Parameter Estimation and Stress-Strength Model of Power Lomax Distribution: Classical Methods and Bayesian Estimation

In this paper, parameter estimation for the power Lomax distribution is studied with dif-ferent methods as maximum likelihood, maximum product spacing, ordinary least squares, weighted least squares, Cramér–von Mises and Bayesian estimation by Markov chain Monte Carlo (MCMC). Robust estimation of the stress-strength model for the Power Lomax distribution is discussed. We propose that the method of maximum product of spacing for reliable estimation of stress-strength model as an alternative method to maximum likelihood and Bayesian estimation methods. A numerical study using real data and Monte Carlo Simulation is performed to compare between diﬀerent methods.

It was studied by many authors (Kumar et al., 2017;Mokhlis et al., 2017;Almetwaly and Almongy, 2018;Almetwally et al., 2018;El-Sherpieny et al., 2020). Figure 1 represents the different graphs of pdf of the PL distribution for different values of the parameters. Ekström (2014) talked about maximum product of spacing (MPS) as an alternative to the maximum likelihood estimation (MLE) technique. Much of the time, the MPS technique works better than the MLE strategy and alluring properties, for example, consistency and asymptotic productivity of the MPS estimator firmly parallel those of the MLE when the last functions admirably. For more examples, see Singh et al. (2014); Almetwally and Almongy (2019b,a); AHmad and Almetwally (2020); El-Sherpieny et al. (2020).
The stress strength model has been known in the mechanical as follows, the stress is the mechanical loads and forces, while the strength is the physical effort that can resist the loads to perform its required function. Birnbaum (1956) was one of the first researchers who dealt with the model of stress-strength content. When the stress exceeds the strength, the failure occurs. If X represents the strength and Y represents the stress, the main theme of statisticians is to estimate the probability of failure or reliability of this model, it is defined as R = P (Y < X).
Since the reliability concept is general, so the stress strength model can be applied in different fields outside of the scope of mechanics, for more details see Saraçoğlu et al. (2012). Singh et al. (2014) discussed the problem of the estimation of stressed system reliability under both classical and Bayesian paradigms. Mokhlis et al. (2017) presented characterizations, associated with the stress strength reliability of distributions with some general exponential and general inverse exponential forms. There are two main objectives in this article: Firstly a comparison study of different estimation methods for the power Lomax distribution to conclude the best estimation method for parameters of PL distribution. Secondly propose the maximum product of spacing method as estimation method of reliability estimation for stress-strength model as an alternative method to maximum likelihood and Bayesian estimation methods, which we think would be of deep interest to statisticians. The rest of the paper is organized as follows. In Section 2, we propose the parameters estimation by using classical estimation and Bayesian estimation. In Section 3, we will use MLE, MPS and Bayesian estimation methods for parameter estimation of the stressstrength model. In Section 4, simulation study are given. Application of real data are discussed in Section 5. Finally, the conclusion and results are given in Section 6.

Parameter Estimation
In this section, we propose the parameters estimation by using classical estimation and Bayesian estimation.

Classical Estimation
In this subsection, we propose the parameters estimation by utilizing traditional estimation. This methods are MLE, Ordinal Least Square (OLS), Weighted Least Squares (WLS), Cramér-von Mises (CVM) and MPS. Rady et al. (2016) discussed the likelihood function of the PL distribution is

Maximum Likelihood Estimation
where Φ = (α, β, λ), and the log likelihood function is given as The normal equations for the unknown parameters have been discussed by Rady et al. (2016). We can use any iterative procedure techniques such as Nelder-Mead type algorithms, to obtain as the numerical solution.

Maximum Product Spacing
As per Cheng and Amin (1983) presented MPS as following where D i can be written as follows such that D i = 1, then the MPS estimators for PL distribution can be composed as pursues The natural logarithm of the product spacing function is To obtain the normal equations for the unknown parameters, we differentiate Equation (8) partially with respect to the parameters Φ and equate them to zero. The estimatorsΦ of Φ can be obtained as the solution of the following equations.
The above nonlinear equations cannot be solved analytically so, we can use any iterative procedure techniques such as Nelder-Mead type algorithms to obtain as the numerical solution. Swain et al. (1988) presented the OLS method for parameter estimation of distribution, it dependent on the observed sample x 1 < · · · < x n from be n ordered random sample of any distribution with CDF, where F (.) denotes the CDF, we get

Weighted and Ordinary Least Square Method
The WLS method can be written as follows, If w i = 1, then the estimate is OLS; if w i = (n+1) 2 (n+2) i(n−i+1) , then the estimate is WLS. The WLS method of the PL distribution can be written as follows After differentiating Equation (11) with respect to parameters Φ and then equating them to zero, we will get the following: The above nonlinear equations cannot be solved analytically so, we can use any iterative procedure techniques such as Nelder-Mead type algorithms to obtain as the numerical solution.

Method of Cramér-von Mises
The Cramér-von Mises estimatesΦ are obtained by minimizing with respect to Φ the function: The estimators can also be obtained by solving the following nonlinear equations: The above nonlinear equations cannot be solved analytically so, we can use any iterative procedure techniques such as Nelder-Mead type algorithms to obtain as the numerical solution.

Bayesian Estimation
In this sectio, the Bayes estimate using square error and Linex loss functions of the unknown parameters Φ of the PL distribution will be obtained. The Bayes estimates is considered under the assumption that the random variables Φ have an independent gamma distribution. Assumed that Φ j ∼ Gamma(a j , b j ) then, the joint prior density of Φ can be written as here all the hyper parameters a j and b j are known and non-negative. For the choice of hyperparameters, the experimenters can incorporate their prior guess in terms of location and precision for the parameter of interest. Such that Combining Equation (3) and Equation (13) to obtain the posterior density of Φ take the following form Therefore, the Bayes estimates of the unknown parameters Φ under square error denoted byΦ; can be calculated through the following equations as follows The Linex loss function has been introduced by Varian (1975). One of the most commonlyused asymmetric loss functions is the Linex loss function For V > 0, the overestimation is more serious than underestimation, for V < 0, the underestimation is more serious than the overestimation, and for V closed to zero, the Linex loss is approximately squared error loss and therefore almost symmetric. Almetwaly and Almongy (2018) used the Bayes estimator of Φ j , denoted byΦ j under Linex loss function has been introduced by Zellner (1986). The Linex loss function is given as following For this situation, we utilize the MCMC strategy to generate samples from the posterior distributions and after that register the Bayes estimators of the individual parameters. In the Markov chain Monte Carlo (MCMC) technique can be used to generate samples from the posterior density function. The joint posterior density functions of Φ j ; j = 1, 2, 3 can be written as For the PL distribution distributions, the full conditional posterior distributions of the parameters are given by Since the full conditional posterior distributions do not have simple forms in perspective of sampling, we use the Metropolis-Hastings algorithm within each Gibbs chain by Metropolis et al. (1953) and see example Almetwally et al. (2018) discussed Bayesian estimation for square error and Linex loss function have been used based on MCMC to estimate parameter of the Weibull Generalized Exponential Distribution under progressive censoring schemes.
In our simulation study presented in the next section, MCMC procedure is used to generate the full conditional posterior distributions. We set the number of periods in the MCMC process to be M = 10, 000.

Stress-Strength Model
For the stress-strength parameter of PL distribution. Let X and Y are the independent strength and stress random variables observed from PL. Then, the stress-strength reliability R is calculated as: Then, we haveR Rady et al. (2016) introduced the stress strength parameter R given as followŝ   We propose the method of MPS for reliability estimation of stress-strength model as alternative method to MLE and Bayesian estimation methods.
We will use possible classical estimation methods and Bayesian estimation methods for parameter estimation of the stress-strength model in case of complete sample of observations.

Maximum Likelihood Estimation
The likelihood function of PL distribution for stress-strength model is and the log-likelihood function is given as To obtain the normal equations for the unknown parameters, we differentiate Equation (16) partially with respect to the parameters B where B = (Φ 1 , Φ 2 ) = (α 1 , β 1 , λ 1 , α 2 , β 2 , λ 2 ) and equate them to zero. The estimatorsB can be obtained as the simultaneous solutions of the equations

Maximum Product Spacing
The Maximum Product Spacing for stress-strength model is denoted as following The natural logarithm of the product spacing function of the exponential distribution for stress-strength model is denoted as following To obtain the normal equations for the unknown parameters, we differentiate Equation (20) partially with respect to the parameters B and equate them to zero. The estimatorsB can be obtained as the solution of the following equations.

Bayesian Estimation
In this subsection, we discuss the Bayesian inference of the unknown parameters of a PL distribution under stress-strength model. For Bayesian parameters estimation we will consider squared error loss function. When the parameters of the model are unknown, a joint conjugate prior for the parameters does not exist. We suggest using independent gamma priors for B having pdfs. The joint prior density of B can be written as By using MCMC technique generate samples from the posterior density function. The joint posterior density functions of B can be written as

Simulation Study
In this portion, we give a complete algorithm of Monte-Carlo simulation study. We clarify our calculation through an application of PL distribution, particularly; we will utilize Monte-Carlo simulation study to discuss two aims. Firstly, in the simulation of Section 2, the classical estimation methods and Bayesian estimation techniques are discussed to obtain the best estimation method of PL distribution. We can generate simulation by using R program. In this case, we must follow the following steps by order: Step 1: Suppose different values of the parameters vector of PL distribution.
Step 3: Generate the sample random values of PL distribution by using quantile function Step 4: Solve differential equations for each estimation methods, to obtain the estimators of the parameters for PL distribution, we calculateΦ.
Step 5: Repeat this experiment (L − 1) times. In each experiment, the same values of the parameters. It is certain that, the values of generating random are varying from experiment to experiment even though n are not changed.
In the end, we have L-values of Bias and MSE, we restricted the number of repeat this experiment to 10000. Take the averages of these values and call them Monte Carlo estimates: whereΦ is the estimated value of Φ, Bias = Mean(Φ − Φ), and the mean squared error (MSE) of the estimator. MSE = Mean(Φ − Φ) 2 .
Step 3: Generate the sample random values of PL distribution by using quantile function Step 4: Solve differential equations in Section 3, to obtain the estimators of the parameters for PL distribution under stress-strength model, we calculateB.
Step 5: Repeat this experiment (L) times. In each experiment, the same values of the parameters.
In the end, we have L-values of mean and MSE, we restricted the number of repeat this experiment to 10000. Take the averages of these values and call them Monte Carlo estimates: whereB is the estimated value of B, and the mean squared error (MSE) of the estimator.
The following conclusions can be drawn from these Table 1-5 and Figure 3-4. 1. All the estimates reveal the property of consistency, i.e., the Bias and MSE decrease when n increase. 2. Keeping β and λ, the Bias and MSE ofα increases and the Bias and MSE ofβ decreases when α increases as shown in Table 1. 3. Keeping α, λ, the Bias and MSE ofβ increases and the Bias and MSE ofα andλ decreases when β increases as shown in Table 2. 4. In most cases, keeping α, and β, the Bias and MSE ofλ increases and the Bias and MSE of β decreases when λ increases as shown in Table 3.

Conclusion
In this paper, parameters estimation for the PL distribution are discussed based on the classical methods and the Bayesian methods. Classical methods are maximum likelihood, maximum product spacing, Ordinary Least Squares, Weighted Least Squares and Cramér-von Mises. In parameter estimation, the estimators based on MPS method behave quite better than the estimators of the classical methods, but the Bayesian estimation is the best one. In Bayesian estimation, the estimators based on LINEX loss function behave quite better than the estimators of the SE loss function, where the MSE is less than from the other methods. In reliability estimation of stress-strength model, we note that when parameters value of stress are small, the estimated model efficiency increases. By checking the previous results, where we note that MPS is better than MLE. We can conclude that the MPS method is a good alternative method to the usual MLE method in many situation, but the Bayesian estimation is the best one in reliability estimation of stress-strength model. Finally, we hope that the finding in this paper will be useful for researchers and statistician.