THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL DISTRIBUTION WITH INDUSTRIAL APPLICATIONS Текст научной статьи по специальности «Математика»

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam BT.. N 4sm

THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D ' , 0 ■ )

DISTRIBUTION WITH INDUSTRIAL APPLICATIONS V0lUme ' December 2024

THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL DISTRIBUTION WITH INDUSTRIAL

APPLICATIONS

Jamilu Yunusa Falgore1, Yahaya Abubakar2, Sani Ibrahim Doguwa3, Aminu Suleiman Mohammed4, and Abdussamad Tanko Imam5

12,34Department of Statistics, Ahmadu Bello University, Zaria-Nigeria. 5Department of Mathematics, Ahmadu Bello University, Zaria-Nigeria. 1 [email protected], [email protected], [email protected], [email protected], [email protected]

Abstract

Based on the limitations of the Inverse Lomax distribution and exponential distribution as outlined in the literature, a new extension of the exponential distribution is introduced in this paper. Some statistical properties of the ILOEED such as mean, variance, skewness, quantile function, moment, moment generating function, as well as kurtosis were demonstrated. The shapes of the hazard function of the proposed distribution suggest that it can be used to fit a dataset with increasing and bath-tube shapes. A simulation study for three different cases was also presented. The result of the simulation for three different cases (I, II, and III) indicated that ILOEED's estimates are consistent. Lastly, an application to Industry datasets was demonstrated based on the ILOEED. Having minimum values of the Goodness-of-fit criteria and Goodness-of-fit statistics, the ILOEED can be recommended to fit these three datasets, in preference to other distributions considered in this paper.

Keywords: Inverse Lomax-G family, Exponentiated Exponential Distribution, Weibull Distribution, Heavy Tail Distribution.

I. Introduction

In their ongoing pursuit of adaptive and flexible statistical models, scientists and researchers have been investigating new distributions that may accurately represent a wide range of real-world data patterns. In this quest, the proposed distribution shows great promise as a more versatile model for a range of phenomena. Interestingly, it leverages the advantages of its parent distributions, the Odd-Exponentiated and the Inverse Lomax distribution (ILD), to produce a distribution that can describe a wide range of datasets, especially those with heavy tails and non-monotone failure rates ([1],[2],[3]). Extreme occurrences or outliers are more likely to occur in a heavy-tail distribution because its tails decay more slowly than those of a normal distribution. This suggests that there is more risk or variability in the data. The Pareto II (ILD), Cauchy, and Student's t-distributions are a few instances of heavy-tail distributions.

Inverse Exponentiated Odd Lomax Exponential distribution was proposed by [4], offering a fresh outlook on statistical modeling and analysis. Their research investigates the statistical properties of this distribution, contributing to a better comprehension of its practical applicability.

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam BT.. N 4sm THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D ' , 0 ■ ) DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_V0lume I9, Pecember, 2U24

A four-parameter Exponentiated Odd Lomax Exponential (EOLE) distribution was proposed by [3], combining an exponentiated odd function with Lomax and exponential elements, thereby enhancing the distribution's versatility and providing a more intricate parameterization for modeling. To give researchers a wider range of tools for a variety of applications, Inverse Exponentiated Lomax Power Series distribution was proposed by [2], which expanded the family of distributions that combines Lomax and exponential components. Inverse Lomax distribution has two major drawbacks. These are: Limited Flexibility in Shape i.e. ILD has some difficulty adjusting its probability density function, especially in the peak and tail areas. This can hinder its ability to precisely model data with certain patterns [5]. A potential constraint in modeling hazard rates i.e. while often used to model non-monotone hazard rates (failure rates that vary over time), the Inverse Lomax distribution may not be able to capture all possible hazard rate shapes that can arise in real-world situations [6]. However, exponential distribution has limitations of constant failure rate and memory-less property. Hence, the need to study the proposed distribution to remedy some of the drawbacks of the ILD and exponential distribution.

To tackle the challenge of modeling the lifespans of electronic devices, which is essential for predicting future failures and achieving energy savings, a reliability model that is based on the inverse power law and generalized inverse Weibull distribution [7]. It demonstrates how successful the proposed distribution is in influencing the average time to failure of the examined capacitor, as opposed to standard models such as the inverse Weibull, using an empirical analysis that focuses on the life cycle of a surface-mounted electrolytic capacitor. To analyze COVID-19 death cases in Europe and China, the Exponentiated Transformation of Gumbel Type-II (ETGT-II) model [8]. This model provides a thorough analysis of statistical features and estimates model parameters using maximum likelihood and Bayesian approaches. The ETGT-II model is shown to be efficient through simulation analysis. It exhibits a promising adaptation to the COVID-19 death data sets, perhaps providing a better fit than other models. The new exponential inverted Topp-Leone (NEITL) distribution is presented by [9]. It is an extension of the inverted Topp-Leone distribution with an extra shape parameter. Its features, estimation methods, and application to actual datasets in the engineering and medical domains are all explored. The generalized log-exponential transformation of Gumbel Type-II (GLET-GTII), which was proposed by [10] as a generator for a generalized version of the Gumbel type-II model, increases modeling flexibility by adding a new parameter. Quantiles, survival function, and reliability are among the statistical attributes that are examined. Maximum likelihood and Bayesian approaches are used as parametric estimation methods, and they show consistency through Monte Carlo simulations and outperform other models in practical implementations, especially when it comes to infectious diseases like COVID-19. Using a new power function and a modified Kies generalized transformation, a novel statistical model and discusses its theoretical characteristics, including the density function, quantile function, and stochastic ordering [11]. The moment exponential distribution is extended by the two-parameter alpha power-transformed moment exponential (APTME) distribution, which shows excellent fit and performance through a variety of estimators and simulation studies. Its practical significance is demonstrated by its application to real-world datasets [12].

Moreover, the flexible four-parameter Kumaraswamy extended exponential (KwEE) distribution is presented by [13]. This model shows that the novel distribution may provide a better fit than current models in several COVID-19 spread analysis situations by evaluating COVID-19 mortality rates in nations such as Italy and the United Kingdom. To represent the dependability metrics of a generalized exponential model based on the inverse power law (IPL), was suggested using a multilayer ANN with Bayesian regularization by [14]. The outcomes show how well ANNs operate as a reliable mathematical tool for evaluating lifetime model reliability, and they are backed by a real-world application. Under the generalized type-I progressive hybrid censoring sample (GTI-PHCS), statistical inference for the Kavya-Manoharan generalized exponential distribution was

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam BT.. N 4sm THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D ' , 0 ■ ) DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_V0lume I9, Pecember, 2U24

proposed by [15]. It does this by examining different estimation techniques, such as maximum

likelihood and Bayesian approaches, and using real-world data analysis and simulations to illustrate

how well the techniques work. Through simulation studies and application to engineering datasets,

the half-logistic modified Kies exponential (HLMKEx) distribution as a flexible three-parameter

model for modeling real-world data was presented [16]. It provides detailed mathematical features,

such as density function forms and estimation methods, and shows its superior fit over competing

distributions. Generalized exponentiated unit Gompertz (GEUG), a unique four-parametric model,

was introduced by [17] to represent clinical trial data of patients with arthritis. By adding new tuning

factors to the unit Gompertz (UG) model, the GEUG model aims to improve the estimate of

distribution parameters, hence increasing the model's adaptability.

Motivation: The inability of Inverse Lomax distribution and exponential distribution to adequately capture some intricate data patterns led to the development of the proposed distribution. To introduce a more adaptable model, that expands on the advantages of its parent distributions; the Odd-Exponentiated and the Inverse Lomax. Key features: More flexibility in defining its density and hazard functions, Non-monotone hazard rates capable of simulating phenomena with fluctuating risk characteristics over time, as well as heavy tails that capture extremes and outliers well. In this article, a new extension of Exponential distribution is introduced. The proposed distribution is formulated based on the Inverse Lomax Odd Exponentiated-G family of distributions. The most important feature of the proposed distribution, with two shape parameters, a scale, and rate parameters represents its ability to provide different density shapes. This means that the proposed distribution can fit various datasets adequately. The proposed distribution has the following desirable properties. (i) The probability density function (pdf) of the ILOEED proposed distribution has a simple closed form. Then, ILOEED can be used for modeling and analyzing reallife data in Industries; (ii) The shape parameters of the proposed distribution make it very flexible to exhibit increasing and bath-tube failure rate shapes; (iii) Additionally, the density of the proposed distribution can also provide more flexible shapes. The paper is organized into six sections. The proposed distribution is defined in Section 2. The Statistical properties of the proposed distribution are presented in section 3. The estimation of the parameters of the proposed distribution using the method of Maximum Likelihood Estimates (MLEs) is introduced in Section 4. In Section 5, a simulation study based on the properties of the MLEs of the proposed distribution is presented. Applications of the proposed distribution to industry datasets are presented In Section 6. Section 7 concludes the paper.

The Inverse Lomax-Odd Exponentiated G (IL-OEG) family was proposed by [1] based on the T-X generator of [18].The cumulative density function (CDF) and probability density function (PDF) of IL-OEG are given as

F (x;A,/,8, A) =

1+ A-

(1 - G (x; A)) G (x; A) ^

And

f (x;A,/,8, A) =

8/Ag (x; A)[1 - G( x; A)]

e-i

[G( x; A)]

8+1

1+ A

; x > OA/,8, A > 0

(1)

(1 - G( x; A)) G( x; A)

-(1+/)

; x > 0,A,/,8, A > 0 (2)

Where A is a vector of parameter(s) for the baseline distribution, G(x; A) = 1 — G(x; A). Based on equations (1) and (2), Exponential distribution is considered to be the baseline distribution. So, the G(x; A) is equivalent to an exponential density.

/

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam BT.. N 4sm THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D ' , 0 ■ ) DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_V0lume I9, Pecember, 2U24

II. Inverse Lomax Odd Exponentiated Exponential distribution (ILOEED)

The exponential distribution is the probability distribution of the time between events in a Poisson point process, that is, a process in which events occur continuously and independently at a constant average rate. It is a subset of the gamma distribution. The CDF and PDF of the exponential distribution are presented in equations (3) and (4).

G(x;v) = 1 - e'"", x,v> 0

And

g(x;v) = ve vx ; x,v> 0 Then, the CDF and PDF of the ILOEED can be given as:

(3)

(4)

F ( x;e,A,y,v) =

-y

1+A<

(1 - evx ),

; x,v,y,A,e> 0

(5)

And

f ( x;0,A,y, v) = ■

r r -vx iel -(1+y)

veyAe-vex 1+a\ e 1 _ 1 (1 - evx ) J_

vx -¡e+i

[1 - e-vx ]

; x,v,y,A,e>0

(6)

Where v is the rate, A is the scale, and e and y are the shape parameters, respectively. Having this combination of parameters, we hope that ILOEED will fit datasets of different shapes. The reliability, hazard, and cumulative hazard functions of the ILOEED are presented in equations (7), (8), and (9).

/•œ

R(x;A,y,e,v) = | f (x;A,y,e,v)dx

= 1 -

1 + A<

(1 - em ) ,

; x,A,y,e,v> 0,

(7)

And

Îx

h(v)dv = -log ( R( x;A,y,e,v))

œ

= -log < 1 -

-y

1 + A<

(1 - evx ),

(8)

■— 1.05.6.0.5.7-0.51-05 ■— U-05.9-1.T-U-0.5 ■— u-05.e-15.T-1.5J.-05 ■— u-05.6-Zï-2.).-05 B— u-05.6=25.i-25,i.-05

Figure 1: PDF and CDF plots of ILOEED at various parameter values

y

Figure (1) shows the various shapes of the ILOEED's PDF. This includes skewed and symmetry. Figure (1) also indicates the various shapes that the ILOEED can take which include constant, Bathtub, and monotone-increasing hazard shapes.

III. The Statistical Properties of the ILOEED

I. The quantile function of the ILOEED

The quantile function of ILOEED can be derived by inverting the CDF of the ILOEED given in equation (5) as follows:

Then,

F ( x;A,r,O,v) = U =

Ur -1 = A<

-r

1 + A<

(1 - ),

(9)

(1 - ),

After simplifying and by making x the subject of the formula, we have

-log | k

x = -

1 + k

v

(10)

Where k =

f -1 \ U r -1

A

and U is uniformly distributed between 0 and 1.

The median of the ILOEED family can be derived by setting U=0.5 in equation (10). II. The moments of the ILOEED

Let X be a random variable that follows ILOEED with parameters (A,y,0,v^, then the C" moment about the origin is given by:

/uc = E (XC ) = P xcf (x;A,y,6,v)dx

J 0

Using some linear representation, we have the moment of the ILOOED as

»

Uc=T V [k2-Ok^ xVev (1 - ev )

[k2-dkl-1]

dx

(11)

k, ,k, =0

(- 1)k!+k2 r(r + k )Y{Okl + k2 )Ak1

Where V =

kxY(y)k2\Y (Ok,)

. By considering the binomial expansion of the term

(1 - ev )

vx\ [k2-Okj -1] _

= Z (-1)j

j=0

Vc =

f [k2 -Ok, -1]^

j )

e j . Then, equation (11) becomes

C * k,

Y k j l

» Y -C\

xce~v(j) xdx = ■ j

[v(1 + j)]c

(12)

O

»

Where Yk ] = £ Wv[k2 -0k, -1](-1)j

j .k1.k2

f [k2 -0kx -1]

A

j

. The mean of the ILOEED can be

derived by setting c=1 in equation (12). Moreover, the second moment can also be derived by setting c=2, and then using the relation Var(X) = f - [f ]2 to find the variance.

III. The Characteristic and Moment Generating Functions of the ILOEED The characteristic function of the ILOEED can be given as

L(x;v,0,y,l) = nlog(ly0v)-v0£ xt - (1 + y)£log

i=1

i=1

1+l

( e-v* ^

1- e~m

-(1+0)£log[1 -e~m'] (13) i=1

And the moment generating function of the ILOEED can be given as Mx (t) = f etxf (x;X,y,O,u)dx

J 0

a

= Z " Oh ]j etxue-^x (1 - )k-Ok -1] dx

pa

= ^J0 e~(t-"(l+'))Xdx =

Y

rX ,-(i-v(1+j))x,_ Y k,j

o(1 + j) -1

Table 1: Mean, Variance, Skewness, and Kurtosis of ILOEED at various parameter values

Parameter E(X) Var(X) SK(X) KUR(X)

v = O = y = l = 0.5 1.0694 6.6518 4.1394 26.2247

v = 1,0 = 0.5, y = 1,l = 0.5 0.9794 2.8142 3.325 14.6679

v = 1.5,0 = 0.5,y = 1.5,l = 0.5 0.9061 1.6187 2.4044 10.900

v = 2,0 = 0.5,y = 2,l = 0.5 0.8449 1.0637 2.0973 9.0716

v = 2.5,0 = 0.5,y = 2.5, l = 0.5 0.7927 0.7549 1.8992 8.0158

v = 3,0 = 0.5,y = 3,l = 0.5 0.7476 0.5639 1.7499 7.3406

v = 3.5,0 = 0.5,y = 3.5,l = 0.5 0.7081 0.4372 1.6408 6.8795

v = 4,0 = 0.5,y = 4,l = 0.5 0.6320 0.3486 1.5570 6.5505

(14)

Table (1) presents some basic statistics based on the moments of the ILOEED at varying parameter values. It's evident from the table that as the shape parameters increase, all the values of the statistics decrease. The skewness is positive and the tails tend towards the right. Positive kurtosis also indicates the tails are heavy.

IV. Maximum Likelihood Estimates (MLE)

In this section, we used the maximum likelihood method to estimate the parameters of the ILOEED. Let

x, x2, x3, x4,x„ be a random sample independently drawn from ILOEED family. Then, the log-likelihood function L (u, O, y, X} of equation (6) is given as

L(x;v,0,y,l) = nlog(lyOv)-vO£x -(1 + y)£log

1 + l

1 - e~

-(1 + 0)£log [1 - e

(15)

=1

=1

Taking the partial derivatives of equation (15) with respect to v, 0, y, and A, yields:

SL n n n

SL = n-8Yxt-Y

Su u ¡=1 ¡=1

(1 + y)0A

( eu

1 - e-uxi v1 e J

e1

xe

(1-e ")!

1+ A

( e~uxi \

1 -eu v1 e J

SL n " " — =--u Y xi- Y

se e Y 1 Y

(1+y)A

Y f

1 - eu v1 e J

log

1 - e"

1+ A

f eu ^

1 - e-uxi v1 e J

-yio§[1

i=1

- e

SL=n-fklg

Sy y i=1

1+ A

' e-Vi y v1 - e ux' j

(1+y)

c e~vx' ^

1 - eu v1 e J

1 + A

^ e-vx' ^

1 - e u v1 e J

(16)

(17)

(18)

(19)

The MLEs of the u, e, y, and A can be determine by solving the following non-linear equations (16), (17), (18), and (19) with respect to each parameter.

V. The Simulation Studies of the ILOEED

e

Here, we highlighted five (5) steps on how to do a Monte Carlo simulation study as follows: Step 1: Clearly state the pseudo-population that can be used in generating random samples usually by writing code in a specific method. In this study, the pseudo-population is the quantile function of ILO EED given in equation (10).

Step 2: Sample from the population of interest (depending on your objective).

Step 3: Estimate the parameter of interest from the sample and keep it in a vector.

Step 4: Repeat the previous steps i.e 2 and 3 n-times (n is the number of trials).

Step 5: Create a relative frequency distribution of resulting values that is a Monte Carlo

approximation of the distribution of samples under the conditions defined by the pseudo-population

and the procedures of sampling. Based on the above procedure, we carry out a simulation studies as

explained below:

i). For known parameter values i.e 0 = (0,v,y,A)T, we simulated a random sample of size n from the ILOEED using equation (10).

ii). We then estimated the parameters of the ILOEED by the method of maximum likelihood. iii). Perform 1,000 replications of steps i through ii.

iv). For each of the four (4) parameters of the ILOEED, we compute the mean, bias, and Root mean squared error (RMSE) from the 1,000 parameter estimates. The statistics are given by

i 1,000 i 1,000 I 1 1,000

0 =-y 0 , Bias(0) =-y (0 -0), RMSE(0) = -y (0 -0)2 (20)

1,000 y ' 1,000 y ' \ 1,000 y '

Where 0 = (0, v, y, A) is the MLE for each iteration (n=10, 20, 30, 50, 70, 90, 150, 170). Table (2) reports Case I. R software by [19] was used for the simulation. 123 was set as the seed for reproducibility. Three cases were considered for the simulation. Case I:( v = 0.3, 0 = 1, y = 0.9 and A = 0.4), Case II:( v= 0.6, 0 = 0.5, y = 0.9 and A = 0.5), and Case III: (v= 2, 0 = 5, y = 0.5 and A = 0.3). Tables (3) and (4) are for Case II and Case III, respectively.

Table 2: Simulation Results for Case I

N Estimates Bias RMSE n Estimates Bias RMSE

10 0.4219 0.1219 0.2651 70 0.3335 0.0335 0.1254

1.1481 0.1481 0.4772 1.0131 0.0131 0.1942

1.0841 0.1841 0.4819 0.9849 0.0849 0.305

0.5721 0.1721 0.5426 0.4487 0.0487 0.2497

20 0.3735 0.0735 0.2062 90 0.3275 0.0275 0.1124

1.0835 0.0835 0.348 1.0048 0.0048 0.1701

1.0434 0.1434 0.4297 0.9808 0.0808 0.2909

0.5046 0.1046 0.4076 0.4373 0.3730 0.2225

30 0.3540 0.0540 0.1713 150 0.3204 0.0204 0.0853

1.0533 0.0533 0.2797 0.0998 -0.0021 0.1396

1.0039 0.1039 0.3742 0.9600 0.0600 0.2393

0.4951 0.0951 0.3606 0.4293 0.0293 0.1763

50 0.3378 0.0378 0.1405 170 0.3174 0.0174 0.0823

1.0324 0.0324 0.2324 0.9969 -0.0031 0.1315

0.9905 0.0905 0.3365 0.9549 0.0549 0.2261

0.4550 0.0550 0.2725 0.4219 0.0219 0.1635

Table 3: Simulation Results for Case II

n Estimates Bias RMSE N Estimates Bias RMSE

10 0.7954 0.1954 0.4988 70 0.6469 0.0469 0.2290

0.6110 0.1110 0.3080 0.5111 0.0111 0.1061

1.0250 0.1250 0.4286 0.9809 0.0809 0.2904

0.6217 0.1217 0.5800 0.5105 0.0105 0.2252

20 0.7301 0.1301 0.3991 90 0.6320 0.0320 0.1907

0.5496 0.0496 0.1884 0.5084 0.0084 0.0931

1.0232 0.1232 0.3936 0.9665 0.0665 0.2571

0.5751 0.0751 0.4643 0.5096 0.0096 0.2034

30 0.7013 0.1013 0.3505 150 0.6243 0.0243 0.1497

0.5289 0.0289 0.1521 0.5032 0.0032 0.0724

0.9932 0.0932 0.3403 0.9512 0.0512 0.2258

0.5607 0.0607 0.3573 0.5070 0.0070 0.1680

50 0.6588 0.0588 0.2572 170 0.6192 0.0192 0.1462

0.5177 0.0177 0.1177 0.5028 0.0028 0.0718

0.9869 0.0869 0.3146 0.9469 0.0469 0.2102

0.5268 0.0268 0.2862 0.5025 0.0025 0.1513

Table 4: Simulation Results for Case III

n Estimates Bias RMSE N Estimates Bias RMSE

10 2.2708 0.2708 0.5049 70 2.1298 0.1298 0.2559

5.4425 0.4425 1.7522 5.0473 0.0473 0.8753

0.7649 0.2649 0.6554 0.5417 0.0417 0.1907

0.5027 0.2027 0.4227 0.4346 0.1346 0.2376

20 2.2121 0.2121 0.3836 90 2.1104 0.1104 0.2116

5.2441 0.2441 1.3917 5.0473 0.0473 0.7814

0.6323 0.1323 0.3960 0.5322 0.0322 0.1673

0.4837 0.1837 0.3432 0.4188 0.1188 0.2100

30 2.1779 0.1779 0.3433 150 2.0906 0.0906 0.1679

5.2282 0.2282 1.2953 5.0259 0.0259 0.6080

0.5810 0.0810 0.3174 0.5159 0.0159 0.1181

0.4822 0.1822 0.3395 0.4053 0.1053 0.1776

50 2.1505 0.1505 0.2937 170 2.0785 0.0785 0.1473

5.1003 0.1003 1.0516 5.0160 0.0160 0.5512

0.5593 0.0593 0.2506 0.5135 0.0135 0.1033

0.4484 0.1484 0.2640 0.3933 0.0933 0.1581

Tables (2), (3), and (4) presents the simulation results. As the value of the sample size (n) increases, the simulation results of the ILOEED show:

• Stability of the MLES,

• The bias of the MLEs approach zero, and

• Decrease in the RMSEs of the MLEs.

VI. Applications of the ILOEED to Industry Datasets

Odd Exponentiated Inverse Lomax Distribution (ILOEED) was fitted to three datasets. This includes datasets with increasing and bathtub hazard shapes. The Goodness-of-fit criteria used are the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). A statistical metric called the Akaike Information Criterion (AIC) by [20] was employed in the selection of models. AIC offers a quantitative method for weighing a model's complexity concerning it's goodness of fit, intending to choose the model that most accurately describes the data while preventing overfitting. Bayesian Information Criterion (BIC), a statistical metric for model selection, was developed by [21]. Similar to AIC, BIC takes a Bayesian approach to balancing model fit and complexity. Moreover, three Goodness-of-fir statistics were used. These are the Kolmogorov-Smirnov (K-s), Anderson-Darling (A-D), and Cramer-Von Mises (C-vM). By comparing the cumulative distribution functions (PDFs) of two datasets, the Kolmogorov-Smirnov test determines whether they have the same continuous distribution. It is a widely applicable test that helps to compare theoretical and empirical distributions and is especially helpful when parametric assumptions are not met. The A-D test is useful for determining fit, especially in situations with extreme values or interesting tail behavior. It does this by calculating the goodness-of-fit between the sample's empirical distribution function and the specified distribution's cumulative distribution function. This test is sensitive to deviations in the distribution's tails. The C-vM test, named for

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam BT.. N 48m THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D , b° _( j DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_V°lume I9, December, 2024

Carl von Mises and Harald Cramer, measures the difference between the sample's empirical distribution function and the specified distribution's cumulative distribution function to determine whether the sample fits the distribution. This test is favored in some circumstances because it is easy to compute and provides a good measure of goodness-of-fit. Finally, The negative log-likelihood (-ll), which is frequently minimized in maximum likelihood estimation, measures how well a model fits observed data by esti-mating the probability of observing the data given the model parameters. It is favored for its stability and ease of use in parameter estimation and is essential to many different disciplines, including biology, econometrics, and machine learning. Fitdistrplus package by [22] in R was used in fitting the three datasets in this section.

I. Application to the Breaking Stress of Carbon Fibres Dataset

These data as reported by the [23], represent the breaking stress of carbon fibres of 50mm length in Gpa. The data is symmetry and has an increasing hazard shape. The dataset is as follows: 0.39, 0.85, 1.08, 1.25, 1.47, 1.57, 1.61, 1.61, 1.69, 1.80, 1.84, 1.87, 1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41, 2.43, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.56, 3.60, 3.65, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90.

ILOEED was fitted alongside the Inverse Exponentiated Odd Lomax Exponential Distribution (IEOLED) by [4], Alpa Power Exponential Distribution (APED) by [24], as well as the Exponential Distribution (ED).

Table 5: MLEs and Goodness-of-fit Criteria for the fitted ILOEED and other comparators for the comparators for the Breaking Strengths of Carbon Fibres Dataset

Distributions Estimates Standard Error

ILOEED (v,0, y,A) 0.6586 1.3019

3.4545 5.7285

0.4797 0.3904

1047.5537 4727.8614

IEOLED (v,0,y,A) 0.0854 0.0274

3.369 0.2963

1.03E+07 1.67E+04

1.69E+12 2.37E+04

APED (v,0) 2.99E+05 1.19E+04

1.10E+00 5.46E-02

ED (v) 3.63E-01 4.46E-02

-ll

AIC

BIC

84.6124 177.2247 185.9833

86.1725 180.3251 189.1037

92.3964 188.7927 193.172

Table (5) presents the MLEs, log-likelihoods, AICs, and BICs of the ILOEED and others. The Table indicates that ILOEED is the best with minimum values of AIC and BIC. Furthermore, Table (6) indicated that the ILOEED fitted the data well with small values of the Goodness-of-fit statistics. These are the K-S, C-vM, and A-D.

Table 6: The Goodness-of-fit statistics of the ILOEED and others for the Breaking Strengths of Carbon

Fibres Dataset

Distributions K-S C-vM A-D

ILOEED 0.0565 0.033 0.2298

IEOLED 0.085 0.0884 0.5055

APED 0.1353 0.2826 1.6089

ED 0.3581 2.871 14.0343

Figure 3: The TTT-Plot for the Carbon Fibre Dataset

The Total Time on Test (TTT) plot for the Carbon Fibre Dataset indicates an increasing hazard rate (concave shape), as seen in Figure (3).

4

è о ф

£ сч

о о

ILOEED IEOLED APED ED

data

Figure 4: The Fitted PDFs and CDFs of the ILOEED and others for the Carbon Fibre Dataset

Figure 5: The Fitted PDFs and CDFs of the ILOEED and others for the Carbon Fibre Dataset

Figures (4) and (5) indicates that the Carbon Fibre data is symmetrical. ILOEED fitted the data better compared with the other comparators.

II. Application to the Strengths Dataset

The following dataset reported by [25], is the Strengths reading in GPa of individual carbon fibres that were put to stress at 20mm gauges and the values are:

1.359, 1.382, 1.382, 1.426, 1.434, 1.435, 1.478, 1.490, 1.511, 1.514, 1.535, 1.554, 1.566, 1.570, 1.586, 1.629, 1.633, 1.642, 1.648, 1.684, 1.697, 1.726, 1.770, 1.773, 1.800, 1.809, 1.818, 1.821,1.848, 1.880, 1.954, 2.012, 2.067, 2.084, 2.090, 2.096, 2.128, 2.233, 2.433, 2.585, 2.585.

Table 7: MLEs and Goodness-of-fit Criteria for the fitted ILOEED and other comparators for the comparators

for the Strengths Dataset

Distributions

Estimates Standard Error

-ll

AIC

BIC

ILOEED (v,0, y,X) 0.7639 1.2163 86.0082 180.0163 188.775

3.1315 4.4416

0.4570 0.3044

1943.9353 5787.6972

IEOLED (v,0,y,X) 0.1017 0.0311 88.2736 184.5472 193.3058

3.3417 0.2939

786928.5 12792.55

APED (u,0) 6.4E+10 1.79E+04

59738.7189 17013.25 95.0118 194.0236 198.4029

ED (u) 1.0525 0.0556

0.5642 0.0881 96.0231 195.0986 197.0178

Table (7) presents the MLEs, log-likelihoods, AICs, and BICs of the ILOEED and others. The

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam BT.. N 4sm THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D , , ° ,( j DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_V0lume I9, P^^i; 2024

Table indicates that ILOEED is the best with minimum values of AIC and BIC. Furthermore,

Table (8) indicated that the ILOEED fitted the data well with small values of the Goodness-of-fit

statistics.

Table 8: The Goodness-of-fit statistics of the ILOEED and others for the Strengths Dataset

Distributions K-S C-vM A-D

ILOEED 0.0581 0.046 0.2382

IEOLED 0.0876 0.0949 0.6043

APED 0.1461 0.3327 1.9048

ED 0.3447 2.7639 13.5301

Figure 6: The TTT-Plot for the Strengths Dataset

The Total Time on Test (TTT) plot for the Strengths Dataset indicates an increasing hazard rate

(concave shape), as seen in Figure (6).

Empirical and theoretical CDFs

iL Û

u

00 0

t

o

o o

1.4

1.6

1.8

2.0

2.2

ILOEED

IEOLED

y APED

ED

2.4

2.6

data

Figure 7: The Fitted CDFs of the ILOEED and others for the Strengths Dataset

Figure 8: The Fitted PDFs of the ILOEED and others for the Strengths Dataset

Figures (7) and (8) indicates that the Strengths data is symmetry. ILOEED fitted the data well compared with the other comparators.

III. Application to the Times to Failure Dataset

This dataset has a Bathtub-shape hazard rate as reported by [26]. The data is about the Times to Failure of 50 Devices that were on life test at time 0. The dataset is as follows:

0.1, 0.2, 1, 1, 1, 1, 1, 2, 3 , 6, 7, 1, 1, 12, 18, 18, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63,

67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85, 85, 86, 86.

Table 9: MLEs and Goodness-of-fit Criteria for the fitted ILOEED and other comparators for the comparators

for the Times to Failure Dataset

Distributions Estimates Standard Error -ll AIC BIC

ILOEED (v,0, y,X) 0.1025 0.0478 234.3902 476.7803 484.5076

0.3363 0.2143

1.4327 1.7004

2.3317 4.4526

IEOLED (v,0,y,Ä) 0.1104 0.2196 243.9899 495.9799 503.7072

0.8589 0.1104

2968.220 5982.6273

APED (v,d) 490198 1.4053

1.9690 1.4053 244.2936 492.5873 496.4509

ED (v) 0.0259 0.0048

0.0224 0.0031 244.7001 491.4002 493.3321

Table (9) presents the MLEs, log-likelihoods, AICs, and BICs of the ILOEED and others. The Table indicates that ILOEED is the best with minimum values of AIC and BIC. Furthermore, Table (10) indicated that the ILOEED fitted the data well with small values of the Goodness-of-fit statistics.

RT&A, No 4(80) Volume 19, December, 2024

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL

DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_

Table 10: The Goodness-of-fit statistics of the ILOEED and others for the tomes to Failure

Dataset

Distributions K-S C-vM A-D

ILOEED 0.1503 0.2817 1.9053

IEOLED 0.2012 0.5934 3.8944

APED 0.1799 0.5331 4.7647

ED 0.1904 0.5727 4.5609

Figure 9: The TTT-Plot for the Times to Failures Dataset

CO c

<D □

(D O

O O

The Total Time on Test (TTT) plot for the Fatigue Fracture Dataset indicates a decreasing and then increasing hazard rate (Bath-tub shape), as seen in Figure (9). CM

ILOEED IEOLED APED ED

20

40

60

80

data

Figure 11: The Fitted PDFs of the ILOEED and others for the Times to Failure Dataset

Figure 11: The Fitted CDFs of the ILOEED and others for the Times to Failure Dataset

Figures (10) and (11) indicates that the Times to Failure data is skewed to the right. ILOEED fitted the data well compared with the other comparators.

VII. Conclusion

In this research, we suggest and investigate a novel probability distribution that is a combination of the Inverse Lomax and exponential distributions, combining the properties of the two. This merger is required if the data in issue combines both the Inverse Lomax and the exponential distributions' features described in Section 1. We looked into some of its statistical properties, such as moments, the moment generating function, the characteristic function, and the quantile function. The parameters were determined using the maximum likelihood technique. According to the simulation studies, as the sample size grows, the estimations of the Biases and RMSEs approach zero, indicating that the estimates are more accurate. Three cases of parameter combination were considered for the simulation studies. The estimates were stable. Exemplifications of real-world datasets demonstrate the ILOEED's significance. For the three datasets used, the proposed distribution is the best with minimum values of the Goodness-of-fit criteria and Goodness-of-fit statistics. This means ILOEED can be used to fit datasets with increasing and bath-tube hazard rates. Based on these facts, we hope that the ILOEED will be preferred above the other models considered in this study. Only datasets from the industry were considered to fit the proposed distribution. We suggest that other areas should be explored in terms of the application of the proposed distribution. Also, other methods of estimation can be considered in further studies.

References

[1] Falgore, J. Y. and Doguwa, S. I. (2020). Inverse lomax-exponentiated-g (il-eg) family of distributions: properties and applications, Asian Journal of Probability and Statistics, 9 (4):48-64.

[2] Hassan, A. S., Almetwally, E. M., Gamoura, S. C., and Metwally, A. S. (2022). Inverse exponentiated lomax power series distribution: Model, estimation, and application, Journal of Mathematics, 1-21.

[3] Dhungana, G. P. and Kumar, V. (2022). Exponentiated odd lomax exponential distribution with application to covid-19 death cases of nepal, PloS one, 17 (6) e0269450.

[4] Chaudhary, A. K., Telee, L. B. S., and Kumar, V. (2022). Inverse exponentiated odd lomax

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam BT.. N 4sm

THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D ' , ° j

DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_V°lume I9, Pecember, 2U24

exponential distribution: Properties and applications, Nepalese Journal of Statistics, 6:29-50.

[5] Al-Marzouki, S., Jamal, F., Chesneau, C., Elgarhy, M. (2021). Half logistic inverse lomax distribution with applications, Symmetry, 13 (2): 309.

[6] Ogunde, A. A., Chukwu, A. U., and Oseghale, I. O. (2023). The kumaraswamy generalized inverse lomax distribution and applications to reliability and survival data, Scientific African 19: e01483.

[7] Sindhu, T. N. and Atangana, A. (2021). Reliability analysis incorporating exponentiated inverse weibull distribution and inverse power law, Quality and Reliability Engineering International, 37 (6): 2399-2422.

[8] Sindhu, T. N., Shafiq, A., and Al-Mdallal, Q. M. (2021). Exponentiated transformation of gumbel type-ii distribution for modeling covid-19 data, Alexandria Engineering Journal 60 (1): 671689.

[9] Metwally, A. S. M., Hassan, A. S., Almetwally, E. M., Kibria, B. G., and Almongy, H. M. (2021). Reliability analysis of the new exponential inverted topp-leone distribution with applications, Entropy, 23 (12):1662.

[10] Sindhu, T. N., Shafiq, A., and Al-Mdallal, Q. M. (2021). On the analysis of number of deaths due to covid- 19 outbreak data using a new class of distributions, Results in Physics 21: 103747.

[11] Shafiq, A., Sindhu, T. N., and Alotaibi, N. (2022). A novel extended model with versatile shaped failure rate: Statistical inference with covid-19 applica-tions, Results in Physics, 36: 105398.

[12] Shrahili, M., Hassan, A. S., Almetwally, E. M., Ghorbal, A. B., and Elbatal, I. (2022). Alpha power moment exponential model with applications to biomedical science, Scientific Programming, 2022(1):6897405.

[13] Hassan, A. S., Mohamed, R. E., Kharazmi, O., and Nagy, H. F. (2022). A new four parameter extended exponential distribution with statistical properties and applications, Pakistan Journal of Statistics and Operation Research, 18(1):179-193.

[14] Sindhu, T. N., Colak, A. B., Lone, S. A., and Shafiq, A. (2023). Reliability study of generalized exponential distribution based on inverse power law using artificial neural network with bayesian regularization, Quality and Reliability Engineering International 39 (6): 2398-2421.

[15] Abdelwahab, M. M., Ghorbal, A. B., Hassan, A. S., Elgarhy, M., Almetwally, E. M., and Hashem, A. F. (2023). Classical and bayesian inference for the kavya-manoharan generalized exponential distribution under generalized progressively hybrid censored data, Symmetry 15 (6):1193.

[16] Alghamdi, S. M., Shrahili, M., Hassan, A. S., Gemeay, A. M., Elbatal, I., and Elgarhy, M. (2023). Statistical inference of the half logistic modified kies exponential model with modeling to engineering data, Symmetry, 15 (3):586.

[17] Sindhu, T. N., Shafiq, A., and Huassian, Z. (2024). Generalized exponentiated unit gompertz distribution for modeling arthritic pain relief times data: classical approach to statistical inference, Journal of Biopharmaceutical Statistics, 34 (3):323-348.

[18] Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions, Metron, 71 (1):63-79.

[19] Team, R. C. (2013). R: A language and environment for statistical computing. r foundation for statistical computing, (No Title).

[20] Akaike, H. (1974). A new look at the statistical model identification, IEEE transactions on automatic control, 19 (6):716-723.

[21] Schwarz, G. (1978). Estimating the dimension of a model, The annals of statistics, 461-464.

[22] Delignette-Muller, M. L. and Dutang, C. (2015). fitdistrplus: An r package for fitting distributions, Journal of statistical software 64:1-34.

[23] Yousof, H. M., Alizadeh, M., Jahanshahi, S., Ramires, T. G., Ghosh, I., and Hamedani, G. (2017). The transmuted topp-leone g family of distributions: theory, characterizations and

J. Y. Falgore, Y. Abubakar, S. I. Doguwa, A. S. Mohammed, A. T. Imam 4 8 THE INVERSE LOMAX ODD-EXPONENTIATED EXPONENTIAL D , b° 4; DISTRIBUTION WITH INDUSTRIAL APPLICATIONS_Volume 19, December, 2024

applications, Journal of Data Science, 15 (4):723-740.

[24] Mahdavi, A. and Kundu, D. (2017). A new method for generating distributions with an application to exponential distribution, Communications in Statistics-Theory and Methods 46 (13):6543-6557.

[25] Nichols, M. D. and Padgett, W. (2006). A bootstrap control chart for weibull percentiles, Quality and reliability engineering international, 22 (2):141-151.

[26] Aarset, M. V. (1987). How to identify a bathtub hazard rate, IEEE Transactions on Reliability, 36 (1):106-108.