A New Extension of the Kumaraswamy Exponential Model with Modeling of Food Chain Data

: Statistical models are useful in explaining and forecasting real-world occurrences. Various extended distributions have been widely employed for modeling data in a variety of ﬁelds throughout the last few decades. In this article we introduce a new extension of the Kumaraswamy exponential (KE) model called the Kavya–Manoharan KE ( KMKE ) distribution. Some statistical and computational features of the KMKE distribution including the quantile ( QUA ) function, moments ( MO m s), incomplete MO m s ( INMO m s), conditional MO m s ( COMO m s) and MO m generating functions are computed. Classical maximum likelihood and Bayesian estimation approaches are employed to estimate the parameters of the KMKE model. The simulation experiment examines the accuracy of the model parameters by employing Bayesian and maximum likelihood estimation methods. We utilize two real datasets related to food chain data in this work to demonstrate the importance and ﬂexibility of the proposed model. The new KMKE proposed distribution is very ﬂexible, more so than numerous well-known distributions.


Introduction
The so-called "food chain", a university topic taught in Germany since 1987, provides the foundation for the relatively new academic field that describes the intricate connections between food and environment [1]. It is included in environmental science, agricultural science or "agroecology" in other European nations [2]. However, the "grey" literature initially appeared in the 1970s [3], and one of the earliest scientific articles from the 1980s [4] was inspired by the 1972 United Nations environment summit in Stockholm. Since then, different food products have undergone refinement and application of environmental evaluation methods in order to assist producers and companies in improving food production from an environmental standpoint. Recently, many papers have discussed modeling of food chain data, such as [5][6][7][8][9][10].
The Kumaraswamy (K) model was first known as the double-bounded model. It was first mentioned by [11]. The cumulative distribution function (cdf) in this model is closed-form. The K-G family of distributions was presented by reference [12] as a novel distributions were discussed in [41][42][43][44][45][46]. Refs. [47,48] studied the Weibull model under a repetitive group sampling plan based on truncated tests and progressively censored group sampling, among others. Kavya and Manoharan [49] just presented a novel transformation, the KM transformation class of statistical models. The cdf and pdf are provided in the next two equations and f (z) = e e − 1 g(z)e −G(z) , z ∈ R.
Recently, [9] introduced the sine exponentiated Weibull exponential (SEWE) model to fit food data in the United Kingdom (UK) and the SEWE model was found to have an excellent fit for these data. However, in this article, we hope that the suggested model gives a better fit to the food data used by [9]. In addition, Figure 1 offers a comprehensive description of the work. The following considerations provide sufficient motivation to investigate the suggested model. It is stated as follows: • The new KMKE distribution gives more flexibility than the SEWE model and other well-known statistical models for food chain data as we prove in Section 7. • The new recommended distribution is quite versatile and comprises three sub-models. • The shapes of the pdf for the KMKE model can be decreasing, right skewness and uni-modal. However, the hazard rate function (hrf) for the KMKE model can be decreasing, increasing and j-shaped. • Numerous statistical and computational characteristics of the recently proposed model are investigated. • The parameters of the KMKE model are estimated utilizing maximum likelihood and Bayesian techniques.
The rest of this article is structured as follows: some relevant literature for some extensions of the K model and their modeling to real data are discussed in Section 2. We provide the novel proposed model designated the KMKE model and its sub-models in Section 3. Several statistical and computational features of the KMKE including the QU A function, MO m s, I N MO m s, COMO m s and MO m generating functions are computed in Section 4. The parameters of the KMKE model are estimated utilizing maximum likelihood and Bayesian techniques in Section 5. In Section 6, the numerical simulations used to evaluate the efficiency of the various estimation approaches are described. In Section 7, we apply the KMKE model to two real datasets to demonstrate its usefulness and applicability. Eventually, in Section 8, some final thoughts are offered.

Relevant Literature
Statistical models are very useful for describing and predicting real-world events. Various extended distributions have been extensively used for data modeling in a wide range of areas throughout the last few decades. Many authors have used Equation (1) to generate new extensions from the K model and used these statistical models in modeling for different real datasets, such as: engineering, physics, medicine, failure times, reliability, survival, income and COVID-19. Table 1 shows some relevant literature for some extensions of the K model and their modeling to real data. We note that all previous authors who studied extensions of the K model did not use their models to fit food chain data. However, in this article, we try to generate a new extension of the K model and hope to give a good fit to the food chain data.

Model Modeling Authors
The new suggested model (KMKE model) Food chain data New K-Weibull model Failure times data [50] K-generalized Rayleigh model Engineering data [51] K-modified Weibull model Failure times data [52] K-transmuted exponentiated modified Weibull model Medical data [53] K-transmuted modified Weibull model Failure times data [54] K-Gompertz Makeham model Physics data [55] K-Gumbel model Engineering data [56] K-generalized gamma model Industrial and medical data [57] K-generalized power Lomax model Physics data [58] K-Burr XII model Engineering, physics and medical data [59] K-generalized inverse Lomax model Reliability and survival data [60] K-Dagum model Income and lifetime data [61] Modified K model Engineering data [62] Transmuted K-Lindley model Medical data [63] K-Marshall-Olkin exponential model Medical data [64] K-half logistic model Physics and medical data [65] K-log logistic model Medical data [66] K-Marshall-Olkin log-logistic model Physics data [67] Modified K Weibull model Reliability and engineering data [68] K-inverted Topp-Leone model COVID-19 data [69] Kavya-Manoharan-K model COVID-19 and physics data [70] Transmuted K model Medical and environmental data [71]

Model Modeling Authors
Generalized inverted K-G Physics data [72] Topp-Leone generalized inverted K model Physics data [73] K log-logistic Weibull model Failure times data [74] Exponentiated inverse K model Economic data [75] Beta K Burr Type X model Physics and medical data [76] Marshall-Olkin extended inverted K model Physics, failure and medical data [77] K generalized Kappa model Geological data [78] Cubic rank transmuted K model Food and industrial data [79] K Marshall-Olkin log-logistic model Physics data [67] Odd generalized exponential K model Geological and environmental data [80] K exponentiated U-quadratic model Medical data [81] K odd Burr-G Physics and engineering data [82] Exponentiated generalized K model Environmental, agriculture and engineering data [83] Size-biased K model Engineering data [84] K generalized power Weibull model Engineering data [85] Exponentiated K-Dagum model Income and lifetime data [61]

The Construction of the Kavya-Manoharan Kumaraswamy Exponential Model
In this section, we create the Kavya-Manoharan Kumaraswamy exponential (KMKE) model by entering Formula (3) into Formula (4), and we obtain the cdf as shown below where β and γ are two shape parameters and α is scale parameter. The pdf of the KMKE model can be investigated by inserting Equations (3) and (2) into (5) as The reliability function, the hazard rate function (hrf), and the reversed and cumulative hrfs (see [86]) for the KMKE model are The KMKE model is very flexible and has three sub-models, see Table 2.

Statistical and Computational Features
In this section, we focus on the statistical and computational characteristics of the KMKE model, particularly the QU A function, MO m s, I N MO m s, COMO m s and MO m generating functions.

Quantile Function
The quantile function of the KMKE model is a useful tool to perform a simulated sample and it can be calculated by inverting Equation (6) where u ∼ Uni f orm(0, 1), then After some simplification, we can obtain the quantile function of the KMKE model as The median of the KMKE model is investigated by putting u = 0.5 in Equation (8),

Moments
In this subsection, we derive the w th moment (MO m ) (see [87]) for the KMKE model. The first four MO m s are the most important to describe the shape and monotonicity of the distribution curve. Suppose Z via a RVr that follows KMKE(α, β, γ), then the w th MO m about the zero of the KMKE model is The proof of Equation (9) is mentioned in Appendix A. By putting w = 1, 2, 3 and 4 into Equation (9) we will obtain the first four MO m s As a consequence, the mean and variance of the KMKE model are calculated via and The moment-generating function (see [87]) of the KMKE model can be computed from the next equation After some simplification we obtain Then the moment-generating function of the KMKE model is The m th incomplete MO m of the KMKE model can be computed from the next equation After some simplification we obtain Then the m th incomplete MO m (see [87]) of the KMKE model is The m th conditional MO m (see [87]) of the KMKE model can be computed from the next equation After some simplification we obtain Then the m th conditional MO m of the KMKE model is Figure 5 shows the mean, variance (var), skewness (SK), kurtosis (KU), coefficient of variation (CV) and index of dispersion (ID) (see [87]).

Estimation Methods
The maximum likelihood and Bayesian methods are the most famous. Based on Bayes' theorem, Bayesian statistics is a method for analyzing data and estimating parameters. The prior and data distributions, which are a special feature of Bayesian statistics, are given to all observable and unobserved parameters in a statistical model. In this section, maximum likelihood estimation and Bayesian estimation have been discussed to estimate the parameters of the KMKE model. Recently, more papers have discussed maximum likelihood and Bayesian estimation methods, such as [88,89].

Maximum Likelihood Estimation
In this section, we focus on how the maximum likelihood technique (see [87]) can be employed to estimate the parameters α, β and γ for the KMKE model. Suppose that z 1 , . . . , z n is a random sample of size n from the KMKE model (7). Then, the total loglikelihood function for Ω = (α, β, γ) is supplied as below The first partial derivatives U n (Ω) = ( ∂L n (Ω) ∂α , and By setting the nonlinear system of equations = 0 and solving these equations simultaneously, we can obtain the MLE( Ω). Because an exact solution is not achievable, these equations can be numerically solved by employing iterative approaches and statistical tools.

Bayesian Estimation
The Bayesian approach is a well-known non-classical inference technique in statistics. It defines uncertainties on the distribution parameters using a joint prior distribution and some proposed symmetric and asymmetric loss functions. It is believed that the three parameters, α, β and γ, are independent and follow gamma prior distributions: The hyper-parameters will be elicited using the parameters priors ∇ j , w j ; for more information, see [90]. The mean and variance of the KMKE distribution's α and γ maximum likelihood estimates will be compared to the mean and variance of the α j , β j and γ j considered priors (gamma priors), where j = 1, . . . , N and N is the number of samples available from the KMKE distribution. By equating α, β and γ with the mean and variance of gamma priors, we may calculate their respective means and variances.
The estimated hyper-parameters can now be stated as follows after solving the preceding two equations: and The likelihood function and the joint prior function Equation (14) can be used to express the joint posterior distribution. Consequently, Ω's joint posterior density function is In actuality, the posterior density's normalization constant C is often intractable, requiring an integral over the parameter space. The squared-error loss function (SELF) is the symmetric loss function: The average is then the Bayesian estimator of Ω under SELF.
The two most well-known loss functions-LINEX and entropy-have been covered. Varian [91] introduced a useful asymmetric loss function, which has recently been used in several publications by [92][93][94]. The linear exponential LINEX loss function describes this function. Assuming that the minimal loss happens atΩ = Ω, the LINEX loss function can be expressed as follows: where c is the shape parameter andΩ is any estimate of the parameter Ω. The shape of this loss function depends on the value of c. When the entropy loss function is used, the Bayes estimator of Ω isΩ According to Calabria and Pulcini [95], the entropy loss function is a decent asymmetric loss function. The form's entropy loss function is thought of as whose minimum is found atΩ = Ω. When the entropy loss function is used, the Bayes estimator of Ω isΩ Since it is challenging to solve these integrals analytically, the MCMC method will be used. The most important sub-classes of MCMC algorithms are Gibbs sampling and the more general Metropolis-within-Gibbs samplers. This algorithm was first presented by Metropolis et al. [96] As with acceptance-rejection sampling, the Metropolis-Hastings (MH) algorithm treats a candidate value produced from a proposal distribution as normal for each iteration of the process.

Simulation
Monte Carlo simulations are used to compare the performance of the suggested estimators for the KMKE parameters model. In this section, the estimation of the KMKE parameters are discussed using Bayesian and likelihood estimation techniques, comparing the results using a simulation study. In the Bayesian technique, symmetric and asymmetric loss functions are obtained. LINES and ELF are used as asymmetric loss functions.

Simulation Study
We investigate several sample sizes with n = 40, 75 and 150 for different α, β and γ parameter selections. We take 5000 random samples from the KMKE distribution. For each estimate, we calculate the bias values, mean square error (MSE) and length of confidence interval (LCI). The LCI of MLE is an asymptotic CI which can be denoted as LACI. The LCI of the Bayesian technique is the credible CI which can be denoted as LCCI.
Bias, MSE and LCI are used to quantify the efficacy of various estimators, with bias and MSE values close to zero indicating the most efficient techniques. The simulation results are obtained using the R programming language. The "maxLik" package computes the MLE using the Newton-Raphson approach. Additionally, the "CODA" package is used to perform the Bayesian estimation with various loss functions. This package evaluates the Markov chain Monte Carlo (MCMC) outputs and diagnoses lack of convergence. The estimated bias, MSE and LCI parameters of the KMWE distribution are displayed in Tables 3-6.

Figures 6-8 show heatmaps of MSE for parameters of the KMKE distribution,
where the X-axis shows the MSE based on different estimation methods with each parameter α, β and γ, respectively (MLE1 is a MSE for α, MLE2 is a MSE for β and MLE3 is a MSE for γ), while the Y-axis shows the MSE based on different cases and sample sizes, for example: C1n40 is an actual value of the parameter in Table 3 where α = 0.5, β = 0.4, γ = 0.5 and n = 40; C1n70 is an actual value of the parameter in Table 3 where α = 0.5, β = 0.4, γ = 0.5 and n = 70; and C2n70 is an actual value of the parameter in Table 3 where α = 0.5, β = 0.4, γ = 1.7 and n = 70.
By simulation in Tables 3-6  • The Bayesian estimation is superior to the MLE in every situation, we observe. • The Bayesian estimation with positive weight asymmetric loss function is superior to the Bayesian estimation with negative weight asymmetric loss function, as we note. • We note that the Bayesian estimation method with positive weight asymmetric loss function is better than the other estimation method. • The Bayesian estimation with symmetric loss function is superior to the Bayesian estimation with negative weight asymmetric loss function, in some simulations. • Bayesian credible and HPD intervals are the shortest LCI.      Figure 8. Heatmaps of MSE values for parameters of KMWE distribution with different sample cases:
Below tables discussed estimates of MLE and various measures of fit with provide statistics for all models fitted based on two real datasets, including different measures such as Kolmogorov-Smirnov discrete (KSD) with P-value of KS (PVKS), Cramer von Mises (CVM) and Anderson-(AD) Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent AIC (CAIC), and Hannon and Quinn's information criterion (HQIC). These tables also contain the MLE of the parameters for the models being examined.
Firstly: The food chain in the UK from 2000 to 2019 is shown in the first dataset, which can be found at https://www.gov.uk/government/statistics/food-chain-productivity and was accessed on 18 July 2022. Furthermore, this data has been cited in [9]. The data are as follows: "102. Secondly, as one component of factor total productivity (FTP), food and drink wholesaling in the UK from 2000 to 2019, see https://www.gov.uk/government/statistics/foodchain-productivity, accessed on 18 July 2022. Furthermore, this data has been cited in [9]. The data are as follows: "101.  Figures 10 and 14 show the two datasets that were fitted to the KMWE model using pdf, cdf, PP-plot and QQ-plot, respectively. Figures 9 and 13 confirm the estimators have maximum plot and unique values of the KMWE model for two datasets, respectively.
A total of 10,000 MCMC samples are produced using the MCMC algorithm that is discussed in Section 5. The MLEs and BEs of the unknown parameters of the KMWE distribution were determined using two datasets in Tables 8 and 10, respectively. Furthermore, generated and provided in Tables 8 and 10 are two-sided 95% ACI/HPD credible intervals for MLE and Bayesian estimations, respectively. They demonstrate how closely the point estimates of the unknown parameters that the MLE and Bayesian estimations obtain are to one another. Additionally, there are similarities in the interval estimates determined by 95% ACI/HPD credible intervals.

Concluding Remarks
The SEWE model [9] was introduced to fit the food data in the United Kingdom (UK), and the SEWE model gave an excellent fit for these data. However, in this article, we investigate a new lifetime model called the KMKE model which gives a better fit than the SEWE model for the food data. The KMKE model has three special models that are proposed and discussed. Some important statistical and computational features of the new model are investigated, such as the QU A function, MO m s, I N MO m s, COMO m s and MO m generating functions. Classical maximum likelihood estimation and Bayesian estimation approaches are utilized to estimate the parameters of the KMKE model. The simulation experiment examines the accuracy of the model parameters by employing Bayesian and maximum likelihood estimation methods. In this article, we use two real datasets related to food to show the relevance and flexibility of the suggested model. The KMKE model gives the best fits for food data and we compare it with the SEWE model, which was introduced by [9] for fitting food data, and also compare it with various known statistical models. This allows it to be used to predict the future dataset of food and drink wholesaling sales, and the extent of its validity and expected risks when using different quantities of food and beverages. By studying the KMKE model for food chain data, we can say that the KMKE model is the best model for evaluating and appropriating almost in-depth food data and avoiding erroneous conclusions, by using the previous prior information of parameters of the proposed model (Bayesian) as gamma distribution, where the Bayesian estimation method has the smallest SE values of parameters. The limitation of our new suggested model is that we estimate its parameters with complete samples only. Future works can use our new model to study the statistical inference for parameters using different censored schemes and different ranked set sampling. Some authors may study the stressstrength model using our model because the KMKE model is very simple and has two parameters only.

Conflicts of Interest:
The authors declare no conflict of interest.

Z
Random variable G(z) Cumulative distribution function of Kumaraswamy generated family H(z) Cumulative distribution function of exponential distribution g(z; α, β, γ) Probability density function of Kumaraswamy exponential distribution G(z; α, β, γ) Probability density function of Kumaraswamy exponential distribution α Scale parameter β Shape parameter γ Shape parameter F(z; α, β, γ) Cumulative distribution function of the Kavya-Manoharan Kumaraswamy exponential distribution f (z; α, β, γ) Probability density function of the Kavya-Manoharan Kumaraswamy exponential distribution S(z; α, β, γ) Reliability function of the Kavya-Manoharan Kumaraswamy exponential distribution h(z; α, β, γ) Hazard rate function of the Kavya-Manoharan Kumaraswamy exponential distribution τ(z; α, β, γ) Reversed hazard rate function of the Kavya-Manoharan Kumaraswamy exponential distribution H(z; α, β, γ) Cumulative hazard rate function of the Kavya-Manoharan Kumaraswamy exponential distribution Q(u) Quantile function µ w The w th moment M Z (t) Moment generating function η m (t) The m th incomplete moment τ m (t) The m th conditional moment ln L By inserting Equation (7) into Equation (A1), we can rewrite the above Equation as By applying the next exponential expansion to the above equation (see [105]) Employing the next binomial expansion to the last term of the previous equation By employing the last binomial expansion in Equation (A2) we have Again employing the next binomial expansion to the last term of the previous equation By inserting the previous expansion in Equation (A4), then we obtain then the w th MO m about the zero of the KMKE model is