Mixture of Akash Distributions: Estimation, Simulation and Application

: In this paper, we propose a two-component mixture of Akash model (TC-MAM). The behavior of TC-MAM distribution has been presented graphically. Moment-based measures, including skewness, index of dispersion, kurtosis, and coefﬁcient of variation, have been determined and hazard rate functions are presented graphically. The probability generating function, Mills ratio, characteristic function, cumulants, mean time to failure, and factorial moment generating function are all statistical aspects of the mixed model that we explore. Furthermore, we ﬁgure out the relevant parameters of the mixture model using the most suitable methods, such as least square, weighted least square, and maximum likelihood mechanisms. Findings of simulation experiments to examine behavior of these estimates are graphically presented. Finally, a set of data taken from the real world is examined in order to demonstrate the new model’s practical perspectives. All of the metrics evaluated favor the new model and the superiority of proposed distribution over mixture of Lindley, Shanker, and exponential distributions.


Introduction
In most reliability scenarios, data are modelled using a single parametric model.However, in certain circumstances, a population can be split into many subgroups, each showing a particular category of collapse.Finite mixture models serve a significant role in modelling such diverse data.Biology, business, engineering, healthcare, genetics, marketing, real-world applications, and social sciences all benefit from finite mixture models.Mixture models are created by varying the proportions of two or more models to generate a new distribution with novel properties.Consequently, it is essential to examine the statistical characteristics of the suggested mixture model and employ the suitable methods for estimating the unexplained parameters.Mixture models are used in a diversity of applications, such as clustering and classification [1][2][3][4].Sultan et al. [5] proposed a mix of inverse Weibull models and utilized density and hazard function graphs to study some of its features.The conventional characteristics of the concoction of Burr XII and Weibull distributions were examined by [6].Recently, the authors [7,8], Ateya [9], Mohammadi et al. [10], and Al-Moisheer et al. [11] are among the scientists who study mixture modelling in a variety of contexts.
Axioms 2022, 11, 516 2 of 20 Many applied sciences, including medical, engineering, insurance, and finance, rely on lifetime data modelling and analysis.Some of the continuous distributions used to explain lifetime data are Weibull, gamma, exponential, lognormal, and Lindley, as well as their generalizations.Since many investigators have employed Lindley distribution to predict lifetime data, and Hussain [12] has demonstrated that Lindley model is effective for stressstrength dependability modelling, Lindley model may not be suitable for describing realworld data in many cases.Shanker [13] developed a novel model by using a two-component concoction of an exponential model (ϑ) and a gamma model (3, ϑ) to have a unique distribution that is more flexible than Lindley and exponential distribution for modelling lifespan data in terms of dependability and hazard rate shapes.Shanker et al. [14] have devised and addressed the concept of modelling lifetime data using one parameter families of distributions, such as Akash, exponential, and Lindley distributions.Many lifetime datasets are employed to exhibit its adaptability over the exponential distribution.Shanker and Shukla [15] examined the two-parameter Akash model and determined its statistical characteristic, estimation problem, and application to it.As a reason, the Akash distribution can be used as an alternate lifetime model in reliability analysis.
The maximal likelihood estimation (MLE) is well-known estimation approach.Despite the fact that MLE is efficient and has strong conceptual features, there is confirmation that it does not work well, especially with small samples.As a result, different estimation approaches have been offered in the studies as options to conventional method.The weighted least-squares estimation (WLSE), L-moments estimator (LME), percentile estimator (PCE) and least squares estimator (LSE) are among the most frequently recommended.These approaches, in general, do not possess desirable theoretical features, but they can offer better estimates of unknown parameters in specific instances than the MLE.Various estimating approaches for many models have been investigated in the studies, as illustrations [16][17][18][19][20][21].The goal of this research is to give a mechanism for expert statisticians to choose the best evaluation method for the Two-Component Mixture of Akash Model (TC-MAM).In this investigation, we estimate the TC-MAM using LSE and WLSE, in conjunction to MLE.
Our goal in this investigation is to develop a novel mixture model for modeling real lifespan datasets from various disciplines of knowledge that is better fitting than mixture of Shanker, exponential and Lindley distributions.The TC-MAM model has an advantage over the Shanker and exponential models because the exponential distribution has a constant hazard rate function and the Shanker model has an increasing hazard rate function, whereas the failure rate function for a TC-MAM model exhibits monotonically increasing, modified declining, decreasing-increasing-decreasing (DID), declining-increasing (DI), and upside-down bathtub behavior.The novel TC-MAM is being developed in particular to offer a novel flexible parametric model for modeling complex data that emerges in dependability research, investigation of lifespan, quality control, statistical mechanics, economics, biological investigations, and other fields.The purpose is to provide a novel model for lifespan analysis that can handle various types of failure rates, as well as various close form features of novel model with simple physical interpretations.
The originality of this research is due to the fact that we present a thorough explanation of the statistical aspects of TC-MAM in the hopes of attracting more applications in lifespan analysis.Additionally, as far as we know, no investigation has been performed to evaluate all of these estimators of the TC-MAM, as well as their mathematical and statistical features and assessment methods to estimate of unexplained parameters of TC-MAM.For various sample sizes and parametric values, we demonstrate how alternative frequentist estimators of the suggested distribution work.

The Two-Component Mixture of Akash Model
A r.v.T is stated to have a TC-MAM if its PDF and CDF can be integrated as: (1) and where = (λ 1 , λ 2 , δ) and δ is a positive mixing parameter, whereas λ i are positive scale parameters.

Mode
By tackling the given non-linear equation with respect to t, the mode of the TC-MAM( ) is derived

Median
The median of TC-MAM( ) is given here.Let F( t| ) be CDF of TC-MAM( ) the median is at 50th quantiles that is Q 0.5 .The median (t * ) can, therefore, be determined by resolving given equation for t.
Numerical strategies like Newton-Raphson approach can be utilised to find t * from Equation (7).
Several graphs of PDF and CDF of TC-MAM, as well as both component densities, for various parametric values are shown in Figures 1 and 2. It should be indicated that input parameters were selected at random until a wide range of patterns could be examined.The PDF exemplifies its adaptability.The PDF curves of TC-MAM( ) indicate that it can be monotonically decreasing, positively skewed, inverted U, and decliningincreasing-decreasing (DID), as well as modified monotonically decreasing with platykurtic, mesokurtic, and leptokurtic curves.As a result, it can be used to model a diverse set of data.

Median
The median of TC-MAM($) is given here.Let F (tj $) be CDF of TC-MAM($) the median is at 50 th quantiles that is Q 0:5 .The median (t ) can therefore be determined by resolving 5  1 Numerical strategies like Newton-Raphson approach can be utilised to …nd t from Eq. ( 7).The PDF exempli…es its adaptability.The PDF curves of TC-MAM($) indicate that it can be monotonically decreasing, positively skewed, inverted U, and declining increasing decreasing (DID), as well as modi…ed monotonically decreasing with platykurtic, mesokurtic, and leptokurtic curves.As a result, it can be used to model a diverse set of data.

mth Moments about Origin
For a r.v.T, the mth moments of TC-MAM( ) are as: The mean of the TC-MAM( ) is: while the variance is given by Graphs of the mean and variance of TC-MAM ( ) for a variety of parameter values that can be identified in Figures 3 and 4. The mean of TC-MAM ( ), shows a monotonically decreasing behavior for fixed value of λ 1 and δ and varying values of λ 2 (see Figure 3a).
The escalating changes of mixing parameter δ enhanced the mean concentration, according to this analysis.We draw mean graphs (see Figure 3b) to demonstrate the behavior of the mean for fixed values of λ 2 and δ and varying values of λ 1 .It reveals the characteristics of component parameter λ 1 in relation to a mean profile.The boosting attitude of δ reduce the concentration of mean for the all varying values of λ 1 .The significances of the mixing parameter versus the mean profile are defined in Figure 3c for various levels of λ 1 .From these drawn lines, it can be deduced that the concentration of the mean profile is a deteriorating function for parameter λ 1 .The variance exhibits the same behavior as the mean in all scenarios (see Figure 4a-c).
The mean of the TC-MAM($) is: while the variance is given by For a r.v. T , the m th moments of TC-MAM($) are as: The mean of the TC-MAM($) is: while the variance is given by

Graphs of the mean and variance of TC-MAM ($) for a variety of parameter values
that can be identi…ed in Figs 3 and 4. The mean of TC-MAM ($), shows a monotonically decreasing behavior for …xed value of 1 and and varying values of 2 (see Fig. 3a).
The escalating changes of mixing parameter enhanced the mean concentration, according to this analysis.We draw mean graphs (see Fig. 3b) to demonstrate the behavior of the mean for …xed values of 2 and and varying values of 1 .It reveals the characteristics of 7 In particular moments about origin and the moments about mean of the TC-MAM( ) are: Axioms 2022, 11, 516 The φCV (Coefficient of Variation), ΨSk (Skewness) and ψK (Kurtosis) of TC-MAM( ) are: and Index of Dispersion (ID) is The TC-MAM( ) is readily explained to be over-distributed when µ 2 > µ, equidispersed µ 2 = µ, as well as under-dispersed µ 2 < µ.
Graphs of the ID of TC-MAM ( ) for various parameter settings are illustrated in Figure 5.The boosting attitude of δ rise the concentration of mean for the all varying values of λ 2 (see Figure 5a).However, boosting attitude of δ reduces the concentration of mean for the all varying values of λ 1 (see Figure 5b).The effects of δ against the concentration of the mean profile are shown in Figure 5c.The concentration of the mean profile is a decreasing function for parameter λ 1 according to these depicted lines.Figures 6 and 7 explain the nature of Ψ Sk and ψK in relation to λ 1 , λ 2 and δ.The coefficient of skewness and kurtosis of TC-MAM ( ), shows a decreasing behavior for fixed value of λ 1 and δ and varying values of λ 2 (see Figures 6a and 7a).
To expose the behavior of Ψ Sk and ψK for fixed value of λ 2 and δ and varying values of λ 1 (see Figures 6b and 7b).The escalating changes of mixing parameter δ enhanced Ψ Sk and ψK concentration, according to this analysis.The significances of the mixing parameter versus the coefficient of skewness and kurtosis profile are defined in Figures 6c and 7c for various levels of λ 1 .From these drawn lines, it can be deduced that the concentration of the skewness and kurtosis profile is a deteriorating function for parameter λ 1 .
and Index of Dispersion (ID) is The TC-MAM($) is readily explained to be over-distributed when 2 > , equi-dispersed 2 = ; as well as under-dispersed 2 < :

Moment Generating Function
The MGF of TC-MAM( ) is specified as:

Reliability Measures
In reliability framework, lifetime models are classified using the reliability/survival function and the failure/hazard rate function.TC-MAM( ) is currently being studied for its reliability properties.

Reliability Function
The reliability function (R( t| )) of TC-MAM( ) is.

Hazard Function
The failure rate function h( t| ) of the TC-MAM( ) is described as follows.
In Figure 8, the HRF of TC-MAM( ) shows monotonically increasing, modified de- creasing, decreasing-increasing-decreasing (DID), decreasing-increasing (DI), and upside down bathtub behavior.Figure 8a,b,d signifies that the reduction in failure rate function profile and is noted by enlarging the value of mixing parameter δ, and for λ 1 < λ 2 .Figure 8c,e,f exhibits the diversion in the failure rate function for various values of δ.It is found in Figure 8 that the failure rate distribution is expanding due to higher the value of mixing parameter δ, and for λ 1 > λ 2 .

Mills Ratio
Mills ratio is an another method of quantifying reliability due to its relation to failure rate.Mills ratio Υ( t| ) of TC-MAM( ) is

Cumulative Hazard Rate Function
The CHRF of TC-MAM( ) is It is a risk indicator: the stronger the H( t| ) estimate, the greater the chance of failure by t-time.It must be stated that R( t| ) = e −H( t| ) and f ( t| ) = h( t| )e −H( t| ) . (32) So,

Reversed Hazard Rate Function
The RHRF of a random life of TC-MAM( ) is defined as exp(−λ 2 t) . (34)

Mean Time to Failure (MTTF)
The expected time for which the device performs efficiently is given by the mean time to failure (MTTF).If TC-MAM( ) then reliability function is used to express MTTF, which is as follows: and R(t) is provided in Equation ( 28).Thus

Estimation Inference via Simulation
Given that the parametric vector is undetermined, certain statistical properties of the TC-MAM( ) are presented to this section.The evaluation of parametric vector is accomplished by three widely known estimation mechanisms, such as MLE, LSE, and WLSE.From now, t 1 , t 2 , . . ., t n signify n determined values from T and their ascending sorting values t (1) ≤ t (2) ≤ . . .≤ t (n) .

Maximum Likelihood Estimation (MLE)
The MLE method is the best methodology for parameter assessment.The popularity of the approach stems from its many advantageous characteristics, such as consistency, normality, and asymptotic efficiency.Let t 1 , t 2 , . . ., t n be n determined values from the Equation ( 2) and be the vector of undetermined parameters.The evaluations of MLEs of can be given by optimizing the likelihood function with respect to λ 1 , λ 2 , and δ given by ) or likewise the log-likelihood function for is So, by partially differentiating l( t| ) in terms of each parameter (λ 1 , λ 2 , δ) and placing the results to zero, the MLEs of the relevant parameters are determined as As a consequence, the MLE is found by evaluating this non-linear set of equations.However such equations cannot be handled analytically, we can use statistical software to solve them using an iterative methodology namely the Newton method or fixed point iteration methods.

Least Square Estimators (LSE)
The ordinary least square approach [22] is widely used for assessing undetermined parameters.The LSEs of λ 1 , λ 2 , and δ, indicated by λ1LSE , λ2LSE , and δLSE , can be determined by minimizing Equation ( 42) with respect to λ 1 , λ 2 , and δ, where F(•)is given by Equation( 4).They may be determined in the similar way by solving the non-linear equations below: where

Simulation Study
The simulation study is used to evaluate the various estimating methodologies outlined in the preceding subsection.Monte Carlo simulations are performed with a variety of mixing proportion δ and distribution parameters.The performance of MLE, LSEs, and WLSEs of the TC-MAM( ) parameters is evaluated using four simulation experiments.The proficiency of the MLEs, LSEs, and WLSEs is discussed using the bias and MSE indicators.In terms of n, the efficiency of each parameter estimation strategy for the TC-MAM( ) model is examined.The simulation algorithm is subdivided into six steps: 1.
For parametric set-I, the estimated bias of parameter 1 under LSE, WLSE is negative, for Set II 1 and under all three estimation methods and for set III, the estimated bias of parameter is negetive (see Fig. 9, 11 and 13).
Figs. 9 and 10 show the bias and MSE of ~ 1 ; ~ 2 ; and ~ , for parametric Set-I and the WLSE always has the smallest value of bias and MSE of all estimators.
In the second scenario, the MLE estimators of 2 is over-estimated, while MSE of 2 is highest among the three considered estimators (see Figs. 11-12).
The estimators of 1 are over-estimated in the third scenario, however the over and under estimation of 2 and are seen among the three investigated estimators, and the WLSE always has the minimum value of bias of all estimators (see Fig. 13).In the second scenario, the MLE estimators of λ 2 is over-estimated, while MSE of λ 2 is highest among the three considered estimators (see Figures 11 and 12).• The estimators of λ 1 are over-estimated in the third scenario, however the over and under estimation of λ 2 and δ are seen among the three investigated estimators, and the WLSE always has the minimum value of bias of all estimators (see Figure 13).• Among the three estimators evaluated, the MSE of λ 1 is the greatest (see Figure 14).

•
The MSE of λ2 is strongly stimulated and higher under MLE and LSE estimation methods when n < 50 (see Figure 10).The final conclusion drawn from the foregoing figures is that, as n rises, estimated bias and MSE graphs for estimators λ1 , λ2 , and δ finally approach zero for all estimating methods.This demonstrates the accuracy of both the estimating methods and the numerical computations for the TC-MAM parameters.

Applications
We demonstrate the flexibility of the TC-MAM in this section by examining a real dataset.The TC-MAM distribution is compared to competing models, such as the two component mixture of Shanker distribution (2C-MSM), the two component mixture of exponential model (2-CMEM), and the two component mixture of Lindley distribution (2C-MLM) using the R function maxLik().The -Log-likelihood (-LL), the AIC, BIC, and AICC have all been used to compare these models.The model having the least quantities of above-mentioned goodness-of-fit (GoF) measures may be the best fit for the real dataset.
Dataset: There are 56 observations in this dataset pertaining to the burning velocity of various chemical substances.The laminar flame speed at the specified composition, temperature, and pressure circumstances is the burning speed/velocity.It lowers as the inhibitor concentration rises, and it may be observed directly by analysing the pressure distribution in the spherical vessel and monitoring the flame propagation.We consider a real-life dataset which represents the burning velocity (cm/s) of several chemical compounds to show the TC-MAM distribution's suitability.15.Figures 15 and 16 show a graphical illustration of MLE existence and uniqueness, respectively.To summarize, the TC-MAM emerges as the better model for the dataset, indicating its usefulness in a real-world setting.We can deduce from this graphical representation and results obtain from Table 1 that the TC-MAM is a better fit for the dataset in consideration.Hazard Rate Function and Mean Residual Life.To investigate and assess the estimating approaches'performance, a simulation study with 1000 iterations was done and it was noted that when n increases, the estimated MSEs of parameters ~ 1 ; ~ 2 ; and ~ under the MLE estimation technique rapidly decrease, illustrating the e¢ ciency of the MLE procedure.As a result, we found that estimating model unknown parameters with regards of accuracy and consistency, the MLE approach surpassed the rest.Furthermore, we used real datasets to explain the utility of the underlying mixture model.
Data Availability Statement: All the data is avaiable in this manuscript.
Con ‡ict of Interest.Authors declare that they do not have con ‡ict of interest.Hazard Rate Function and Mean Residual Life.To investigate and assess the estimating approaches'performance, a simulation study with 1000 iterations was done and it was noted that when n increases, the estimated MSEs of parameters ~ 1 ; ~ 2 ; and ~ under the MLE estimation technique rapidly decrease, illustrating the e¢ ciency of the MLE procedure.As a result, we found that estimating model unknown parameters with regards of accuracy and consistency, the MLE approach surpassed the rest.Furthermore, we used real datasets to explain the utility of the underlying mixture model.
Data Availability Statement: All the data is avaiable in this manuscript.
Con ‡ict of Interest.Authors declare that they do not have con ‡ict of interest.

Conclusions
In this investigation, we used three estimated techniques: MLE, LSE, and WLSE to work on two component mixtures of Akash models.In particular, the Akash mixing model's statistical and reliability features were achieved, such as central moments, Cumulants, Cumulant Generating Function, Probability Generating Function, Mean Time to Failure, Factorial Moment Generating Function, Coefficient of variation, Mills ratio, skewness and kurtosis, Reversed Hazard Rate Function, and Mean Residual Life.To investigate and assess the estimating approaches' performance, a simulation study with 1000 iterations was performed and it was noted that when n increases, the estimated MSEs of parameters λ1 , λ2 , and δ under the MLE estimation technique rapidly decrease, illustrating the efficiency of the MLE procedure.As a result, we found that estimating model unknown parameters with regards of accuracy and consistency, the MLE approach surpassed the rest.Furthermore, we used real datasets to explain the utility of the underlying mixture model.

Fig. 2 .
Fig. 2. Behaviour of F 1 (tj 1 ) (…rst component CDF); F 2 (tj 2 ) (second component CDF) and CDF of TC-MAM (F m (tj $)) with against t.Several graphs of PDF and CDF of TC-MAM, as well as both component densities, for various parametric values are shown in Figs.1 and 2. It should be indicated that input parameters were selected at random until a wide range of patterns could be examined.

Fig. 11 .
Fig. 11.Behaviour of bias of estimators with di¤erent methods under parametric set II against n.

Figure 11 .
Figure 11.Behavior of bias of estimators with different methods under parametric set II against n.

Fig. 11 . 18 Fig. 12 .
Fig. 11.Behaviour of bias of estimators with di¤erent methods under parametric set II against n.

Fig. 13 .
Fig. 13.Behaviour of bias of estimators with di¤erent methods under parametric set III against n.

Figure 12 .
Figure 12.Behavior of MSE of estimators with different methods under parametric set II against n.

Fig. 12 .
Fig. 12. Behaviour of MSE of estimators with di¤erent methods under parametric set II against n.

Fig. 13 .
Fig. 13.Behaviour of bias of estimators with di¤erent methods under parametric set III against n.

Figure 13 .
Figure 13.Behavior of bias of estimators with different methods under parametric set III against n.

Fig. 14 .
Fig. 14.Behaviour of MSE of estimators with di¤erent methods under parametric set III against n.

Figure 14 .
Figure 14.Behavior of MSE of estimators with different methods under parametric set III against n.•Under all three estimation procedures, the estimated bias of parameters 1 , λ 2 , δ reduces as n grows.•Forparametric set-I, the estimated bias of parameter λ 1 under LSE, WLSE is negative, for Set II λ 1 and δ under all three estimation methods and for set III, the estimated bias of parameter δ is negative (seeFigures 9,11 and 13).• Figures 9 and 10 show the bias and MSE of λ1 , λ2 , and δ, for parametric Set-I and the WLSE always has the smallest value of bias and MSE of all estimators.•In the second scenario, the MLE estimators of λ 2 is over-estimated, while MSE of λ 2 is highest among the three considered estimators (see Figures11 and 12).•The estimators of λ 1 are over-estimated in the third scenario, however the over and under estimation of λ 2 and δ are seen among the three investigated estimators, and the WLSE always has the minimum value of bias of all estimators (see Figure13).• Among the three estimators evaluated, the MSE of λ 1 is the greatest (see Figure14).•TheMSE of λ2 is strongly stimulated and higher under MLE and LSE estimation methods when n < 50 (see Figure10).• Figures 13 and 14 demonstrate the influence of choice of parameters on the estimation approaches, here bias and MSEs are comparatively low among the selected set of parameters.• Some big shifts in MSEs of considered estimators under MLE, LSE, and WLSE are observed when n < 50.• In terms of bias, the WLSE's performance is relatively favorable.• Furthermore, when n increases, the MSE for all three estimating strategies decreases, satisfying the consistency criteria (Figures 10, 12 and 14).• In all estimating methodologies, the difference between estimates and stated parameters reduces as n rises.• As n approaches infinity, WLSE estimation is frequently better in terms of bias and MSE when likened to other estimation methods for all given parameter values.• The estimated MSEs of parameters λ1 , λ2 , and δ under the MLE estimation technique decrease quickly as n increases, demonstrating the effectiveness of the MLE procedure.
• Figures 13 and 14 demonstrate the influence of choice of parameters on the estimation approaches, here bias and MSEs are comparatively low among the selected set of parameters.• Some big shifts in MSEs of considered estimators under MLE, LSE, and WLSE are observed when n < 50.• In terms of bias, the WLSE's performance is relatively favorable.• Furthermore, when n increases, the MSE for all three estimating strategies decreases, satisfying the consistency criteria (Figures 10, 12 and 14).• In all estimating methodologies, the difference between estimates and stated parameters reduces as n rises.• As n approaches infinity, WLSE estimation is frequently better in terms of bias and MSE when likened to other estimation methods for all given parameter values.• The estimated MSEs of parameters λ1 , λ2 , and δ under the MLE estimation technique decrease quickly as n increases, demonstrating the effectiveness of the MLE procedure.

Figure 15 .
Figure 15.The plots of PLLF for Dataset.

Figure 16 .
Figure 16.The graphs of score functions cross the horizontal axis at λ1 , λ2 , and δ of Dataset.

Table 1 .
MLEs, and GoF statistics for the Dataset I.