A Study on Alpha Power Lomax Distribution
Rana A. Bakoban ^{1} , E. A. Farag ^{1,2}, Najwa S. Alsulami ^{1}
^{1}^{ }Department of Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabi
^{2} Faculty of Science, Mathematics Development, Helwan University, Egypt

ABSTRACT 

In this paper,
we refer to the new distribution an alpha power Lomax distribution. Various
properties of the proposed distribution are obtained including mode,
quantiles, entropies, and order statistics are obtained. Parameters of the
proposed distribution are estimated using maximum likelihood, ordinary least
squares and weighted least squares. Simulation study is conducted to compare
between estimators. 

Received 15 September 2022 Accepted 16 October 2022 Published 31 October 2022 Corresponding Author Rana A. Bakoban, rabakoban@uj.edu.sa
DOI 10.29121/IJOEST.v6.i5.2022.393 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or notforprofit sectors. Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CCBY, authors retain the copyright, allowing anyone to download,
reuse, reprint, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. 

Keywords: Alpha Power
Lomax Distribution, Maximum Likelihood Estimation, Renyi
Entropies, Stress–Strength Reliability and Order Statistics 
1. INTRODUCTION
Improvement over standard distributions has gained popularity in statistical theory over the past few years. Typically, new distributions are created by combining existing distributions or adding a new parameter using generators. Alpha power transformation (APT) a novel approach for integrating an additional parameter in continuous distribution, was introduced by Mahdavi and Kundu (2017). Basically, the concept was put forth to add skewness to the baseline distribution. Then: Let g(x) be the probability density function (PDF) of a continuous random variable X.:
Equation 1
The corresponding cumulative distribution function (CDF)
Equation 2
Where α is the shape parameter and G(x) and g(x) are the CDF and the PDF of the baseline distribution.
Mahdavi and Kundu (2017) developed a brandnew method for creating distributions by adding a second shape parameter to a wellknown baseline distribution. They used the exponential baseline distribution to introduce the alpha power exponential distribution, and they researched the fundamental characteristics as well as parameter estimation for the suggested distribution. Alpha power Weibull distribution (APWD) was introduced by Nassar et al. (2017). The capacity of APWD to simulate monotone and nonmonotone failure rate functions, which are often used in reliability studies, is what gives its significance. Moments, quantiles, entropy, order statistics, mean residual life function, and stressstrength were among the attributes of APWD that were discovered. The parameters were estimated using the maximum likelihood method. The significance of the APWD was demonstrated using of actual data sets. Devi et al. (2017) introduced the entropy of the Lomax probability distribution. They derived entropy of a Lomax probability distribution as well as their order statistics which is used in business, economics, actuarial modelling, waiting problems, and biological sciences.
This study’s objective is to provide a new more adaptable distribution. The structure of this article is as follows: We go over the distribution of Alpha power Lomax (APL) in Section 2. In Section 3, new statistical properties are investigated. The Renyi entropy is derived in Section 4. In Section 5, the stressstrength reliability is described. Section 6 presents the order statistics from the Alpha power Lomax. Section 7 examines estimate techniques. Section 8 provides a simulation study. In Section 9, application is offered.
2. ALPHA POWER LOMAX DISTRIBUTION
In this section, we will define a new model is called alpha power Lomax distribution, denoted (APL), with three parameters (α, λ, β).
According to ElHoussainy et al. (2016) cumulative’s distribution function of Lomax distribution with scale parameter λ > 0 is
Equation 3
where, β > 0 and λ > 0.
The probability density function corresponding to (3) reduces to
Equation 4
The CDF of APL is generated by renaming Equation 3 as a baseline cumulative function in Equation 2.
Equation 5
and corresponding PDF of APL is defined as follows
Equation 6
where λ is the scale parameter and α is the shape parameter.
The PDF curves for the APL distribution are shown in Figure 1, Figure 2, Figure 3 with various parameter values.
Figure 1
Figure 1 The APL’s PDF curves with λ = 2, β = 3 
Figure 2
Figure 2 The APL’s PDF curves with α = 3, λ = 2 
Figure 3
Figure 3 The APL’s PDF curves with α = 3, β = 2 
Figure 1, Figure 2, Figure 3 show that for various values of the parameters, the APLD density curve is uni modal and positively skewed.
2.1. The survival and the hazard functions
The following forms could be used to represent the survival and hazard rate functions for the APL distribution, respectively.
Equation 7
and
Equation
8
Figure 4, Figure 5 display the survival function of APLD and the hazard function for different values of parameters.
Figure 4
Figure 4 The APL’s survival function curves with λ = 2, β = 2 
Figure 5
Figure 5 The The APL’s hazard function curves
show with λ = 2 and 
3. Statistical Properties
3.1. Quantile function
Let u = F (x) where U follows uniform (0, 1). By using the transformation method, we consider the random variable X of APL as follows
Equation 9
3.1.1. THE MEDIAN
The median of the APL distribution can be obtained by putting u = 0.5 in Equation 9 as follows
Equation 10
3.2. The mode
The mode of the distribution can be found by solving the following equation
By taking the derivative of Equation 6 and equating it to zero and solving for x, mode becomes
Equation 11
This is equation is not linear. The NewtonRaphson method can be used to solve it numerically.
3.2.1. Skewness and kurtosis
Because the APL’s moments are not exist, we’ll apply an alternate forms using quartiles. Bowley skewness was considered as an alternate measure to determine asymmetry (Kenney and Keeping (1962)) of a distribution that takes the form
where Q represents the quantile function from Equation 9. Moors (1988) proposed a different method to determine the kurtosis of the distribution based on octiles, and it has the following form
Table 1
Table 1 The Mode, Median, Skewness and Kurtosis of APL Distribution for Different Values of the Parameters. 

α 
λ 
β 
Mode 
Median 
Skewness 
Kurtosis 
4.5 
0.5 
2 
0.000679 
0.373801 
0.341431 
1.58388 
8.5 
0.5 
2 
0.097225 
0.458852 
0.327832 
1.57484 
12.5 
0.5 
2 
0.148810 
0.512296 
0.321915 
1.57231 
4.5 
1 
2 
0.001358 
1.024560 
0.341431 
1.58388 
4.5 
3 
2 
0.004074 
3.07377 
0.341431 
1.58388 
4.5 
6 
2 
0.008149 
6.147550 
0.341431 
1.58388 
4.5 
0.5 
2.5 
0.014549 
0.281491 
0.309599 
1.50100 
4.5 
0.5 
3 
0.020491 
0.225435 
0.288054 
1.45078 
4.5 
0.5 
3.5 
0.022919 
0.187873 
0.272518 
1.41727 
Table 1 the following:
1) For constant values of λ and β, the median, mode increasing as α increases, while the kurtosis and skewness decrease.
2) For constant values of α and β, the median and mode increase while λ is increasing but the kurtosis and skewness remain stable.
3) For constant values of λ and α, the mode increasing as β increases, while the median, kurtosis, and skewness decrease.
4. Renyi Entropy of APL
Renyi entropy was invented by Renyi (1959). The variation of the uncertainty is measured by a random variable’s entropy X. Additionally, Song (2001) has effectively used the theory of entropy to a variety of applications, including information theory, engineering, and physics.
Theorem 1: The Renyi entropy of APL
Equation 12
Proof. For the density function f(x), the Renyi entropy is defined as:
Equation 13
Using Equation 6, we obtain
by substituting z = (1 + x) −β and using the series representation,
Equation 14
Then we have
Hence, the theorem is proved.
5. The StressStrength Reliability
In literature related to engineering, reliability, and bio statistics, the stressstrength model has attracted a lot of interest. Let X_{1} represents stress and X_{2} represents strength. The stressstrength model’s most basic version states that a system failure occurs when the stress exceeds the strength. Kotz and Pensky (2003). The stressstrength reliability for the APL is given by the ensuing theorem.
Theorem 5.1. The form of APL’s stressstrength reliability
Equation 15
Proof. The stressstrength reliability can be defined as
Equation 16
using the Equation 5 and Equation 6 of APL distribution, stress strength R, can be obtained as
by substituting z = (1 + x) −β and using the series representation
Equation 17
Hence, the theorem is proved.
6. Order Statistics of APL
Consider denotes the order statistic of a random sample of size n from APL distribution with CDF, F(x), and PDF, f(x), the PDF of Xr is given by Balakrishnan and Cohen (2014)
Equation 18
By using Equation 5, Equation 6 the PDF the rth order statistic from APLD is given by:
Equation 19
The PDF of the smallest order statistic, X_{1}, is as follows
Equation 20
And the PDF of the largest order statistic, X_{n}, is given by
Equation 21
7. METHODS OF ESTIMATION
In the section, we will study classical estimation methods to estimate unknown parameters of the APL distribution. Estimation methods that used are: (1) maximum likelihood method, (2) Ordinary least squares, (3) Weighted least squares.
7.1. MAXIMUM LIKELIHOOD ESTIMATION OF APL
Let X_{1}, X_{2}, ...X_{n} be a random sample of size n from the APL distribution. By using (6), we obtain the likelihood function L (x, α, λ, β) as
Equation 22
and the loglikelihood function is
Equation 23
Equation 24
Then by deriving the loglikelihood function with respect to α, λ and β we get
Equation 25
Equation 26
Equation 27
After solving Equation 25, Equation 26 and Equation 27 simultaneously using the NewtonRaphson method in Mathematica 12.0, the MLE of α, β and λ could befound.
7.2. Ordinary Least Squares Estimation of APL
Swain et al. (1988) First suggested the leastsquares estimators and
weighted leastsquares estimators. Assuming a random sample of size n from the APL distribution, with the observations
being in the following order: x1:n <
x2:n <
…. < xn:n we may obtain the ordinary least
squares (OLS) estimates α , λ and β as follows.
Equation 28
Then by deriving the S (α, λ, β) function with respect to α, λ, β, we get
Equation 29
Equation 30
Equation 31
After solving Equation 29, Equation 30, and Equation 31 simultaneously using the NewtonRaphson method in Mathematica 12.0, the OLS of α, β and λ could be obtained.
7.3. Weighted Last Squares Estimation of APL
Weighted least squares (WLS) estimates are represented using the following α, β and λ form
Equation 32
Then by deriving the W (α, λ, β) function with respect to α, λ, β, we get
Equation 33
Equation 34
Equation
35
After solving Equation 33, Equation 34 and Equation 35 simultaneously using the NewtonRaphson method in Mathematica 12.0, the WLS of, and could be obtained.
8. SIMULATION STUDY
To make sense of the theoretical findings, simulation using Mathematica 12.0 have been carried out, of the estimating issue presented in the earlier parts. Simulated research has been carried out for mean square error (MSE) and average MLEs.
The estimator computation algorithm is
Step 1: Create a random sample of size n using Equation (9) for the starting parameter values (α, β, λ).
Step 2: Use the NewtonRaphson method to solve the equations in to estimate the MLE, LSE and
WLSE of the parameter α to solve the equations given in (25), (29) and (33), respectively.
Step 3: Compute the estimator of entropy from (12), using the estimates in the previous step.
Step 4: 1000 times, repeat the first three steps.
Step 5: Calculate the MSE.
Different sample sizes of the ML estimate for α, entropy and MSE for the real parameter values are shown in Table 2. The LSE estimations for α, entropy and MSE for true parameter values are presented in Table 3. Additionally, the WLSE estimates for α, entropy, and MSE for the genuine parameter val ues are shown in Table 4
Table 2
Table 2 The MLE and MSE for the Parameters α. 

Parameters 
n 
Mean (α) 
MSE (α) 
MLE Entropy 
MSE Entropy 
α = 2 
30 
2.10996 
0.055305 
0.794999 
0.001414 
β = 4 
50 
2.10507 
0.050296 
0.794381 
0.001292 
λ = 3 
200 
2.09217 
0.049327 
0.792177 
0.001273 
ρ = 2 
500 
2.07664 
0.041060 
0.789856 
0.001079 

1000 
2.04933 
0.031904 
0.785526 
0.000860 
α = 3.5 
30 
3.49168 
0.081989 
0.901258 
0.000717 
β = 3 
50 
3.47555 
0.080602 
0.899818 
0.000716 
λ = 2 
200 
3.47064 
0.078377 
0.899322 
0.000695 
ρ = 2 
500 
3.47054 
0.074389 
0.893239 
0.000648 

1000 
3.44995 
0.065321 
0.893229 
0.000566 
α = 5.5 
30 
5.49820 
0.084452 
1.815170 
0.000363 
β = 3 
50 
5.49050 
0.083072 
1.814670 
0.000359 
λ = 2 
200 
5.48660 
0.082376 
1.814400 
0.000352 
ρ = 2 
500 
5.47801 
0.079926 
1.813870 
0.000345 

1000 
5.47643 
0.075352 
1.813790 
0.000326 
Table 3
Table 3 The LSE and MSE for the parameters α. 

Parameters 
n 
Mean (α) 
MSE (α) 
MLE Entropy 
MSE Entropy 
α = 2 
30 
2.167800 
0.059513 
0.804842 
0.001468 
β = 4 
50 
2.166970 
0.057549 
0.804195 
0.001427 
λ = 3 
200 
2.133420 
0.045056 
0.799589 
0.001126 
ρ = 2 
500 
2.087780 
0.029733 
0.792361 
0.000760 

1000 
2.008250 
0.010686 
0.779427 
0.000294 
α = 3.5 
30 
3.464680 
0.087933 
0.898712 
0.000789 
β = 3 
50 
3.453120 
0.083220 
0.897577 
0.000744 
λ = 2 
200 
3.385090 
0.082102 
0.891249 
0.000713 
ρ = 2 
500 
3.277550 
0.079814 
0.881001 
0.000697 

1000 
3.171800 
0.051935 
0.870616 
0.000321 
α = 5.5 
30 
5.482420 
0.084139 
0.885941 
0.000360 
β = 2 
50 
5.457720 
0.083646 
0.884451 
0.000351 
λ = 1 
200 
5.412630 
0.083646 
0.881737 
0.000321 
ρ = 2 
500 
5.307800 
0.073992 
0.874050 
0.000316 

1000 
5.192590 
0.073594 
0.868265 
0.000207 
Table 4
Table 4 The WLSE and MSE for the Parameters α 

Parameters 
n 
Mean (α) 
MSE (α) 
MLE Entropy 
MSE Entropy 
α = 2 
30 
2.176210 
0.062294 
0.806236 
0.001535 
β = 4 
50 
2.172040 
0.060366 
0.805593 
0.001495 
λ = 3 
200 
2.171150 
0.059024 
0.805502 
0.001487 
ρ = 2 
500 
2.167290 
0.059968 
0.804785 
0.001462 

1000 
2.164850 
0.056621 
0.804519 
0.001402 
α = 3.5 
30 
3.476270 
0.089437 
0.899785 
0.000800 
β = 3 
50 
3.463550 
0.087462 
0.898549 
0.000778 
λ = 2 
200 
3.46067 
0.084138 
0.898370 
0.000747 
ρ = 2 
500 
3.453780 
0.083912 
0.897605 
0.000754 

1000 
3.451360 
0.082047 
0.897472 
0.000734 
α = 5.5 
30 
5.483850 
0.086180 
0.886005 
0.000312 
β = 2 
50 
5.483810 
0.085978 
0,886004 
0.000311 
λ = 1 
200 
5.480090 
0.084704 
0.885813 
0.000308 
ρ = 2 
500 
5.474100 
0.082359 
0.885431 
0.000301 

1000 
5.458950 
0.081502 
0.884541 
0.000295 
From Table 2, Table 3, Table 4, we observed that the MSE of the estimate α and entropy decreases as the sample size increase. Also, the mean of the estimate α and the MLE of entropy decreases as the sample size increase.
9. Applications
In this section. The inverse Lomax distribution (ILD) introduced by Hassan and Mohamed (2019) and the exponential lomax distribution (ELD) introduced by Abdo et al. (2015) will be used to compare the fit of the APLD with other lifetime models that are wellknown. Additionally, we take into account the model selection parameters, such as the Akaike information criterion (AIC), loglikelihood (), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and HannanQuinn information criterion (HQIC). The lowest values of AIC, BIC, CAIC, HQIC and greatest value of, all point to a better fit of the data, Where
k is the number of parameters; n is the sample size and (θˆ) is the loglikelihood function evaluated at the highest likelihood estimates.
Data I: COVID19 cases every day in Saudi Arabia
From 1 April to 2 May, this information reveals the daily COVID19 instances in Saudi Arabia. These statistics were obtained from the Saudi Ministry of Health’s website, located at https://covid19.moh.gov.sa. These findings are
110, 157, 165, 154, 140, 206, 138, 272, 137, 355, 364, 382, 429, 472,435, 493, 518, 762, 1132, 1088, 1122,
1147, 1141, 1158, 1172, 1197,1223, 1289, 1266, 1325, 1351, 1344, 1362.
Table 5 displays descriptive statistics for this data.
Table 5
Table 5 Descriptive Statistics for the Number of Daily COVID19 Cases in Saudi Arabia 

Measure 
Value 
Measure 
Value 
N 
33 
Minimum 
110 
Maximum 
1362 
Mean 
727.455 
Q1 
255.5 
Q3 
1178.25 
Median 
518. 
Skewness 
0.0350687 
Kurtosis 
1.26721 
Variance 
230684.0 
Standard deviation 
480.295 

The performance of the APLD in comparison to other models is shown in Table 6 along with the MLEs for the model parameters.
Table 6
Table 6 MLEs of the Model Parameters and the Statistics of the AIC, BIC, CAIC, HQIC and for the Number of Daily COVID19 Cases in Saudi Arabia 

Distributions 
MLE α 
MLE entropy 

AIC 
BIC 
CAIC 
HQIC 
APLD 
34.7454 
2.73066 
243.261 
492.532 
497.012 
493.350 
494.033 
ELD 
0.04329 
0.39915 
425.628 
857.257 
861.746 
858.085 
858.768 
ILD 
182.650 
2.95628 
253.035 
512.070 
516.560 
512.898 
513.581 
Table 6 shows that, among the models taken into consideration, the APLD has the smallest AIC, CAIC, BIC, and HQIC values. Which shows that the APLD seems to be a model that might work well with the data set.
Data II: (Natural phenomena) Badr (2019) presented the consist of 30 successive March precipitation (in inches) observations.
77, 52, 90, 174, 162, 205, 81, 131, 120, 32, 195, 95, 120, 81, 47,281, 143, 187, 337, 118, 220, 135, 300,
475, 309, 248, 151, 96, 210,189.
Table 7 displays descriptive statistics for this data.
Table 7
Table 7 Descriptive Statistics for the of 30 Successive March Precipitation (in inches) 

Measure 
Value 
Measure 
Value 
N 
30 
Minimum 
32 
Maximum 
475 
Mean 
168.7 
Q1 
95. 
Q3 
210. 
Median 
147. 
Skewness 
1.12002 
Kurtosis 
4.30607 
Variance 
9786.15 
Standard deviation 
98.925 

The performance of the APLD in comparison to other models is shown in Table 8 along with the MLEs for the model parameters.
Table 8
Table 8 MLEs of the Model Parameters and the Statistics of the AIC, BIC, CAIC, HQIC and for the of 30 Successive March Precipitation (In Inches) 

Distributions 
MLE α 
MLE entropy 
,e 
AIC 
BIC 
CAIC 
HQIC 
APLD 
66.161 
2.83245 
200.811 
401.622 
412.201 
408.422 
409.183 
ELD 
0.0370132 
0.351725 
476.589 
959.178 
963.757 
959.978 
960.739 
ILD 
440.78 
3.040180 
296.266 
598.532 
603.111 
599.322 
600.093 
Table 8 is results show that while, e has the largest value and the APLD has the smallest AIC, CAIC, BIC, and HQIC values. As a result, the APLD performs better than other models.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
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