Power-Modified Kies-Exponential Distribution: Properties, Classical and Bayesian Inference with an Application to Engineering Data

We introduce here a new distribution called the power-modified Kies-exponential (PMKE) distribution and derive some of its mathematical properties. Its hazard function can be bathtub-shaped, increasing, or decreasing. Its parameters are estimated by seven classical methods. Further, Bayesian estimation, under square error, general entropy, and Linex loss functions are adopted to estimate the parameters. Simulation results are provided to investigate the behavior of these estimators. The estimation methods are sorted, based on partial and overall ranks, to determine the best estimation approach for the model parameters. The proposed distribution can be used to model a real-life turbocharger dataset, as compared with 24 extensions of the exponential distribution.


Introduction
Exponential distribution is analytically tractable and, due to its lack of memory, is considered one of the important classical distributions. However, due to its constant hazard rate and unimodal density function, it has limited applications and cannot be adopted to a model phenomenon showing decreasing, increasing, or bathtub-shaped hazard rates. Hence, the statistical literature contains several extensions of the exponential distribution to increase its applicability and flexibility. One example is the modified Kies-exponential (MKE) introduced by Al-Babtain et al. [1] with the cumulative distribution function (CDF) and probability density function (PDF) (for t > 0) where α is a shape parameter, and λ is a scale parameter. The hazard rate function (HRF) of the MKE distribution can be decreasing, increasing, or bathtub-shaped. Interestingly, the two-parameter MKE model has a bathtub-shaped hazard function, whereas most distributions with this bathtub shape have problems related to algebraic complexity, an increasing number of parameters, and/or estimation problems. Hence, it can be adopted to a model phenomenon showing decreasing, increasing, or bathtub-shaped hazard rates and thus becomes more flexible than the exponential distribution for analyzing real-life data.
We propose a flexible extension of the MKE distribution named the power-modified Kies-exponential (PMKE) distribution, which provides more accuracy and flexibility for fitting real-life data. This distribution is generated based on the power transformation (P-T). The P-T has attracted attention over the years for its mathematical properties, which sometimes lead to surprising physical consequences, and for its appearance in a diverse range of natural and man-made phenomena. In fact, the generated power distributions have applications to a broad variety of different branches of human endeavor, including physics, earth sciences, biology, ecology, paleontology, computer and information sciences, engineering, and the social sciences.
The PMKE distribution has a very flexible density that can be symmetric, negatively skewed, positively skewed, or reverse-J-shaped, and it can allow for greater flexibility in the tails. It is also capable of modeling monotonically increasing, monotonically decreasing, and bathtub-shaped hazard rates. Furthermore, the CDF of this distribution has a closed form expression, which makes it ideal for applications in various fields such as engineering, reliability, life testing, survival analysis, and biomedical studies. A real data application from engineering science shows that the PMKE distribution is very competitive, with 19 extensions of the exponential distribution, including beta exponential (BE) [11], Marshall-Olkin logistic-exponential (MOLE) [12], exponentiated-exponential (ExE) [13], Harris extended-exponential (HEE) [14], Marshall-Olkin exponential (MOE) [15], and inverse-Pareto exponential distributions.
The rest of this paper is organized as follows. In Section 2, we introduce the PMKE distribution. In Section 3, we derive some of its mathematical properties. Actuarial measures of the new distribution are discussed in Section 4. Some classical methods of estimation along with detailed simulation results are reported in Section 5. Section 6 is devoted to Bayesian estimation of the parameters under different loss functions. A real-life data application is presented in Section 7. Finally, some conclusions and major findings are addressed in Section 8.

The PMKE Distribution
By applying the PT transform X = T 1 β to (1), we obtain the CDF of the PMKE distribution (for x > 0): where α and β are shape parameters, and λ is a scale parameter. Henceforth, we denote by X ∼PMKE(α, λ, β) a random variable with CDF (2). The PDF and HRF of X are and Plots of the PDF and HRF of X are displayed in Figures 1 and 2, respectively. These plots reveal that the density of X can be left-skewed, reverse-J-shaped, or right-skewed, and its HRF can be bathtub-shaped, increasing, or decreasing.

Quantile Function
The quantile function (QF) of X follows by inverting the CDF (2) as

Moments
The rth ordinary moment of X can be expressed in terms of the complete gamma function We obtain the first four ordinary moments of X by setting r = 1, 2, 3, and 4. The central moments and cumulants of X are easily obtained from these ordinary moments.
The sth incomplete moment of X takes the form where γ β+s β , λ(m − kα)t β denotes the lower incomplete gamma function. The important application of the first incomplete moment is related to the Bonferroni and Lorenz curves defined by L(p) = α 1 (x p )/µ 1 and B(p) = α 1 (x p )/(pµ 1 ), respectively, where x p = Q(p) can be evaluated numerically by Equation (4) for a given probability p. These curves are very useful in economics, demography, insurance, engineering, and medicine. Another application of the first incomplete moment refers to the mean residual life (MRL) and the mean waiting time given by m 1 (t) = [1 − α 1 (t)]/S(t) − t and M 1 (t) = t − α 1 (t)/F(t), respectively.

Actuarial Measures
We discuss the theoretical and computational aspects of some important risk measures, which play a crucial role in portfolio optimization under uncertainty.
The VaR of a random variable is the qth quantile of its CDF given by VaR q = Q(q) (see Artzner [16]). Therefore, the VaR of X can be obtained from (4).
The TVaR is used to quantify the expected value of the loss given that an event outside a given probability level has occurred. The TVaR of X is given by which follows as where E p (z) = ∞ 1 t −p exp (−tz)dt is the exponential integral. The expected shortfall (ES) is a risk measure sensitive to the shape of the tail of the distribution of returns on a portfolio, namely, Some numerical values of VaR, TVaR, and ES for four distributions are reported in Table 1. The values of these measures are obtained for four distributions at the same parameter values to investigate the tails of these models. The values of VaR, TVaR, and ES for the PMKE distribution are greater than those for the MKE distribution and the other two models, thus showing that the proposed distribution has a heavier tail than its competing models. Hence, the additional parameter β provides greater flexibility for the PMKE distribution over the MKE model.

Methods of Estimation
In this section, we discuss seven methods to estimate the parameters θ = (α, λ, β) of the PMKE distribution and compare them by means of Monte Carlo simulations.
The AdequacyModel package for the R statistical computing environment provides a comprehensive and efficient general meta-heuristic optimization method for maximizing or minimizing an arbitrary objective function, which can be used to find the estimates of θ in the following methods. The data is accessed on 9 May 2021 and its details are available at https://rdrr.io/cran/AdequacyModel/.

Methods
Let x 1 , . . . , x n be a random sample of size n from the PDF (3). The log-likelihood function for θ reduces to The maximum likelihood estimate (MLE) of θ can be obtained by maximizing . Let x 1:n , . . . , x n:n be the corresponding order statistics. The ordinary least-squares estimates (OLSEs) of the parameters are determined by minimizing the function Alternatively, the weighted least-squares estimators (WLSEs) can be calculated by minimizing Further, the Anderson-Darling estimates (ADEs) are obtained by minimizing the function whereas the Cramér-von Mises estimates (CVMEs) are determined by minimizing The maximum product of spacing estimates (MPSEs) are based on the uniform spacing

and they follow by maximizing
Finally, the percentile estimates (PCEs) follow by minimizing where p i = i/(n + 1) is an estimate of F(x i:n ).

Monte Carlo Simulations
We explore here the performance of the aforementioned estimation methods in estimating the PMKE parameters using simulation results. We use the sample sizes n = {20, 50, 100, 200, 350} and some parameter values. We generate N = 1000 random samples from the PMKE distribution and calculate the average absolute biases (BIAS), mean square errors (MSEs), and mean relative estimates (MREs) using R software.
The BIAS, MSEs, and MREs have the forms where θ i can represent α, λ, and β. Tables 2-6 report the simulation results, including the BIAS, MSEs, and MREs from the seven estimation methods. We can note that they show small BIAS, MSEs, and MREs for all parameter combinations. All seven estimators have the consistency property, where these quantities decrease when the sample size increases for all scenarios. Further, we conclude that the MLEs, ADEs, CVMEs, LSEs, MPSs, PCEs, and WLSEs are close to the true PMKE parameters.
The simulation results in Tables 2-6 show the ranks of the estimates among all approaches by the superscripts in each row, and the partial sum of the ranks by ∑ Ranks. The partial and overall ranks of these estimates reported in Table 7 indicate the performance ordering of all estimators. According to Table 7, the performance ordering of all methods is MPSEs, MLEs, ADEs, WLSEs, PCEs, CVMEs, and LSEs. In summary, the MPSEs outperform all estimates from the other approaches for the PMKE distribution with an overall score of 38. Furthermore, the maximum likelihood can be considered a rival approach for the MPS method with an overall score of 69.5.

Bayes Estimation Method and Simulations
We obtain here the Bayes estimators of the parameters of the PMKE distribution using the symmetric and asymmetric loss functions. We have to choose a prior density function that covers our belief about the data and choose appropriate hyper-parameter values. Based on a complete sample, we adopt the square error (SE), general entropy (GE), and linear exponential (Linex) loss functions to obtain the estimates and consider that α, λ, and β are independent. We choose gamma-independent priors for the parameters, namely, The gamma prior encourages researchers to feel confident in the data. If we do not have any belief about the data, we must adopt non-informative priors by setting the following values, so that µ i tends to zero and λ i tends to infinity (i = 1, 2, 3). In this way, we can change informative priors into non-informative priors. After this, we can find the form of the joint PDF prior of α, λ, and β as By multiplying the last two equations, we obtain According to the SE loss function, the Bayes estimator of B = B(θ), where θ = (α, λ, β), iŝ where π * (θ) is given by Equation (6). The Bayes estimator under the LINEX loss function is the value ofB such that E θ [exp(−cθ)] exists. The Bayes estimateΘ GE under the GE loss function iŝ such that E θ [θ −q ] exists. We cannot find a result for the integrals in Equations (7)-(9). Thus, we use the Markov Chain Monte Carlo (MCMC) technique to approximate these integrals and consider the Metropolis-Hastings algorithm as an example of the MCMC technique to find the estimates.

The MCMC Method
We adopt the MCMC method here because we do not have a well-known distribution for the posterior density function. We then calculate the BEs of α, λ, and β under the conditional posterior distribution functions for these parameters: Therefore, we do not have closed forms for the conditional posterior distributions for these parameters since they do not represent any known distribution. We use the Metropolis-Hasting algorithm below to explain the steps required to compute the Bayes estimates for B = B(α, λ, β) under the SE loss function.

The Metropolis-Hastings Algorithm
The Metropolis-Hastings algorithm can be considered as an MCMC method for generating data from any CDF. These generated samples can be used to approximate the distribution or to compute an integral (e.g., an expected value). We use the MCMC algorithm because it is sometimes difficult to obtain samples and the posterior comes from an unknown distribution.

1.
The starting values are as follows: Set i = 1.
Obtain the BEs of α using MCMC under the SEL function asα SE = ∑ N The simulated results in Tables 8 and 9 show the ranks of the estimates under different loss functions by the superscripts in each row, and the partial sum of the ranks by ∑ Ranks. The partial and overall ranks of the explored estimates are listed in Table 7, indicating the performance ordering of all estimators. According to Table 10, the Bayesian estimates' performance ordering is BGE, BLN, and BSE.     q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q A comparison of the PMKE distribution with its MKE sub-model using the likelihood ratio statistic (LR) is performed to check the hypotheses H 0 : β = 1 vs. H 1 . The LR statistic is equal to 7.124 and its p-value = 0.0076, which rejects H 0 . Hence, the new PMKE distribution yields a superior fit to these data than the MKE distribution.

Conclusions
We have introduced a new continuous model called the power-modified Kies-exponential (PMKE) distribution and have derived some of its mathematical properties. The new density function can take different shapes. Furthermore, the PMKE failure rate function can be monotonically increasing, monotonically decreasing, or bathtub-shaped. We have also calculated some of its actuarial measures. We considered seven classical and Bayesian methods to estimate the parameters based on a complete sample. An extensive simulation study has been conducted to compare the performance of the estimates from the seven estimation methods. Based on our study, the classical maximum product of the spacing approach is recommended to estimate the PMKE parameters. The Bayesian approach provides more accurate estimates under general entropy and linear exponential loss functions than the square error loss function. A real data analysis shows that the new distribution provides a better fit than other distributions.