The problem of skewness is common among clinical trials and survival data , which has be en the research focus derivation and proposition of different flexible distributions. Thus, a new distribution called Extended Rayleigh Lomax distribution is constructed from Rayleigh Lomax distribution to capture the excessiveness of some survival data. We derive the new distribution by using beta logit function proposed by Jones (2004). Some statistical properties of the distribution such as density, cumulative density, reliability rate, hazard rate, reverse hazard rate, moment generating and likelihood functions; skewness, kurtosis and coefficient of variation are obtained. We also performed the expected estimation of model parameters by maximum likelihood; goodness of fit and model selection criteria , including Anderson Darling, CramerVon Misses, Kolmogorov Smirnov (KS), Akaike Information, Bayesian Information , and Consistent Akaike Information Criterion is employed to select the better distribution from those models considered in the work. The results from the statistics criteria show that the intend ed distribution perform s well and has a good representation of the States in Nigeria ’s Covid-19 death cases data than other competing models.
In survival analysis, problems are encountered in the analysis of clinical data because distributions proposed are not flexible enough to follow the movement of the data to give accurate results. In light of this, there is a need to develop a more flexible parametric model using Covid-19 data for example. In recent times, there was the outbreak of the third wave of Covid-19 pandemic called Delta Variant after the second wave generating a global outcry. Many researches/works have been done by several researchers since the breakup of the pandemic in December 2019 from various fields, such as Medicine, Statistics, Economics, etc., with different ideas, models, methods and approaches in their respective works. These include Badmus et al. (2020), Dey et al. (2020), WHO, (2020), Yoo, (2020) amongst others [
Zheng and Bobasera [
The main goal of the extensions of the proposed model is to enhance the robustness and flexibility of the classical model. Also, motivating factors, including the model, are a family of beta generalized models in such that when equating one or more parameter(s) to one (1), the model becomes the baseline distribution. In addition, the model has capability to increase and decrease shapes due to heavily skewed and heavy tail probability distribution that plays an important role in modeling skewed data like clinical data.
Most clinical data are always skewed, thus a new distribution is constructed and generated from a parent distribution called Rayleigh Lomax (RL) distribution by Kawsar et al. [
The paper is arranged in the following order. Section 2 contains material and methods which involve the properties of the proposed Extended Rayleigh Lomax distribution. In Section 3, we obtain the moments and generating function. Section 4 includes an estimation of parameters, analysis of secondary data obtained from Covid-19 weekly report in Nigeria, results and discussion. Section 5 includes based on the findings.
There are several methods in literature which have been used by many researchers. In this study, we consider beta logit function introduced by Jones [
Density Function
The density function of the above distribution is obtained using the beta link function given as:
g ( z , p , q ) = c ( z ) [ C ( z ) ] p − 1 [ 1 − C ( z ) ] q − 1 B ( p , q ) (1)
z , p , q , θ , λ , β ≃ E R L ( z , p , q , θ , λ , β > 0 )
C ( z ) = 1 − e − β 2 ( θ θ + z ) − 2 λ
and
c ( z ) = β λ θ ( θ θ + z ) − 2 λ + 1 e − β 2 ( θ θ + z ) − 2 λ x ≥ − θ and θ , λ , β > 0
where, C(z) and c(z) are cdf and pdf of the parent distribution respectively, a and b are additional shape parameters to the parent distribution.
g ( z , p , q , θ , λ , β ) = 1 B ( p , q ) [ C ( z ) ] p − 1 ( 1 − C ( z ) ) q − 1 c ( z ) < z , p , q , θ , λ , β > 0 (2)
where, θ, λ and β are initial scale and shape parameters, while a and b are new parameters called shape, introduced to the distribution. Then (2) becomes extended Rayleigh Lomax (ERL) Distribution and it plots are shown in
Some new and existing distributions from the extended Rayleigh Lomax distribution are mentioned below:
1) When p = 1 in (2), we have the Lehmann Type II Rayleigh Lomax Distribution (New)
2) If q = 1 in (2), we get the pdf of exponentiated Rayleigh Lomax distribution (New)
3) When β = 1 in (2), this consists of beta Lomax distribution (New)
4) When β θ = β , λ = 1 and λ = 1 in (2), it then becomes beta Rayleigh distribution (New)
5) If p = q = 1 in (2), it yields Rayleigh Lomax distribution which is the parent distribution [
6) When q = β = 1 in (2), this consists exponential Lomax distribution by El-Bassiouny et al. [
7) If p = q = λ = θ = 1 in (2), it yields Rayleigh Lomax distribution which is the parent distribution [
The associative distribution function cdf in (2) is given as
P ( Z ≤ z ) = ∫ 0 z f ( c ) d c = G ( z , p , q , θ , λ , β ) = ∫ 0 z 1 B ( p , q ) [ C ( z ) ] p − 1 ( 1 − C ( z ) ) q − 1 c ( z ) d c (3)
We set
k ( x ) = 1 − e − β 2 ( θ θ + x ) − 2 λ (4)
d c d z = β λ θ ( θ θ + z ) − 2 λ + 1 e − β 2 ( θ θ + z ) − 2 λ [ 1 − e − β 2 ( θ θ + z ) − 2 λ ]
d c d z = β λ θ ( θ θ + z ) − 2 λ + 1 e − β 2 ( θ θ + z ) − 2 λ − [ e − β 2 ( θ θ + z ) − 2 λ ] 2
Also
d z = d c β λ θ ( θ θ + z ) − 2 λ + 1 e − β 2 ( θ θ + z ) − 2 λ [ 1 − e − β 2 ( θ θ + z ) − 2 λ ]
Putting dx in Equation (2), we realize:
g ( z , p , q , θ , λ , β ) = 1 B ( p , q ) ∫ 0 ∞ [ 1 − e − β 2 ( θ θ + z ) − 2 λ ] p − 1 [ e − β 2 ( θ θ + z ) − 2 λ ] q − 1 d c (5)
and c in Equation (4) becomes
c ( z ) = d c ( z ) d z = β λ θ ( θ θ + z ) − 2 λ + 1 e − β 2 ( θ θ + z ) − 2 λ
Equation (5) can be expressed as
g ( z , p , q , θ , λ , β ) = 1 B ( p , q ) C a − 1 ( 1 − C ) b − 1 d c d z (6)
Now, putting (2) in (6), we get
G ( z , p , q , θ , λ , β ) = P ( Z ≤ z ) = ∫ 0 z 1 B ( p , q ) C p − 1 ( 1 − C ) q − 1 d c d z = ∫ 0 z 1 B ( p , q ) C p − 1 ( 1 − C ) q − 1 d c d z
where, B ( z , p , q ) = ∫ 0 z C p − 1 ( 1 − C ) q − 1 d c and it is called the incomplete beta function.
G ( z , p , q , θ , λ , β ) = ∫ 0 z C p − 1 ( 1 − C ) q − 1 d c B ( p , q ) = B ( c , p , q ) B ( p , q ) (7)
(7) becomes the cumulative distribution function of ERL distribution.
The reliability function of BRL distribution is given by
S u r ( z , p , q , θ , λ , β ) = 1 − G ( z , p , q , θ , λ , β ) = 1 − ∫ 0 z f ( c ) d c
1 − ∫ 0 z C p − 1 ( 1 − C ) q − 1 d c B ( p , q ) = 1 − B ( c , p , q ) B ( p , q )
S u r ( z , p , q , θ , λ , β ) = B [ ( p , q ) − ( z , p , q ) ] B ( p , q ) (8)
h a z ( z , p , q , θ , λ , β ) = g ( z , p , q , θ , λ , β ) 1 − G ( z , p , q , θ , λ , β ) = C p − 1 ( 1 − C ) q − 1 c B ( p , q ) B [ ( p , q ) − ( z , p , q ) ] B ( p , q ) = C p − 1 ( 1 − C ) q − 1 c B [ ( p , q ) − ( z , p , q ) ] (9)
R h a z ( z , p , q , θ , λ , β ) = g ( z , p , q , θ , λ , β ) G ( z , p , q , θ , λ , β ) = C p − 1 ( 1 − C ) q − 1 c B ( c , p , q ) B ( p , q ) = B ( p , q ) C p − 1 ( 1 − C ) q − 1 c B ( c , p , q ) (10)
The ERL distribution is a probability density function with the use of:
∫ 0 ∞ g E R L ( z ) d z = 1 (11)
Jones (2004) in his generalized beta distribution of first kind is given by:
g z ( z , p , q , u ) = [ B ( p , q ) ] − u [ C ( y ) ] p u − 1 [ 1 − C ( y ) u ] q − 1 K ( y ) 0 < y < 1
where p , q and u > 0 , therefore differentiating (p) above, we obtain
g E R L D ( z ) = [ B ( p , q ) ] − u G p u − 1 ( 1 − G u ) q − 1 d G d y
∫ − ∞ ∞ g E R L D ( z ) = ∫ − ∞ ∞ u B ( p , q ) G p u − 1 ( 1 − G u ) q − 1 d G
Putting M = G u , then differentiating M with respect to G
d M d G = U G u − 1
d G = d M U G u − 1
G = M 1 u
∫ − ∞ ∞ g E R L D ( z ) d z = ∫ 0 1 u B ( p , q ) ( M 1 u ) p u − 1 ( 1 − M ) q − 1 d M U G u − 1 = [ B ( p , q ) ] − 1 ∫ 0 1 M p − 1 u ( 1 − M ) q − 1 M 1 − 1 u d M = [ B ( p , q ) ] − 1 ∫ 0 1 M p − 1 ( 1 − M ) q − 1 d M
∫ 0 1 M p − 1 ( 1 − M ) q − 1 d M = B ( p , q )
Therefore, g E R L D ( z ) = B ( p , q ) B ( p , q ) = 1 .
Hence, the g E R L distribution has a true continuous probability density function.
In this section, we derive and obtain mgf of the distribution m ( t ) = E ( e t z ) and the general rth moment of a beta generated distribution defined by Hosking [
μ r 1 = B ( p , q ) − 1 ∫ 0 1 [ F − 1 ( z ) ] r z p − 1 [ 1 − z ] q − 1 d z (12)
Cordeiro et al. [
m ( t ) = B ( p , q ) − 1 ∑ j = 0 ∞ ( − 1 ) j ( q − 1 j ) e ( a , r ; p j − 1 ) (13)
where,
e ( a , r ) = ∫ − ∞ ∞ e t z [ F ( z ) ] m f ( z ) d z
then,
M z ( t ) = B ( p , p ) − 1 ∑ j = 0 ∞ ( − 1 ) j ( q − 1 j ) ∫ − ∞ ∞ e t z [ F ( z ) ] p ( j + 1 ) − 1 f ( z ) d z (14)
Putting pdf and cdf of the Extended Rayleigh Lomax distribution into Equation (14), we get
M E R L D ( z ) ( t ) = B ( p , q ) − 1 ∑ j = 0 n ( − 1 ) j ( q − 1 j ) ∫ − ∞ ∞ e t z [ F ( z ) ] p ( j + 1 ) − 1 f ( z ) (15)
If p = q = 1 in Equation (14) that becomes the moment generating function of the baseline distribution.
Hence, the rth moment of the ERL distribution is obtained, since the moment generating function of the parent distribution is given by
M z ( t ) = ∑ j = 0 ∞ t j j ! ∫ 0 ∞ z j f ( z , β , λ , θ ) d z ∑ c = 0 ∞ t k k ! ⋅ ∑ j = 0 ∞ ( q K j ) θ j ( 2 β ) 1 2 λ ( − θ ) q − j r ( j 2 λ + 1 ) (16)
Equation (16) can be re written as
M E R L D ( z ) ( t ) = B ( p , q ) − 1 ∑ j = 0 n ( − 1 ) j ( q − 1 j ) ∫ − ∞ ∞ e t z [ F ( z ) ] p ( j + 1 ) − 1 ∑ i = 0 ∞ t i i ! ⋅ ∑ h = 0 ∞ ( i K h ) θ h ( 2 β ) h 2 λ ( − θ ) i − h r ( h 2 λ + 1 ) = B ( p , q ) − 1 ∑ i = 0 ∞ ∑ j = 0 ∞ ∑ h = 0 ∞ ( − 1 ) j ( q − 1 i ) t i i ! ( i K h ) θ h ( 2 β ) h 2 λ ⋅ ( − θ ) i − h r ( h 2 λ + 1 ) ⋅ [ F ( z ) ] p ( j + 1 ) − 1 (17)
and the rth moment of ERL distribution is obtained from Equation (17)
μ E R L D ( r ) i = E ( z r ) = B ( p , q ) − 1 ∑ j = 0 ∞ ∑ h = 0 ∞ ( − 1 ) j ( q − 1 i ) ⋅ [ F ( z ) ] p ( j + 1 ) − 1 ⋅ t i r ! ( r h ) θ h ( 2 β ) h 2 λ ( − θ ) r − h r ( h 2 λ + 1 ) (18)
Letting p = q = 1 in (18) gives the rth moment of the baseline distribution by Kawsar et al. [
μ r i = E ( z r ) = ∑ h = 0 ∞ ( r h ) θ h ( 2 β ) h 2 λ ( − θ ) r − h r ( h 2 λ + 1 )
Other measures such as the Skewness (SKERLD) (p, q, β, λ, θ) and Kurtosis (KTERLD) (p, q, β, λ, θ) are also obtained below:
The rth moment of the ERL distribution is written as:
μ E R L D ( r ) i = ∫ 0 ∞ z r F E R L D ( z ) d z
That is,
μ E R L D ( r ) i = ∫ 0 ∞ z r ( [ B ( p , q ) ] − 1 ( C ( z ) ) p − 1 ( 1 − C ( z ) ) q − 1 d c ( z ) )
where, C ( z ) = ( 1 − w ( z ) ) − u
I.e. W ( z ) = e − β 2 ( θ θ + z ) and u = 2 λ
therefore,
μ E R L D ( r ) i = ( r K h ) θ h ( 2 β ) h u ( − θ ) r − h r ( h u + 1 ) B ( a , b ) ⋅ ∑ j = 0 ∞ ∑ h = 0 ∞ ( − 1 ) j ( q − 1 i ) ( [ ( 1 − w ( z ) ) − u ] p ( j + 1 ) − 1 ) = X ( ( r h ) θ h ( 2 β ) h u ( − θ ) r − h r ( h u + 1 ) ) (19)
where,
X = ∑ j = 0 ∞ ∑ h = 0 ∞ ( − 1 ) j ( q − 1 i ) ( [ ( 1 − w ( z ) ) − u ] p ( j + 1 ) − 1 ) B ( p , q )
At the same time, the first four central moments μ r i = 1 , 2 , 3 , 4 are obtained through (17) as:
Furthermore, the mean and second to fourth moments of the ERL distribution are given as follows:
μ = μ 1 | , μ 2 = μ 2 | − μ 2 , μ 3 = μ 3 | − 3 μ μ 2 | + 2 μ 3 , and μ 4 = μ 4 | − 4 μ μ 3 | + 6 μ 2 μ 2 | − 3 μ 4
μ 1 | = X ( ( 1 h ) θ h ( 2 β ) h u ( − θ ) 1 − h r ( h u + 1 ) ) (20)
μ 2 | = X ( ( 2 h ) θ h ( 2 β ) h u ( − θ ) 2 − h r ( h u + 1 ) ) (21)
μ 3 | = X ( ( 3 h ) θ h ( 2 β ) h u ( − θ ) 3 − h r ( h u + 1 ) ) (22)
μ 4 | = X ( ( 4 h ) θ h ( 2 β ) h u ( − θ ) 4 − h r ( h u + 1 ) ) (23)
Other measures such as skewness, kurtosis and coefficient of variation of the ERL distribution are given below.
The skewness is a means of measuring non symmetry of the distribution. The skewness is given by:
S K E R L D = μ 3 μ 2 1.5 (24)
The kurtosis is another measure that measures the peak of the distribution. The kurtosis of the BRL distribution is given as:
K T E R L D = μ 4 μ 2 2 − 3 (25)
This is also a measure of variability of a probability distribution. The CV of the ERL distribution is given as:
C V E R L D = μ 2 μ (26)
We made attempt to derive the maximum likelihood estimates (MLEs) of the ERL distribution parameters including: θ, λ, β, a and b which are scale and shape parameters. According to Cordeiro et al. [
L ( φ ) = n log [ B ( p , q ) ] + ∑ i = 1 n log [ C ( z , δ ) ] + ( p − 1 ) ∑ i = 1 n log [ 1 + C ( z , δ ) ] + ( p − 1 ) [ C ( z , δ ) ] (27)
φ = ( p , q , k , δ ) and δ = ( θ , λ , β ) are vectors
If c = 1, it becomes Equation (27) which leads to
L ( φ ) = c o n s t − n log [ B ( p , q ) ] + ∑ i = 1 n log [ C ( z , δ ) ] + ( p − 1 ) ∑ i = 1 n log [ 1 + C ( z , δ ) ] + ( p − 1 ) [ C ( z , δ ) ] (28)
c ( z , δ ) and C ( z , δ ) have been stated at the beginning.
The log likelihood function of ERL distribution is given as:
L B R L D ( φ ) = − n log [ B ( p , q ) ] + ∑ i = 1 n [ c ( z ) ] + ( p − 1 ) ∑ j = 1 n log [ C ( z ) ] + ( q − 1 ) ∑ i = 1 n log [ 1 − C ( z ) ] (29)
Taking the differentiation in respect to p, q, θ, λ and β give the following:
∂ L ( φ ) ∂ p = − n r | ( p ) r ( p ) + n r | ( p + q ) r ( p + q ) + ∑ z = 1 n log ( 1 − C ( Z , δ ) ) (30)
∂ L ( φ ) ∂ q = − n r | ( q ) r ( q ) + n r | ( p + q ) r ( p + q ) + ∑ z = 1 n log ( C ( Z , δ ) ) (31)
∂ L ( φ ) ∂ θ = ∑ j = 1 n [ ∂ [ C ( Z , δ ) ] ∂ θ C ( Z , δ ) ] + ( p − 1 ) ∑ z = 1 n [ ∂ [ 1 − C ( Z , δ ) ] ∂ θ 1 − C ( Z , δ ) ] + ( q − 1 ) ∑ z = 1 n [ ∂ [ C ( Z , δ ) ] ∂ θ ( Z , δ ) ] (32)
∂ L ( φ ) ∂ λ = ∑ j = 1 n [ ∂ [ C ( Z , δ ) ] ∂ λ C ( Z , δ ) ] + ( p − 1 ) ∑ z = 1 n [ ∂ [ 1 − C ( Z , δ ) ] ∂ λ 1 − C ( Z , δ ) ] + ( q − 1 ) ∑ z = 1 n [ ∂ [ C ( Z , δ ) ] ∂ λ ( Z , δ ) ] (33)
∂ L ( φ ) ∂ β = ∑ j = 1 n [ ∂ [ C ( Z , δ ) ] ∂ β C ( Z , δ ) ] + ( p − 1 ) ∑ z = 1 n [ ∂ [ 1 − C ( Z , δ ) ] ∂ β 1 − C ( Z , δ ) ] + ( q − 1 ) ∑ z = 1 n [ ∂ [ C ( Z , δ ) ] ∂ β ( Z , δ ) ] (34)
The data used for the analysis is a secondary data obtained from Covid-19 situation weekly epidemiological report 39; 5th-11th July, 2021 (NCDC website state the website) [
The summary of goodness of fit statistics is used to check for normality of the data; skewness, kurtosis, Anderson Darling (AD), Kolmogorov Smirnov (KS) and Cramer-Von-Mises (CVM) shown in
function (ecdf) plot show the data is skewed. For instance, non-linearity by scatter and quantiles plots, outliers by boxplot and skewness by histogram and density plots. The minimum and maximum values in the data set are inclusive.
The results obtained in
| Mini | Maxi | Skewness | Kurt | AD | KS | CVM |
|---|---|---|---|---|---|---|
| 2.00 | 456.00 | 3.55 | 14.39 | 3.883e−12 | < 2.2e−16 | 3.591e−09 |
| Model-Par/Model Sel | ERLD* | ExpLD | LRLD | BRD | RLD | ExpRLD | BLD |
|---|---|---|---|---|---|---|---|
| a | 0.801 (0.015) | 2.203 (0.044) | - | 1.801 (0.038) | - | 1.802 (0.040) | 0.801 (0.015) |
| b | 5.500 (0.132) | - | 2.500 (0.056) | 0.999 (0.019) | - | - | 2.501 (0.057) |
| θ | 0.025 (NA) | 0.002 (NA) | 0.008 (NA) | 0.048 (NA) | 0.000 (NA) | 0.016 (NA) | 0.010 (NA) |
| λ | 0.113 (0.003) | 0.109 (0.002) | 0.117 (0.003) | - | 0.116 (0.003) | 0.110 (0.002) | 0.117 (0.003) |
| β | 0.501 (0.014) | - | 0.500 (0.014) | 2.500 (0.063) | 2.900 (0.078) | 1.500 (0.037) | - |
| −2LogL | 1298.939 | 1629.850 | 2003.402 | 2201.424 | 2391.883 | 2498.683 | 2775.822 |
| AIC | 2607.878 | 3265.700 | 4014.804 | 4410.848 | 4789.766 | 5005.366 | 5559.644 |
| CAIC | 2609.813 | 3266.427 | 4016.054 | 4412.098 | 4790.493 | 5006.616 | 5560.916 |
| HQIC | 2610.718 | 3267.404 | 4017.076 | 4413.120 | 4791.470 | 5007.638 | 5561.916 |
| BIC | 2615.933 | 3270.533 | 4021.248 | 4417.292 | 4794.599 | 5011.810 | 5566.088 |
bracket for all the models. The model ERLD is compared with other six models: ExpLD, LRLD, BRD, RLD, ExpRLD and BLD. Also, model selection criterion is performed on all models considered in the study. From the results, ERLD has the smallest values in all as we can see bold and starred where AIC = 2607.878, CAIC = 2609.813, HQIC = 2610.718 and BIC = 2615.933, which indicates that it is a robust and flexible model.
Despite the level of Nigerian Covid-19 death cases data set, the ERL distribution follows the movement of the data and has better representation of the data than any of the other existing distributions. The proposed distribution, being flexible and versatile, can accommodate increasing, decreasing, bathtub and unimodal shape hazard function. It is therefore useful and effective in the analysis of clinical and survival data.
The authors declare no conflicts of interest regarding the publication of this paper.
Badmus, N.I., Faweya, O. and Ige, S.A. (2023) Parametric Modeling Approach to Covid-19 Pandemic Data. Open Journal of Statistics, 13, 61-73. https://doi.org/10.4236/ojs.2023.131005