https://orcid.org/0009-0005-2777-6129
Figures
Abstract
This paper presents a novel extension of the exponentiated inverse Rayleigh distribution called the half-logistic exponentiated inverse Rayleigh distribution. This extension improves the flexibility of the distribution for modeling lifetime data for both monotonic and non-monotonic hazard rates. The statistical properties of the half-logistic exponentiated inverse Rayleigh distribution, such as the quantiles, moments, reliability, and hazard function, are examined. In particular, we provide several techniques to estimate the half-logistic exponentiated inverse Rayleigh distribution parameters: weighted least squares, Cramér-Von Mises, maximum likelihood, maximum product spacings and ordinary least squares methods. Moreover, numerical simulations were performed to evaluate these estimation methods for both small and large samples through Monte Carlo simulations, and the finding reveals that the maximum likelihood estimation was the best among all estimation methods since it comprises small mean square error compared to other estimation methods. We employ real-world lifetime data to demonstrate the performance of the newly generated distribution compared to other distributions through practical application. The results show that the half-logistic exponentiated inverse Rayleigh distribution performs better than alternative versions of the Rayleigh distributions.
Citation: Kamnge JS, Chacko M (2025) Half logistic exponentiated inverse Rayleigh distribution: Properties and application to life time data. PLoS ONE 20(1): e0310681. https://doi.org/10.1371/journal.pone.0310681
Editor: Srinivasa Rao Gadde, UDOM: The University of Dodoma, TANZANIA, UNITED REPUBLIC OF
Received: June 5, 2024; Accepted: September 4, 2024; Published: January 16, 2025
Copyright: © 2025 Kamnge, Chacko. 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 and its Supporting information files.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The practice of introducing additional parameters to create new distributions has become increasingly prevalent in statistical theory in recent decades [1]. By considering the relevance of probability theory and statistics, there is often a need for a generalised lifetime model that can effectively analyse data in the field of survival and reliability analysis [2]. Anwar and Bibi [3] derived a novel probability distribution by employing the generalized weibull distribution function on the half logistic family of distribution. Alzaatreh et al. [4] introduced T-X family of continuous distributions by interchanging the probability density function (pdf) of any continuous random variable with the pdf of beta distribution. Lee et al. [5] devised a method for producing continuous distributions with a single variable. Jones [6] employed the Beta distribution to analyse the set of distributions introduced by Eugene et al. [7].
In line with that, this study employed the half logistic transformation method to the exponentiated inverse Rayleigh (EIR) distribution as the base model since this distribution was derived from the Rayleigh distribution, which was used in the field of engineering, physics, and medical side in modeling the survival and reliability data. Additionally, this base model was proven to be flexible in the sense that when the shape of EIR is one, this distribution returns to inverse Rayleigh (IR), which was the base for EIR [8]. Furthermore, this base distribution for this current study, i.e., EIR, was found to be useful in the area of quality control to assess the quality of coating weights of iron sheets data [8]. Despite its usefulness, the EIR has the following limitation: it is not able to accurately model complex data structures especially modeling moderately right-skewed or near-symmetrical lifetime data, more specifically this distribution is able to model only non-monotonically hazard rate function. For instance, the Exponential and Weibull distributions are unable to accurately represent real data that follows a non-monotonic failure rate function [9]. Owning that, this paper aims to contribute to the generation of a new probability distribution that models the data with both monotonically and non-monotonically hazard rate functions since data usually come from the field of survival and reliability analysis, especially from engineering, geology, education, economics, and health have monotonic and non-monotonic behavior.
In the Recent past, Cordeiro et al. [10] suggested a new technique, called half-logistic transformation, by including an additional parameter in the life time model. The primary purpose of this family was to utilize the non-symmetrical behavior of the parent distribution. Let X is a continuous random variable with cumulative distribution function (cdf) F(x) then the cdf of type I half-logistic family of distributions is given by
(1)
While the pdf is given by
(2)
where g(x) and G(x) are the pdf and cdf of base distribution respectively.
The half-logistic transformation has been used by many researchers; for example, Anwar and Bibi [3] explored the new probability distributions by applying the generalized Weibull distribution to the half-logistic family of distributions. Moreover, using the same half-logistic transformation, Moakofi et al. [11] produced a half-logistic log-logistic Weibull distribution, and Dhungana et al. [12] produced half-logistic inverted Weibull distribution.
The main aim of this paper is to produce a new probability distribution by using the half logistic family of distributions. In this paper, we consider the exponentiated inverse Rayleigh distribution by Rao and Mbwambo [8] as the baseline distribution. A random variable X is said to follow an exponetiated inverse Rayleigh distribution if it possesses the following pdf and cdf respectively.
(3)
and
(4)
The distribution thus obtained is called half logistic exponentiated inverse Rayleigh distribution (HLEIRD). The new distribution is able to model data with both monotonically and non-monotonically hazard rate function. In addition, this article attempt to estimate the parameters of the model by applying different estimations methods, so this study aims to develop a guideline for choosing the best estimation method for the HLEIRD which would be of profound interest to applied statisticians. The choice of the methods of estimation varies among the users and area of applications. The present study is unique because so far no attempt has been made to develop half logistic exponentiated inverse Rayleigh distribution and to compare the aforementioned methods of estimation for parameters. The rest of the paper is organised as follows: In section 2, pdf, cdf, survival function, and hazard function of HLEIRD are introduced together with their graphs. In section 3>, statistical and mathematical properties such as quantile function, median, mode, moment generating function, moments, and order statistics are obtained. Furthermore, in section 4, different estimation methods such as maximum likelihood estimation, Maximum product spacing estimation, least squares estimation, cramér-von mises estimation, and weighted least squares estimation are considered. In section 5, simulation studies are performed to assess the efficiency of different methods for estimation. In section 6, a real data is applied to prove the flexibility and suitability of the model, and lastly, in section 7, which summarises the conclusion of the study.
2 Half logistic exponentiated inverse Rayleigh distribution
By applying the cdf of the exponentiated inverse Rayleigh distribution defined in (4) to the cdf defined in (1), we obtain the cdf for the HLEIRD and is given by
(5)
The pdf corresponding to the above cdf is given by
(6)
We have drawn the graphs of pdf and cdf of HLEIRD for the different parameter values and are given in Figs 1 and 2. Figs 1 and 2 demonstrate that the HLEIRD exhibits greater flexibility when dealing with various shapes, including symmetrical shapes as well as left and right-skewed shapes. This characteristic makes it suitable over wide range of lifetime data.
The survival function and hazard rate function, for HLEIRD are respectively given by:
(7)
and
(8)
We have also drawn the graphs of survival function and hazard rates function of HLEIRD for different parametric values and are given in Figs 3 and 4. The Figs 3 and 4 reveal that this family is capable of producing various shapes, such as reverse J curve, reserve S curve, increasing and decreasing curve. For more details see Table 1 below.
This demonstrates that the HLEIRD family has the potential to effectively accommodate data sets with a wide range of shapes. This implies that distribution is very powerful in modelling survival and reliability data.
3 Statistical and mathematical properties
In this section, we furnish some significant statistical and mathematical properties of the HLEIRD such as moments, moment generating function, ordered statistics, mode, quantiles, median, skewness, and kurtosis.
3.1 Some stochastic ordering results
The HLEIRD enjoys tractable stochastic ordering results involving the corresponding cdf. From a statistical point of view, such results allow a better comprehension of the roles of the parameters in the fitting ability of the HLEIRD model. The most notable of them are presented below.
Proposition 1. If F(x, φ) is the cdf of HLEIRD defined in (5), where φ = (λ, σ, α) then the following inequalities holds:
- 1) For any α1 ≥ α2 and λ, σ, x > 0, we have F(x, λ, α1, σ)≥F(x, λ, α2, σ)
- 2) For any λ1 ≥ λ2 and λ, σ, x > 0, we have F(x, λ1, α, σ)≥F(x, λ2, α, σ)
- 3) For any σ1 ≥ σ2 and λ, α, x > 0, we have F(x, λ, α, σ1)≥F(x, λ, α, σ2)
Proof. The proof is based on monotonic arguments with respect to the parameters. We have
This implies that F(x;φ) is strictly decreasing with respect to α. Therefore for any α1 ≥ α2 and λ, σ, x > 0, we have F(x, λ, α1, σ)≥F(x, λ, α2, σ). With the same methodology, we have
This implies that F(x;φ) is strictly decreasing with respect to λ. Therefore for any λ1 ≥ λ2 and λ, σ, x > 0, we have F(x, λ1, α, σ)≥F(x, λ2, α, σ). Similarly, we have
This implies that F(x;φ) is strictly decreasing with respect to σ. Therefore for any σ1 ≥ σ2 and λ, α, x > 0, we have F(x, λ, α, σ1)≥F(x, λ, α, σ2).
3.2 Linear representation
The following result introduces a useful linear representation for the exponentiated pdf of the HLEIRD with power parameter v > 0.
Proposition 2. Let v > 0. Then, f(x;φ)v can be expressed as the following series expansion:
where
and
Proof. From (6) the pdf of HLEIRD is given by
where φ = (λ, σ, α). Then
Then by using the generalized binomial series formula given by
where |Z| < 1, k > 0 and
represents the generalized binomial coefficient defined by
we get
(9)
After rearrangement, we get the desired result. Hence the proof.
By taking ν = 1 in (9), we get a useful series expansion for the pdf of the HLEIRD, in the sense that we express a sophisticated function as sums of tractable functions, i.e., gk(x, φ, ν). In particular, we will use it in the coming sections to provide measures and functions which are easy to handle from the analytical and numerical point of views.
3.3 Moments of HLEIRD
The rth raw moment of HLEIRD is given by
. By using the Proposition 2 with ν = 1, we obtain the following:
(10)
The mean of the HLEIRD can be obtained by putting r = 1 in (10) and is given by
From (10) it can be shown that HLEIRD has first moment (mean) only.
3.4 Moment generating function
The moment generating function of HLEIRD is given by . By using the Proposition 2 with ν = 1, we have the following
By using the expansion
we get the following
3.5 Entropy measures
The entropy of the HLEIRD can be measured in different ways. Here, we focus our attention on the Rényi entropy by [13] and the q-entropy by [14]. For discussions and applications of these two entropy measures, refer [15], and the references therein. The Rényi entropy of the HLEIRD given by, for δ ≠ 1
Therefore, by referring (11), one can express Iδ as:
Then, the q-entropy of the HLEIRD is defined by, for q ≠ 1
Therefore, by using (11), with a similar approach, we get
3.6 Order statistics
Moments of order statistics have great role in quality control testing and reliability to predict time to fail of a certain item by considering few early failures. Suppose X1:m < X2:m, ⋯, <Xm: m are ordered statistics of a random sample size m drawn from HLEIRD with cdf FX(x) and pdf fX(x) then the pdf of Xk: m is given by
(12)
Putting k = 1 in (12), we obtain pdf of the smallest order statistic as follows
The pdf of largest order statistic is obtained by putting k = m in (12) and is given by
3.7 Quantile and random number generation
Quantile are very needful for estimation purposes basically, quantile estimators and also it is used in simulation. The pth quantile of the HLEIRD is given by
(13)
Quantile can also be used in finding the skewness and kurtosis of the distribution.
The equation (13) can be used in simulation to generate random variable from HLEIRD. Given U∼Uniform (0,1) then the random variable X from HLEIRD is given by
3.9 Skewness and kurtosis using quantile approach
There are different methods which are used to find skewness and kurtosis in a certain distribution. The most common method is the one which use moments of the distribution but for HLEIRD we have first moment only. Due to this reason the appropriate approach of finding kurtosis and skewness is by using quantiles.
where Q1, Q2 and Q3 are obtained respectively by putting
,
and
in 13.
Kurtosis is known as the measure of dispersion of distribution. Moors (1988) suggested a robust alternative measure of kurtosis as follows:
where Ei is the ith octile given by
We have tabulated the value of skewness and kurtosis for different values of λ, α, and σ and given in Table 2. Similarly, we have drawn Figs 5–7 to show the changes in skewness and kurtosis for different values of λ, α, and σ. The findings show that when λ is constant increase in both α, and σ leads to a decrease in skewness and kurtosis. Likewise, for the case when α is constant, findings show that increases in both σ and λ leads to a decrease in skewness and kurtosis. Lastly, when σ is constant, the findings depict the same results as α and λ when they are constant, but in this case, it is found that the kurtosis is above 3 when α=0.5 and λ=0.5, which depicts the presence of leptokurtic. This result indicates that the kurtosis and skewness decrease when these parameters increase, and it reaches a point where the curve becomes a mesokurtic and platykurtic curve, as shown in Figs 5–7. Moreover, the finding depicts that the distribution is positively skewed (right-skewed) as it is evidenced by the presence of positive values for all skewness values calculated under different parametric values.
4 Methods of estimation
In this section, different methods for the estimation of parameters σ, α and λ of the HLEIRD are discussed. The various methods of estimation are maximum likelihood, ordinary and weighted least square, and percentile based estimation.
4.1 Maximum likelihood estimation
Let X1, X2, …, Xn be random sample of size n drawn from HLEIRD, then the likelihood function can be obtained as follows
(14)
The log likelihood is obtained as
(15)
Also
gives
(17)
and
gives
(18)
By solving (16), (17) and (18) all together, we get the estimates of σ, α and λ. We can get the solution of the above equations by using methods like Newton Raphson method or bisection method. Also, ML estimators follows asymptotically normally distribution, that is , Σ is the dispersion matrix and is given by Σ = I−1, where
(19)
(20)
(21)
(22)
(23)
(24)
Then the asymptotic (1-ξ)100% confidence interval for any parameters β is given by
where Σii is the diagonal element of matrix Σ and Zq is the qth quantile of a standard normal distribution.
4.2 Ordinary and weighted least-squares estimators
These least square methods were suggested by Swain et al. [16], for estimating parameters of Beta distribution. This study employed these estimation methods namely ordinary and weighted least-squares estimation method. Suppose X1:n, X2:n, ⋅⋅⋅, Xn: nare the order statistics of a random sample of size n from HLEIRD with pdf given in (6). The least square estimators (LSEs) ,
and
of the parameters σ, α and λ are obtained by minimizing E with respect to σ, α and λ respectively.
By solving the following equations we get the LSEs of σ, α and λ respectively.
(30)
Next we consider the weighted least square estimators (WLSE).
The weighted least square estimators can be obtained by minimizing Q in (31) with respect to unknown parameters σ, α and λ. Thus WLSE of σ, α and λ are respectively obtained by solving the following equations
(32)
4.3 Cramér–Von Mises estimation
The selection of Cramér–Von Mises-type minimum distance estimators was supported by empirical evidence presented by MacDonald [17], which indicated that the estimator’s bias is smaller than that of the other minimum distance estimators.
Cramér–Von Mises estimatiors of HLEIRD parameters are obtained by minimizing (33) with respect to σ, α and λ or by solving the following Eqs 34, 35 and 36 simultaneously.
(34)
(35)
(36)
where ϑ1(xi: n), ϑ2(xi: n) and ϑ3(xi: n) are defined in (27), (28) and (29) respectively.
4.4 Method of maximum product of spacings
The maximum product spacing (MPS) method was first introduced by Cheng and Amin [18, 19] as a substitute for the maximum likelihood method for the estimation of univariate continuous probability distributions. In addition to that, the MPS derived independently as an approximation to the Kullback-Leibler measure of information by Ranneby [20]. The method was proven efficient and consistent as MLE under more general conditions.
By refering the notation applied in section 4.2 the uniform spacings of a random sample from the HLEIRD distribution is defined as
where F(x0|σ, α, λ) = 0, F(xn + 1: n|σ, α, λ) = 1 and
.
The maximum product spacings estimates ,
and
of the parameters σ, α and λ are obtained by maximizing G(σ, α, λ) with respect to σ, α and λ respectively, where
(37)
(38)
Then the estimates are obtained maximizing H(σ, α, λ) with respect to σ, α and λ respectively are obtained by or by solving the following Eqs 39, 40 and 41 simultaneously.
(39)
(40)
(41)
where ϑ1(.), ϑ2(.) and ϑ3(.) were defined in (27), (28) and (29) respectively.
5 Simulation study
In this section, a Monte Carlo simulation study is conducted to evaluate the performance of diferent estimation methods described in the previous sections. The performance of the diferent estimators is evaluated in terms of mean square error (MSE). The simulation is conducted by using R-sofware, 10000 random samples of size n from HLEIRD was generated for n = (20, 40, 70, 100) and (σ, α,λ) = (0.5,0.5,1), (1,0.5,1), (0.5,1,1), (1,1,1), (0.5,0.5,2), (1,0.5,2), (0,5,1,2), and (1,1,2).
Average bias and MSE values of the estimates obtained by the method of MLE, LSE, WLSE, CVME and MPS are shown in Tables 2–7. All methods show that they have consistency property since the values of average bias and MSE decrease as the sample size increases. Based on the value of MSE, the method of MLE shows good performance since it has small value of MSE compared to other methods.
The findings in Table 8 show the different ranks of the estimation method of HLEIRD from the different parametric combinations. The result showed that the MLE method outperforms all other estimation methods with an overall score of 54, which is less compared to all other estimation methods. Therefore, as evidenced by the simulation study, the MLE method performs best for HLEIRD.
6 Application
In this section, we fit the proposed HLEIRD to real data sets and compare them with their sub-models such as exponentiated inverse Rayleigh distribution(EIRD), inverse Rayleigh distribution(IRD),Rayleigh distribution(RD) and inverse weibull distribution(IWD) distributions, respectively. The first data set represents the strength measured in GPA for single-carbon fibers and impregnated 1000-carbon fiber tows. The single fibers were tested under tension at a gauge of 20 mm. Data were initially reported by Badar and Priest (1982) and later applied by Kundu et al. [21].
The data is: 1.312,1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128, 3.233, 3.433, 3.585, 3.585.
The second data set was extracted from Dey et al. [22], and the data set represents the strength measured in GPA for single-carbon fibers and impregnated 1000-carbon fiber tows, but for this case, single fibers were tested under tension at a gauge of 10 mm.
The data is: 0.562, 0.564, 0.729, 0.802, 0.950, 1.053, 1.111, 1.115,1.194, 1.208, 1.216, 1.247, 1.256, 1.271, 1.277, 1.305, 1.313, 1.348, 1.390, 1.429, 1.474, 1.490, 1.503, 1.520, 1.522, 1.524, 1.551, 1.551, 1.609, 1.632, 1.632, 1.676, 1.684, 1.685, 1.728, 1.740, 1.761, 1.764, 1.785, 1.804, 1.816, 1.824, 1.836, 1.879, 1.883, 1.892, 1.898, 1.934, 1.947, 1.976, 2.020, 2.023, 2.050, 2.059, 2.068, 2.071, 2.098, 2.130, 2.204, 2.262, 2.317, 2.334, 2.340, 2.346, 2.378, 2.483, 2.683, 2.835, 2.835.
6.1 Model validity and selection criteria
The Akaike information criteria (AIC), Consistent Akaike information iriterion (CAIC), Hannan-Quinn information criterion (HQIC), and Bayesian information criteria (BIC) are employed to verify that this HLEIRD model is appropriate for the datasets that have been taken. In addition to that, K-S distance and P-value were employed to verify the suitability of the model over the other four probability distribution models employed.
The expression of AIC, CAIC, HQIC and BIC are given by
(42)
(43)
(44)
and
(45)
where
denotes the log-likelihood at MLEs, q is the number of parameters, and n is the sample size.
Tables 13 and 14 provide the goodness-of-fit measures, and their corresponding P-values for the fitted models of the two data sets employed in this study.
Table 9 show summary of the data set one which depict that the median is 2.478 which is higher than mean 2.451 this indicate presence of negative skewness. Additionally, this evidenced the skewness -0.0282. Also data is Platykurtic since the kurtosis is less than 3. Therefore, these findings provide a picture that the data is asymmetric in nature and it has a heavier tail to the left side.
Figs 8 and 9 give the Violin plot, Histogram, and Box plot for data set one, which show that there is a presence of left-skewness. This is evidenced by the Box plot and Violin plot. Also, if the Histogram plot covers more left part, indicates left-skewed presence. The TTT plot is concave, indicating a decreasing failure rate. Early failures are more frequent, and the rate of failures decreases over time. This is evidenced by a plot bends downward, indicating a decreasing failure rate. This suggests that the data set is appropriate for additional study since, as it was observed from the pdf plot, HLEIRD is capable of modeling asymmetric data or skewed data sets.
Table 10 shows summary of the data set one which depict that the median is 1.728 which is higher than mean 1.701 this the indicate presence of negative skewness. Additionally this evidenced the skewness -0.0285. Also data is Platykurtic since the kurtosis is less than 3. Therefore, these findings provide a picture that the data is asymmetric in nature and it has a heavier tail to the left side.
Figs 10 and 11 give the Violin plot, Histogram, and Box plot for data set one, which show that there is the presence of a left-skewed. This is evidenced by the Box plot and Violin plot. Also, if the Histogram plot covers more left part, indicates left-skewed presence. The TTT plot is concave, indicating a decreasing failure rate. Early failures are more frequent, and the rate of failures decreases over time. This is evidenced by a plot bends downward, indicating a decreasing failure rate. This suggests that the data set is appropriate for additional study since, as it was observed from the pdf plot, HLEIRD is capable of modeling asymmetric data or skewed data sets.
The values in Tables 11 and 12 provide a comparison of the new distribution HLEIRD with the existing four distributions: EIRD, IRD, RD, and IWD. The findings in Table 11 show that HLEIRD has a low AIC,HQIC compared to all probability distributions employed, while in the case of BIC and CAIC, the EIRD has a lower BIC and CAIC. Since AIC,HQIC are more powerful criterion for choosing a model, for the first data set, HLEIRD was selected as the best model compared to the other models. In line with that, for the case of data set two, from Table 12, the HLEIRD was selected to be the best model compared to the other four distributions, EIRD, IRD, and IWD, since the HLEIRD was found to have a lower BIC, CAIC, HQIC, and AIC.
6.2 Model goodness of fit
The results for Tables 13 and 14 show goodness of fit for five pdf employed in this study. Similarly, Figs 12 and 13 show the goodness of fit for all pdf employed in this study. The finding in Tables 13 and 14 show that the HLEIRD was fitting the data well compared to other pdf employed, since HLEIRD has the highest P-values for both data sets applied under all goodness fit of tests employed. Moreover, the graphs of PP plot, histogram, and theoretical densities and lastly, empirical and theoretical CDFs given in Figs 12 and 13 verify how well the used data set fits the HLEIRD compared with the other four distributions.
The Tables 15 and 16 give the estimated values of the parameters of HLEIRD using five different estimation methods: maximum likelihood, least square, weighted least square, maximum product spacings, and Cramer-von Mises estimation. The results show that under various estimation methods for data set one and data set two yields the same result as least square method of estimation fit well two both data sets employed. This is because of that the least squares estimation have the highest P-value in the case of the K-S test and the lowest likelihood value.
7 Conclusion
In this paper, a new three-parameter probability distribution was introduced by applying a half-logistic transformation to the exponential inverse Rayleigh distribution. It is called the half-logistic exponential inverse Rayleigh distribution. Some of its statistical and mathematical properties such as stochastic ordering results, linear representation of HLEIRD, raw moments, moment generating function, entropy measure, order statistics, skewness and kurtosis features, quantile, and median, were derived. To estimate the parameters of the proposed distribution, five estimation methods were discussed. Specifically, simulations were performed to evaluate the performance of these estimation methods for both small and large samples through Monte Carlo simulations study. It revealed that the maximum likelihood was the best estimation compared to other estimation methods. In addition, the performance of the proposed distribution was assessed by fitting it to two actual data sets. The findings revealed that the new distribution performs better than alternative versions of the Rayleigh distributions. The limitation of this study is that the model may not be suitable for a very small sample size as well as high peaked data. Furthermore, future studies can develop bivariate or multivariate versions of HLEIRD to study engineering and medical problems in different dimensions. Lastly, this study employed only the classical estimation method, so the future study can employ the Bayesian estimation method to estimate the parameters of this newly proposed model.
Acknowledgments
The authors would like to thank the Editor and the reviewers for their valuable comments which helped to improve the quality of this article.
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