( v ) = E ( | V − μ * | ) = ∫ 0 ∞ | v − μ * | f ( v ) d v (2.36)

ψ 2 ( v ) = E ( | V − M * | ) = ∫ 0 ∞ | v − M * | f ( v ) d v (2.37)

The absolute deviation about the average point can be calculated using

ψ 1 ( v ) = ∫ 0 ∞ | v − μ * | f ( v ) d v = ∫ 0 μ * | μ * − v | f ( v ) d v + ∫ μ * ∞ | v − μ * | f ( v ) d v = 2 μ * G ( μ * ) − 2 ∫ 0 μ * v f ( v ) d v (2.38)

While, the absolute deviation about the mid-point is calculated using

ψ 2 ( v ) = ∫ 0 ∞ | v − M * | f ( v ) d v = ∫ 0 m * | M * − v | f ( v ) d v + ∫ m * ∞ | v − M * | f ( v ) d v

= μ * − 2 ∫ 0 m * v f ( v ) d v (2.39)

= − μ * + 2 ∫ m * ∞ v f ( v ) d v (2.40)

where μ * = E ( V ) is the average of the random variable V, M * is the mid-point of the random variable V and G ( μ * ) = ∫ 0 μ * f ( v ) d v . By using p.d.f. (2.1), expressions (2.41) through (2.44) were obtained as follows:

∫ 0 μ * * v f ( v ) d v = μ − [ θ 5 μ * 5 + [ 2 θ + 5 ] θ 4 μ * 4 + [ 3 θ 2 + 8 θ + 20 ] θ 3 μ * 3 + [ 2 θ 3 + 9 θ 2 + 24 θ + 60 ] θ 2 μ * 2 + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] θ μ * + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] ] e − θ μ * θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.41)

∫ 0 m * v f ( v ) d v = μ * − [ θ 5 m * 5 + [ 2 θ + 5 ] θ 4 m * 4 + [ 3 θ 2 + 8 θ + 20 ] θ 3 m * 3 + [ 2 θ 3 + 9 θ 2 + 24 θ + 60 ] θ 2 m * 2 + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] θ m * + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] ] e − θ m * θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.42)

∫ m * ∞ v f ( v ) d v = [ θ 5 m * 5 + [ 2 θ + 5 ] θ 4 m * 4 + [ 3 θ 2 + 8 θ + 20 ] θ 3 m * 3 + [ 2 θ 3 + 9 θ 2 + 24 θ + 60 ] θ 2 m * 2 + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] θ m * + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] ] e − θ m * θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.43)

∫ 0 μ * * f ( v ) d v = 1 − [ θ 4 μ * 4 + [ 2 θ + 4 ] θ 3 μ * 3 + [ 3 θ 2 + 6 θ + 12 ] θ 2 μ * 2 + [ 2 θ 3 + 6 θ 2 + 12 θ + 24 ] θ μ * + [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] ] e − θ μ * [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.44)

Substituting Equations (2.41) and (2.44) in Equation (2.38), and Equation (2.42) in Equation (2.39) or Equation (2.43) in Equation (2.40), we obtain the expressions for calculating the absolute deviations about the average and mid-points respectively of an Iwueze’s distribution as:

ψ 1 ( v ) = 2 [ [ μ * 2 + μ * + 1 ] 2 θ 4 + 4 [ 2 μ * 3 + 3 μ * 2 + 3 μ * + 1 ] θ 3 + 18 [ 2 μ * 2 +   2 μ * + 1 ] θ 2 + 48 [ 2 μ * + 1 ] θ + 120 ] e − θ μ * θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.45)

ψ 2 ( v ) = 2 [ m * [ m * 2 + m * + 1 ] 2 θ 5 + [ 5 m * 4 + 8 m * 3 + 9 m * 2 + 4 m * + 1 ] θ 4   + 2 [ 10 m * 3 + 12 m * 2 + 9 m * + 2 ] θ 3 + 6 [ 10 m * 2 + 8 m * + 3 ] θ 2   + 24 [ 5 m * + 2 ] θ + 120 ] e − θ m * θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] − μ * (2.46)

The Bonferroni curve, Lorenz curve, Bonferroni index and Gini index [26] have a wide range of applications in economics, medicine, insurance, demography, and reliability. The Bonferroni curve is defined as follows [26]:

B ( c ) = 1 c μ * ∫ 0 q v f ( v ) d v = 1 c μ * [ ∫ 0 ∞ v f ( v ) d v − ∫ q ∞ v f ( v ) d v ] = 1 c μ * [ μ * − ∫ q ∞ v f ( v ) d v ] (2.47)

or alternatively, Equation (2.47) is written equivalently as

B ( c ) = 1 c μ * ∫ 0 p G − 1 ( v ) d v (2.48)

while, the Lorenz curve [26] is defined as

L ( c ) = 1 μ * ∫ 0 q v f ( v ) d v = 1 μ * [ ∫ 0 ∞ v f ( v ) d v − ∫ q ∞ v f ( v ) d v ] = 1 μ * [ μ * − ∫ q ∞ v f ( v ) d v ] (2.49)

or alternatively, Equation (2.49) is written equivalently as

L ( c ) = 1 μ * ∫ 0 p G − 1 ( v ) d v (2.50)

where μ * = E ( V ) and q = G − 1 ( c ) . However, the Bonferroni index [26] is defined as

B = 1 − ∫ 0 1 B ( c ) d c (2.51)

while the Gini index [26] is

G = 1 − 2 ∫ 0 1 L ( c ) d c (2.52)

But using Equation (2.1), we obtain

∫ q ∞ v f ( v ) d v = [ θ 5 q 5 + [ 2 θ + 5 ] θ 4 q 4 + [ 3 θ 2 + 8 θ + 20 ] θ 3 q 3 + [ 2 θ 3 + 9 θ 2 + 24 θ + 60 ] θ 2 q 2 + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] θ q + [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] ] e − θ q θ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] (2.53)

On substituting Equation (2.53) in Equations (2.47) and (2.49), the expressions for calculating the Bonferroni and Lorenz curves respectively of an Iwueze’s distribution are obtained as:

B ( c ) = 1 c [ 1 − [ q [ q 2 + q + 1 ] 2 θ 5 + [ 5 q 4 + 8 q 3 + 9 q 2 + 4 q + 1 ] θ 4   + 2 [ 10 q 3 + 12 q 2 + 9 q + 2 ] θ 3 + 6 [ 10 q 2 + 8 q + 3 ] θ 2   + 24 [ 5 q + 2 ] θ + 120 ] e − θ q [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] ] (2.54)

L ( c ) = 1 − [ q [ q 2 + q + 1 ] 2 θ 5 + [ 5 q 4 + 8 q 3 + 9 q 2 + 4 q + 1 ] θ 4   + 2 [ 10 q 3 + 12 q 2 + 9 q + 2 ] θ 3 + 6 [ 10 q 2 + 8 q + 3 ] θ 2   + 24 [ 5 q + 2 ] θ + 120 ] e − θ q [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] (2.55)

In addition, on substituting Equations (2.54) and (2.55) in Equations (2.51) and (2.52) respectively, the expression for Bonferroni index of an Iwueze’s distribution is obtained as

B = 1 − [ q [ q 2 + q + 1 ] 2 θ 5 + [ 5 q 4 + 8 q 3 + 9 q 2 + 4 q + 1 ] θ 4   + 2 [ 10 q 3 + 12 q 2 + 9 q + 2 ] θ 3 + 6 [ 10 q 2 + 8 q + 3 ] θ 2   + 24 [ 5 q + 2 ] θ + 120 ] e − θ q [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] (2.56)

while, the expression for the Gini index of an Iwueze’s distribution is:

G = 2 [ q [ q 2 + q + 1 ] 2 θ 5 + [ 5 q 4 + 8 q 3 + 9 q 2 + 4 q + 1 ] θ 4   + 2 [ 10 q 3 + 12 q 2 + 9 q + 2 ] θ 3 + 6 [ 10 q 2 + 8 q + 3 ] θ 2   + 24 [ 5 q + 2 ] θ + 120 ] e − θ q [ θ 4 + 4 θ 3 + 18 θ 2 + 48 θ + 120 ] − 1 . (2.57)

Entropy is the most well-known statistical feature and/or quality that accounts for the uncertainty in a continuous random variable’s probability distribution. The most prominent type of entropy, known as the Renyi entropy [27], is defined for V with. f ( v ) as follows:

T R E ( ℑ ) = 1 1 − ℑ log [ ∫ 0 ∞ f ℑ ( v ) d v ] ; ℑ > 0   and   ℑ ≠ 1 (2.58)

By substituting Equation (2.1) into Equation (2.58), we obtain the expression necessary for obtaining the entropy of Iwueze distribution as

T R E ( ℑ ) = 1 1 − ℑ log [ ∫ 0 ∞ θ 5 ℑ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] ℑ [ 1 + v + v 2 ] 2 ℑ e − ℑ θ v d v ] = 1 1 − ℑ log [ θ 5 ℑ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] ℑ ∫ 0 ∞ ∑ i = 0 ∞ ( 2 ℑ i ) v i [ 1 + v ] i e − ℑ θ v d v ] = 1 1 − ℑ log [ θ 5 ℑ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] ℑ ∑ i = 0 ∞ ( 2 ℑ i ) ∑ k = 0 ∞ ( i k ) ∫ 0 ∞ v i + k e − ℑ θ v d v ] (2.59)

But on solving ∫ 0 ∞ v i + k e − ℑ θ v d v and noting the well known gamma relational form ∫ 0 ∞ h η e − θ h d h = Γ ( η + 1 ) θ η + 1 = η ! θ η + 1 ; η > 0 and η is an interger, Equation (2.59) reduces to

T R E ( ℑ ) = 1 1 − ℑ log [ θ 5 ℑ [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] ℑ ∑ i = 0 ∞ ( 2 ℑ i ) ∑ k = 0 ∞ ( i k ) ( i + k ) ! ( ℑ θ ) I + k + 1 ] = 1 1 − ℑ log [ ∑ i = 0 ∞ ( 2 ℑ i ) ∑ k = 0 ∞ ( i k ) ( i + k ) ! θ 5 ℑ − ( i + k + 1 ) [ θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ] ℑ ℑ I + k + 1 ] (2.60)

The stress and strength reliability of an independent strength; say V and stress, say B, of random variables from an Iwueze’s distribution with parameters θ 1 and θ 2 is evaluated and calculated using [28]:

R S S = P ( V < B ) = ∫ 0 ∞ P ( V < B / B = b ) d b = ∫ 0 ∞ f ( v , θ 1 ) G ( v , θ 2 ) d v = ∫ 0 ∞ θ 1 5 [ 1 + v + v 2 ] 2 e − θ 1 v [ θ 1 4 + 2 θ 1 3 + 6 θ 1 2 + 12 θ 1 + 24 ] [ 1 − [ [ [ θ 2 4 + 2 θ 2 3 + 6 θ 2 2 + 12 θ 2 + 24 ] + θ 2 4 v 4 + 2 θ 2 3 ( θ 2 + 2 ) v 3   + 3 θ 2 2 [ θ 2 2 + 2 ( θ 2 + 2 ) ] v 2 + 2 θ 2 [ θ 2 3 + 3 [ θ 2 2 + 2 [ θ 2 + 2 ] ] ] v ] e − θ 2 v θ 2 4 + 2 θ 2 3 + 6 θ 2 2 + 12 θ 2 + 24 ] ] d v = 1 − θ 1 5 [ θ 2 4 t 1 ( θ 1 , θ 2 ) + 2 θ 2 3 t 2 ( θ 1 , θ 2 ) + 6 θ 2 2 t 3 ( θ 1 , θ 2 ) + 12 θ 2 t 4 ( θ 1 , θ 2 ) + 24 t 5 ( θ 1 , θ 2 ) ] [ θ 1 4 + 2 θ 1 3 + 6 θ 1 2 + 12 θ 1 + 24 ] [ θ 2 4 + 2 θ 2 3 + 6 θ 2 2 + 12 θ 2 + 24 ]   [ θ 1 + θ 2 ] 9 (2.61)

where

t 1 ( θ 1 , θ 2 ) = [ [ θ 1 + θ 2 ] 8 + 4 [ θ 1 + θ 2 ] 7 + 20 [ θ 1 + θ 2 ] 6 + 96 [ θ 1 + θ 2 ] 5 + 456 [ θ 1 + θ 2 ] 4     + 1920 [ θ 1 + θ 2 ] 3 + 7200 [ θ 1 + θ 2 ] 2 + 20160 [ θ 1 + θ 2 ] + 40320 ] (2.62)

t 2 ( θ 1 , θ 2 ) = [ θ 1 + θ 2 ] [ [ θ 1 + θ 2 ] 7 + 5 [ θ 1 + θ 2 ] 6 + 24 [ θ 1 + θ 2 ] 5 + 114 [ θ 1 + θ 2 ] 4     + 480 [ θ 1 + θ 2 ] 3 + 1800 [ θ 1 + θ 2 ] 2 + 5040 [ θ 1 + θ 2 ] + 10080 ] (2.63)

t 3 ( θ 1 , θ 2 ) = [ θ 1 + θ 2 ] 2 [ [ θ 1 + θ 2 ] 6 + 4 [ θ 1 + θ 2 ] 5 + 18 [ θ 1 + θ 2 ] 4 + 72 [ θ 1 + θ 2 ] 3     + 264 [ θ 1 + θ 2 ] 2 + 720 [ θ 1 + θ 2 ] + 1440 ] (2.64)

t 4 ( θ 1 , θ 2 ) = [ θ 1 + θ 2 ] 3 [ [ θ 1 + θ 2 ] 5 + 4 [ θ 1 + θ 2 ] 4 + 14 [ θ 1 + θ 2 ] 3 + 48 [ θ 1 + θ 2 ] 2     + 120 [ θ 1 + θ 2 ] + 240 ] (2.65)

t 5 ( θ 1 , θ 2 ) = [ θ 1 + θ 2 ] 4 [ [ θ 1 + θ 2 ] 4 + 2 [ θ 1 + θ 2 ] 3 + 6 [ θ 1 + θ 2 ] 2 + 12 [ θ 1 + θ 2 ] + 24 ] (2.66)

The method of likelihood estimation is a methodology for estimating parameters θ from a random sample of size s ( V 1 , V 2 , V 3 , ⋯ , V s ) that follows an Iwueze’s distribution with a theoretical density function in Equation (2.1) and the likelihood function L * ( θ ) is given as

L * ( θ ) = L * = ( θ 5 θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ) s ∏ i = 1 s [ 1 + v + v 2 ] 2 e − θ ∑ i = 1 s v i (2.67)

The natural logarithm of L ( θ ) is calculated as follows:

L * 1 = ln L * ( θ ) = 5 s ln θ − s ln ( θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ) + ∑ i = 1 s ln [ 1 + v + v 2 ] 2 − θ ∑ i = 1 s v i = 5 s ln θ − s ln ( θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ) + ∑ i = 1 s ln [ 1 + v + v 2 ] 2 − s θ v → (2.68)

Hence,

d L * 1 d θ = 5 s θ − s ( 4 θ 3 + 6 θ 2 + 12 θ + 12 ) θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 − s v → , (2.69)

By equating Equation (2.69) to zero, the estimate for θ ⌢ of θ that maximizes Equation (2.68) was calculated using

5 s θ − s ( 4 θ 3 + 6 θ 2 + 12 θ + 12 ) ( θ 4 + 2 θ 3 + 6 θ 2 + 12 θ + 24 ) − s v → = 0 , (2.70)

Equation (2.70) reduces to

v → θ 5 + ( 2 v → − 1 ) θ 4 + 2 ( 3 v → − 2 ) θ 3 + 6 ( 2 v → − 3 ) θ 2 + 24 ( v → − 2 ) θ − 120 = 0 (2.71)

Where the sample average is v → . The Quintic Equation (2.71) can be solved analytically, numerically and using Mathematical Software such as R, Mathematica etc. Solving the Quintic Equation (2.71) yields five values for θ ⌢ in which only one satisfies θ ⌢ > 0 .

The likelihood estimate for θ (abbreviated MLE θ ⌢ ) was employed to fit Iwueze’s, Lindley [3], Shanker [4], Akash [5], Rama [6], Suja [7], Sujatha [8], Amarendra [9], Devya [10], Shambhu [11], Aradhana [12], Akshya [14], Pranav [15], Ishita [16], Ram Awadh [17], Prakaamy [18], Odoma [19] and Exponential [20] [21] [22] distributions to the real-life data set on relief times (in minutes) of twenty (20) patients receiving analgesic treatment. Iwueze’s and the aforementioned distributions constitute the competing distributions for modeling the real-life data. The real life-time data used in the study is shown in Table 2.

See [1].

Then, the superiority (goodness) of fit for the competing distributions fitted to the real-life data set was compared and judged using the values of the superiority of fit indices. The probability (denoted p), − 2 L * 1 , Akaike Information Criterion (denoted AIC) [29] [30] [31] [32], AICC (Akaike Information Criterion Corrected [30]), BIC (Bayesian Information Criterion) [30] [31] [32] [33] and Kolmogorov-Smirnov (denoted K * − S * ) [31] values were evaluated and utilized as the superiority of fit indices for comparison’s in order to judge the superiority (or goodness) of the fit of the competing distributions. The computational expressions for evaluating AIC, AICC, BIC and K * − S * statistics are:

AIC = − 2 L * 1 + 2 m (2.72)

AICC = AIC + 2 m ( m + 1 ) z − m − 1 (2.73)

BIC = − 2 L ^ 1 + m L n ( z ) (2.74)

D = sup v | G z ( v ) − G 0 ( v ) | (2.75)

where the totality of estimated parameters equals one is m, the totality of all observations is z and G Z ( v ) is the theoretical distribution function. The decision will then be made to accept the distribution with the lower Probability (p) and K * − S * values, as well as the least and/or lowest value for − 2 L ^ 1 , AIC, AICC and BIC among all considered distributions, as the distribution that provides superior fit to the life-time data.