https://orcid.org/0000-0001-8938-9187
Figures
Abstract
The Gull Alpha Power Lomax distribution is a new extension of the Lomax distribution that we developed in this paper (GAPL). The proposed distribution’s appropriateness stems from its usefulness to model both monotonic and non-monotonic hazard rate functions, which are widely used in reliability engineering and survival analysis. In addition to their special cases, many statistical features were determined. The maximum likelihood method is used to estimate the model’s unknown parameters. Furthermore, the proposed distribution’s usefulness is demonstrated using two medical data sets dealing with COVID-19 patients’ mortality rates, as well as extensive simulated data applied to assess the performance of the estimators of the proposed distribution.
Citation: Tolba AH, Muse AH, Fayomi A, Baaqeel HM, Almetwally EM (2023) The Gull Alpha Power Lomax distributions: Properties, simulation, and applications to modeling COVID-19 mortality rates. PLoS ONE 18(9): e0283308. https://doi.org/10.1371/journal.pone.0283308
Editor: Joydeep Bhattacharya, Iowa State University, UNITED STATES
Received: November 20, 2022; Accepted: March 6, 2023; Published: September 7, 2023
Copyright: © 2023 Tolba et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Researchers have been contributing to the theory of probability in recent years in order to overcome some of the limitations of statistical models. The Exponential distribution, for example, cannot handle data characterized by hazard rates that are monotonic or non-monotonic; it can only describe an object’s constant hazard rate. The gamma distribution has the shortcoming that it can only handle data with an increasing failure rate. However, real-life data is characterized by a non-monotonic hazard rate function. In distribution theory, efforts will always be made to generalize distributions. The essence of generalizing distributions is to obtain more robust and flexible models that have a wide range of applications. To achieve this, many methods are applied, as revealed by numerous pieces of literature. Also, the analysis and empirical results obtained greatly depend on how appropriately the chosen distribution fits the data under consideration.
Modifying current probability models to handle hazard rates that are both monotonic and non-monotonic, as well as provide an acceptable fit, is common practice. The new family of distributions was developed by [1] by using the Logit function and studying the Gumbel-Weibull distribution. [2, 3] studied the gamma-X family of distributions and the normal distribution as special cases of the model. For more reading on the developed distributions and their applications in reliability analysis, medical research, and the biological situation, there may be more than one cause of failure competing for the event. The event can be either death or recovery from a certain disease (risk), as referred to [4] introduced a competing risk model with lifetime Weibull sub-distributions, [5] studied statistical inference to the parameter of the Akshaya distribution under competing risk data with the application of HIV infection to AIDS; and [6] discussed statistical analysis of a regression competing risk model with covariates using Weibull sub-distributions. Also, [7] discussed the analysis of the Thymic Lymphoma of Mice application and estimation for the Akshaya failure model with competing risks, and [8] presented the conclusions for the stress-strength reliability model with a partially accelerated life test for its strength variable. More applications for developed distributions have been discussed by [9–25].
The goal of creating a new distribution family is to create a new statistical model in order to solve some of the problems with existing probability distributions. Not only will the proposed distribution handle different types of hazards, but it will also increase flexibility and produce a better fit than alternative probability models. In the literature, there are distributions. In this study, the Gull Alpha power family of distributions is proposed as a novel family of distributions. distribution. The Lomax distribution is used to derive the specific case of this family. Gull Alpha Power Lomax Distribution is also known as Gull Alpha Power Lomax Distribution (GAPL). The GAPL distribution is a modified version of the Lomax distribution that can be used to simulate non-monotonic hazard rate shapes. The hazard function, survival function, and moments of the distribution have been derived. Application to real data sets to demonstrate the versatility of the proposed model is done.
A new family of distribution called the Gull Alpha Power Family (GAPF) was developed by [26]. The CDF and the PDF of the family are given as follows:
(1)
where α is the shape parameter. The PDF of the family is:
(2)
This article is arranged as follows: In Section 2, we present and describe the Gull Alpha power Lomax distribution (GAPL), and its mathematical characteristics are presented in Section 3. Section 4 gives detailed estimation methods such as maximum likelihood, confidence intervals, bootstrap-p, and bootstrap-t for the unknown parameters. Bayesian analysis is discussed in Section 5. The numerical computations were performed to assess the behavior of estimates in Section 6. Also, In Section 7, we present two Applications to COVID-19 data sets. Finally, concluding remarks are mentioned in Section 8.
2 Gull Alpha Power Lomax Distribution (GAPL)
The CDF of the Lomax distribution is used to explain the specific form of GAPF in this section. The [27], also known as Pareto II, distribution has been frequently used in a variety of situations. [28] discussed moments of dual generalized order statistics and characterization for the transmuted exponential model. [29] obtained order statistics of inverse Pareto distribution. The Lomax distribution has been used for reliability modeling and life testing (e.g., [30]), and applied to income and wealth distribution data ([31, 32]), firm size ([33]), and queuing problems ([31, 32]). It’s also been used in the biological sciences, and it’s even been used to estimate the distribution of file sizes on servers ([34]). When the data is heavy-tailed, some authors, such as [35], have advised using this distribution instead of the exponential distribution. The Lomax distribution can be motivated in a number of ways. For example, [36]) show that it arises as the limit distribution of residual lifetime at a great age, and [37] studied the relates of the Lomax distribution to the Burr family of distributions. On the other hand, the Lomax distribution has been used as the basis for several generalizations. For example, [38] extend it by introducing an additional parameter using the [39] approach; [40] use the Lomax distribution as a mixing distribution for the Poisson parameter and derive a discrete Poisson-Lomax distribution, and [41] introduced the double-Lomax distribution and applied it to IQ data. The record statistics of the Lomax distribution have been studied by [42] the implications of various forms of right-truncation and right-censoring are discussed by [3, 43] and others; and sample size estimation has been discussed by [44].
The CDF of the Lomax distribution [44] is given by
(3)
and the probability density function
(4)
where θ and λ are the shape and scale parameters, respectively.
The CDF and PDF of the GAPL distribution are given, respectively:
(5)
and the probability density function
(6)
where θ, α and λ are greater than zero.
The GAPL distribution is characterized by three parameters α, θ, λ. The PDF graphical representations for a different set of parameter values are given in Fig 1.
2.1 Hazard and survival functions
The hazard and survival functions for the GAPL distribution are defined in this section.
(7)
and
where θ, α and λ > 0.
The hazard rate plot is displayed in Fig 2.
3 Statistical properties
In this section, some important statistical properties have been discussed.
3.1 Quantile function
The importance of the quantile function is to get quantiles and assist in the simulation study. The quantile function is given as:
(8)
(9)
For the median, put u = 0.5 in Eq (9). Table 1 gives the quantiles for specified parameter values.
3.3 Order statistics
For an ordered random sample X1, X2, ………, Xn from the GAPL distribution the PDF of the ith minimum and maximum order statistic is given by
(11)
and
(12)
3.5 Renyi entropy
The Renyi entropy of the GAPL distribution is given as:
(15)
From Eq (6), the Renyi entropy RH(x) becomes as
(16)
3.6 Skewness and kurtosis
The Moors Kurtosis and the Galton Skewness of the GAPL distribution are defined as:
(17)
and
(18)
Table 2 gives the values of the Skewness (Sk) and Kurtosis (Kt).
4 Parameter estimation
In this section, estimation methods have been obtained for the parameters of the GAPL distribution.
4.1 Maximum likelihood estimation
Because the probability model’s parameters are unknown, they must be estimated using data gathered from a sample. For a more in-depth look into maximum likelihood estimate, see here [45–47]. The conventional method of maximum likelihood estimates is utilized to determine the parameter estimates in this section. The Likelihood function of the GAPL distribution is given as:
(19)
Substituting from (6) in the Eq (19) expression, we get
(20)
by taking the log function on both sides, we get
Define the binomial expansion
then
(21)
To obtain the estimates of the parameters, the partial derivatives with respect to α, λ, θ are obtained and the results equated to zero.
(22)
(23)
and
(24)
Eqs (22)–(24) are not in closed form. To obtain the solution, numerical methods are proposed.
4.2 Bootstrap confidence intervals
The last section demonstrated how difficult it is to derive second-order derivatives in order to generate ACIs for the unknown model parameters. So, we take bootstrapping into account. In particular, we use the percentile bootstrap (Boot-p) and bootstrap-t (Boot-t) approaches (Tibshirani [48]) and bootstrap-t (Boot-t) (see Hall [49]) respectively.
4.3 Parametric Boot-p CI
Here, we’ll go over the formula for getting confidence intervals using the Boot-p approach. Initially, we get the MLEs of Θ = (α, λ, θ), by solving Eq 20. Also, denoted them by then, the bootstrap sample
has to be generated. We compute
based on x*. Repeat this procedure for Nboot times to get
where
, i = 1, 2, ⋯, Nboot. Next, we arrange
in ascending order and denote them by
. Thus, the (1 − δ)100% approximate bootstrap-p confidence interval for Θ is obtained as (L, U), where
and
.
4.4 Parametric Boot-t CI
Under a small sample size, the Boot-p approach does not perform well; for further information. Because the Boot-t approach is easier to use than the Boot-p method, we will examine it in this subsection. We get , similar to the procedure as mentioned in Boot-p method. Then, based on the bootstrap sample
, we compute the variance-covariance matrix
. For i = 1, 2, ⋯, Nboot, calculate the value of the statistic
. Then, we arrange them in ascending order and get
. Thus, the (1 − δ)100% approximate bootstrap-t confidence interval for Θ is obtained as (L, U), where
and
.
5 Bayesian estimation method
In this section, Bayesian inference was used to estimate the GAPL distribution parameters using an informative prior in order to achieve the correct posterior distributions. For more information and examples of the Bayesian estimation method, see [23, 50–55].
5.1 The model parameter priors
In informative priors, it’s noteworthy to notice that when the three GAPL distribution parameters are unknown, a joint conjugate prior to the parameters does not exist. As a result, we investigate Bayesian inference using independent gamma priors for d and q, as well as the subsequent combined prior distribution:
(25)
The hyper-parameters qi, wi, i = 1, 2, 3 are chosen to reflect prior information about the parameters of the GAPL distribution α, λ and θ, and they should be well-known and positive.
5.2 Posterior distribution
The likelihood function Eq (26) as follows:
(26)
and the joint prior function Eq (25) can express the joint posterior distribution. Consequently, Θ joint posterior density function is
(27)
The posterior density normalization constant , which in practice frequently requires an integral over the parameter space, is typically intractable as follows:
5.3 Loss functions
The squared-error loss function, which is denoted by SELF, is the symmetric loss function. The average is then the Bayesian estimator of Θ under SELF.
(28)
The two most well-known asymmetric loss functions are the LINEX and the entropy loss functions. Varian [56] introduced an extremely helpful asymmetric loss function, which has recently been used in several publications by [52, 57, 58].
The shape of this loss function, where c ≠ 0, depends on the value of c. When the LINEX loss function is used, the Bayes estimator of Θ is
(29)
According to Calabria and Pulcini [59], the entropy loss function is a decent asymmetric loss function.
The entropy loss function’s Bayesian estimation for the constant Θ is
(30)
As expected, the conditional distributions of α, λ, and θ cannot be analytically reduced to any standard distribution comparable to Bayesian inference, in this case, using the loss function approach. As a result, we recommend using the MCMC simulation technique to approximate the Bayesian estimates of α, λ, and θ.
5.4 Markov chain Monte Carlo
The MCMC method will be used since the expectation of loss functions are challenging to answer analytically by mathematical integration. The most important sub-classes of MCMC algorithms are Gibbs sampling and the more general Metropolis-within-Gibbs samplers. This algorithm was discussed in Robert et al. [60]. The Metropolis-Hastings (MH) algorithm, like acceptance-rejection sampling, treats a candidate value generated from a proposal distribution as normal for each iteration of the process. Starting at , the MH method computes an appropriate transition in two steps:
(1) Draw π(Θ*|Θ) from a proposal density while Θ* is a constant.
(2) You can either stick with the current sample Θi+1 = Θ or switch to Θi+1 = Θ*. with acceptance likelihood.
This well-stated transition density ensures that the chain converges to its particular invariant density starting from any initial condition, in addition to ensuring that the target density remains invariant.
6 Simulation studies and results
In this section, the simulation studies and results have been shown.
6.1 Monte Carlo simulation
The average bias, root mean square error, and mean of the parameter estimates were assessed through a simulation study. Different sample sizes and different sets of parameter values were used in the simulation study. The Average Bias (AB) and the Root Mean Squared Error (RMSE) were calculated using the equations below:
(31)
where ϕ is a vector of parameters (λ, α, θ) and
(32)
Figs 3–5 display the Means of the parameter estimates, the RMSE and AB for the MLEs (λ, α, θ) = (0.7, 0.3, 0.8) for increasing sample sizes. As observed, the parameter values tend to be the true value as the sample size increases. For the RMSE and AB, they decrease as the sample size increases.
Tables 3–5 discussed different cases as Case I is α = 1.2, θ = 1.5 with λ is changed as 0.7, 1.5, and 3, Case II is α = 1.2, θ = 3 with λ is changed as 0.7, 1.5, and 3, and Case III is λ = 1.5, θ = 0.7 with α is changed as 0.7, 1.5, and 3. The sample size is changed to 50, 100, and 200. These tables obtained AB, MSE, CI with length as length asymptotic CI (LACI), length credible CI (LCCI), length asymptotic CI (LACI), length bootstrap-p for MLE (LBP.MLE), length bootstrap-t for MLE (LBt.MLE), length bootstrap-p for Bayes (LBP.Bayes), and length bootstrap-t for Bayes (LBt.Bayes).
7 Applications to COVID-19 data set
The data will be applied to illustrate the flexibility and importance of the GAPL distribution with its sub-model (Lomax distribution) and other competing model (power Lomax) and Generalized exponential distributions. The estimation of the unknown parameters will be obtained by the ML method. The values of the models of the statistics log-likelihood, Akaike Information Criterion(AIC), Bayesian Information Criterion (BIC), and Consistent Information Criterion (CAIC) are used to compare the candidate distributions. In general the smaller the values the appropriate the distributions to fit the data. The Mathematical formulas of the criterion are:
where
denotes the log-likelihood function evaluated at the maximum likelihood estimation, h is the number of parameters and n is the sample size. Here we let ϕ denote (λ, θ, α).
The proposed distribution is compared to the following distributions:
- Exponential Lomax distribution [61] with CDF given as
- Lomax distribution [27] with CDF
- The Power Lomax [62] with CDF given as
7.1 Data set I: China COVID-19 survival times
The survival rates of patients affected by the COVID-19 pandemic in China are discussed in this subsection. The data set under consideration shows how long patients lived after being admitted to the hospital until they passed away. A group of fifty-three (53) COVID-19 sufferers were among them. From January to February 2020, they were discovered in hospitals in critical condition [63]. The descriptive statistics for the data are displayed in Table 6. The data is right skewed because of the positive sign of the skewness coefficient.
The data has a modified bathtub failure rate as depicted in the TTT plot in Fig 6.
The MLEs of the parameters of the proposed distribution GAPL and its sub-models are presented in Table 7.
The GAPL distribution provides a better fit than the competing distributions. As indicated in Table 8, the GAPL distribution has the highest log-likelihood and the smallest values of K-S and W* compared to the other models. Considering the formal tests of goodness of fit tests, in order to verify which distributions better fit the china daily COVID-19 cases data, since the GAPL distribution has the lowest values for the K-S, Anderson-Darling, and W* we then conclude that the distribution provides a better fit than the competing distributions.
The plots of the densities of the fitted distributions are shown in Fig 7.
Table 9 discussed Bayesian estimation for parameters of GAPL for China COVID-19 daily cases. By comparing Bayesian and MLE in Table 7, we note that the Bayesian estimation has the smallest SE for parameters. Fig 8 presents the PP and QQ plots of GAPL distribution for China’s COVID-19 daily cases. Fig 9 shows MCMC plots of GAPL parameters for China COVID-19 daily cases.
7.2 Data set II: Netherlands COVID-19 mortality rates
In this subsection, the data set under consideration shows how long patients lived after being admitted to the hospital until they passed away. This data is available at this link (https://ourworldindata.org/coronavirus/country/netherlands).
The descriptive statistics for the data are displayed in Table 10. The data is right skewed because of the positive sign of the skewness coefficient.
The data has an increasing failure rate as depicted in the TTT plot in Fig 10.
The maximum likelihood estimates of the parameters of the proposed distribution GAPL and its sub-models are presented in Table 11.
The GAPL distribution provides a better fit than its competing distributions. As indicated in Table 12, the GAPL distribution has the highest log-likelihood and the smallest values of K-S and W* compared to the other models. Considering the formal tests of goodness of fit tests, in order to verify which distributions better fit the jet airplane data, since the GAPL distribution has the lowest values for the K-S, Anderson-Darling, and W* we then conclude that the distribution provides a better fit than the sub-models. The plots of the densities of the fitted distributions are shown in Fig 11.
Fig 12 discussed the contour plot of GAPL distribution for the Netherlands COVID-19 mortality dates data. Also, Fig 13 shows the PP and QQ plots of GAPL distribution for the Netherlands COVID-19 mortality dates data. Table 13 discussed Bayesian estimation for parameters of GAPL for Netherlands COVID-19 mortality dates data. By comparing Bayesian and MLE in Table 11, we note that the Bayesian estimation has the smallest SE for parameters. Fig 14 shows MCMC plots of GAPL parameters for the Netherlands COVID-19 mortality dates data.
8 Conclusion
There will always be attempts to generalize distributions in distribution theory. The goal of generalizing distributions is to create more reliable, adaptable models with a variety of applications. Many techniques are used to do this, as numerous pieces of literature have shown. Additionally, the analysis and empirical findings heavily depend on how well the chosen distribution fits the input data. In this article, We suggested the Gull Alpha Power Lomax distribution as a new generalization of the Lomax distribution. The quantile and moments, among other statistical properties of this distribution, have been derived and discussed. The greatest likelihood estimators have been calculated. We performed a simulation study to compare these methods. We have compared estimators with respect to bias and mean-squared error. The simulation results revealed that the Bayesian method is a very competitive method among others. Compared with other distributions based on AIC, BIC, CAIC, HQIC, LL, KS, and probability values, the GAPL distribution proved stronger than the competing distributions. The new distribution is applied to two actual data sets that are related to the daily number of COVID-19 cases in China. The Lomax distribution, exponential Lomax, and power Lomax distributions are compared. We find that the proposed model is extremely competitive in terms of fitting this real data set based on the comparison criterion between all of these models.
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