Contaminated Robust Estimator of Weibull Distribution's Reliability Function

Introduction

The Weibull distribution in 1951 was used by researcher Waloddi Weibull for an experimental demonstration to watch the change in iron expansion and use this distribution to measure the change in the service period spent by radio staff. The two-parameter and multi-parameter Weibull distribution that will be mentioned later have been commonly used in the fields of reliability and survival test experiments, so we see their applications in survival distribution and various failures in addition to building models of reliability. Reliability appeared as an important science dealing with the ages of equipment, especially the possibility of survival, average life, and the possibility that the age of the system is greater than a certain time, interest in the subject of reliability has increased after the widespread spread of industry and the increase in complexities in equipment and research before World War II was concerned with quality control and maintenance of machines, and reliability was not dealt with as an independent field. With the beginning of World War II, the field of reliability became an independent and private entity. The fundamental idea behind reliability is to understand how things will behave when they fail for a set amount of time to lower production and maintenance costs. This is especially true in the context of survival theory, as both survival and reliability have the same characteristic of measuring the duration of life, whether it be for an organism, system, or machine. Due to its emphasis on the lifespan of systems and equipment, their potential for failure, and their sustainability over time, this issue is particularly significant in the context of industry and contemporary technology, Therefore, to examine and analyze phenomena following the nature of statistical data and carry out the estimating process using statistical methods, it is necessary to adopt the theory of dependability, including the least squares method, the moment's method, the contraction method, the Bayes method, the method, and others [5] . The issue of anomalous values has been studied by many scientists and researchers, such as Barnett, Hawkins (1980), Freeman (1980), and Lewis (1978) who dealt with the subject of outliers and their effects on the accuracy of the results. In the year (1999), the researcher Sheikh presented a method for estimating observations in simple linear regression models through the residual column after deleting the observation (i ( and using measures of dispersion and mean. Al-Jabari and Nazi (2002) proposed a new method for detecting and estimating anomalous observations of simple linear regression models. In 2002, Al-Kinani used the nonparametric method to diagnose and estimate outliner values in the case of multiple linear models. The appearance of outliers in a set of real data will affect their analysis. [11] This study aims to calculate the Weibull distribution's reliability function using two parameters contaminated by the exponential distribution using two methods and the height of the statistical scale mean of error squares to compare two methods, specifically the weighted least squares method and the Bayes method to determine the optimal approach to estimate that function. [3,5,12] Reliability is usually used to give a job or work, and the broad meaning of it is that it is a measure of performance, as any functional vocabulary needs certain specifications or performance requirements when performing the work required of it or to continue to perform its work under normal operating conditions and it can be said that this vocabulary is reliable, and so it is clear from this concept that the meaning of reliability is related to a measure of the degree of successful performance of the job required under its working conditions. Reliability can be defined in different forms, including:  The probability that a unit (device) performs a certain work under certain conditions and for a specific period.  Alternatively, it is the possibility that a system will operate for a specific period and under the working conditions for which it is designed.

Reliability Function

It is also the possibility of performing a piece of equipment (unit -device) for which it is designed as necessary and during a certain period and under certain operational conditions. This formula is one of the most acceptable definitions of reliability because it is a function of a type of performance in which the device is successful, that is, does not fail during Term service [6] .

For each period (t, t+∆t), let T be a random variable and be the probability failure function, where F(t) is the failure time (t) cumulative distribution function (cdf).

The Reliability Function, represented by 𝑅(𝑡), has the following definition: [2,13] This distribution is one of the important statistical models widely used in failure tests, life duration tests, life duration tests in the field of theory, life tests or decrease in life rate due to increase, and the expansion using this distribution is due to its flexibility when applied to various data in general and has many applications in Life Data, as well as the use of this distribution in the field of reliability and the subject of quality control in general. The distribution of Weibull can be obtained with two parameters. Through the use of the General Gamma Distribution, which was derived by the scientist Stacy, which is one of the most important distributions. It is worth mentioning that the important characteristic of this distribution is that it is a generalization of the many distributions and has a wide area of interest for researchers and scholars of reliability, especially the Kama distribution the Weibull distribution, and the exponential distribution. [8] For a two-parameter Weibull distribution with α > 0 and β > 0, the probability density function and cumulative distribution function are as follows: [10] 𝑓(𝑡, 𝛽, 𝛼

Statistical distribution 3.1 Weibull Distribution

α: Scale parameter or Characteristic life. β: Shape parameter. and CDF, Reliability

Exponential Distribution

Referred to as the exponential function, one kind of continuous probability distribution is the exponential distribution, this distribution is used to estimate the times between occurrences, when it comes to time measurement issues, the exponential distribution is often used, this covers the time spent using the post office, making phone calls, unloading freighters, the time needed to fix a machine and the wait time for a consumer to receive assistance, the exponential distribution is used in the precise sciences to describe how long radioactive atoms last before decaying. [1] If the PDF provided indicates the distribution of the random variable T is exponential, then

and CDF, Reliability [7] With a ratio of , an exponential distribution has contaminated the Weibull distribution, for us to possess the exponential-Weibull distribution 4. The Estimation of Reliability Function 4.1 Weighted Least Squares (WLS) [13] [14] There are many alternative methods for estimating the parameters of the regression model, including Weighted Least Squares, which depend on the weights W i In this research, the Bergman formula was used for these weights, according to Habeeb (2022) as the following: Let them (𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ) be a sample chosen at random to match the distribution of what G(.), furthermore,𝑦 𝑖 , 𝑖 = 1,2, … , 𝑛 stands for the sample's ranking statistics, following which the distribution's 𝐺(𝑦 𝑖 ) prediction and variation are:

Exponential-Weibull

Referring to the findings of researcher Katz's mentioned in [2] We minimize the sum of the following weighted squares using the definition to obtain a WLS estimator.

The estimation of parameters 𝛽 ̂, 𝛿 ̂, 𝛼 ̂ for the Weibull distribution is contaminated with an exponential distribution using the density function for distributions and to determine the estimate of the reliability function;

Then, the Weibull distribution is represented by f 1 (t), while the contaminated distribution (exponential distribution) is represented by f 2 (t), where τ is the formula's reducible level of contamination ( 6)

When we use the partial derivative to equalize (10) for θ, we obtain Equation ( 10) second derivative is as follows:

To determine a parameter's estimate (β, α, δ) in formula (10), the Newton-Raphson method, which is based on formulas (13), and (14), is used to solve this equation numerically as follows:

The only Weighted Least Squares estimator method parameter of landmarks (β, α, δ) is formula (10) as follows:

Stability is used to evaluate the reliability function as follows:

) 𝛽 ̂𝑊𝐿𝑆 + 𝜏(e -𝛿 ̂𝑊𝐿𝑆 t ) (17) [3] A Bayesian estimator is taken into account inside the essential loss function in Bayesian estimates, these loss functions are asymmetric (square error). Bayesian inference requires the assumption of unknown parameters, it can be given the prior distribution shown below.

Weibull distribution contaminated by the exponential distribution using the Robust Bayes Method

The density of the prior

For β, α, and δ, respectively, let the parameters (a, b, p, q,.) take on a Gamma (𝑎,b), Gamma (p, q), and U(0,δ) representations. The priors are taken to be independent.

Thus, given β, α, and δ, the prior joint distribution is g(β, α, δ) ∝ 𝑏 𝑎 𝑞 𝑝 𝛽 𝑎-1 𝛼 𝑝-1 𝑒 -(𝑏𝛽+𝑞𝛼) 𝛿 Γ(𝑝)Γ(𝑎) , ∀𝑏, 𝑎, 𝛼, 𝑞, 𝑝, 𝛽, 𝛿 > 0 (18)

The density of the posterior

Bayesian inference is based on the posterior distribution, By replacing likelihood, we calculate the three parameters' Bayesian estimation L (β, α, δ), as well as the prior joint distribution for β, α, and δ. From g (β, α, δ) in (18) in that order, we obtain the matching posterior joint P (β, α, δ), this is just the marginal distribution function divided by the joint density function of the prior distribution.

The estimators of Bayes

The Weibull distribution contaminated exponential distribution's parameters are estimated in this part using the Bayesian method and quadratic loss functions. The β, α, and δ Bayesian estimators (BQ) are as follows:

The related posterior hazards are:

Under the loss functions, the reliability Bayesian estimator is given by:

Use numerical procedures to calculate these estimators.

Simulation

We determine the default values, Choosing default values for β, α, δ, 𝜏 Let 𝜏 = 15%, 30% For simulation n = (25, 50, 100) virtual sample size, with L=1000 being the recurrence of each experiment The WLS method of formula (17) and the RB method of formula (21) were used to evaluate the Weibull model's reliability function.

The two reliability function estimate techniques are compared in this final step of the MSE adoption process using the following formula:

𝑗=0

The reliability function estimate and MSE for each sample volume are displayed in the following tables. a-Random number generation for the Weibull distribution given two parameters (β, α), and one-parameter exponential distribution (δ), Based on the VB.net language's accessible generating function: t = Weibull ~ (β,α) or t = exp ~ (δ) b-Find the estimators:

Currently, the process of estimating the Weibull-contaminated distribution's reliability function using an exponential distribution is addressed in the theoretical section of this paper using estimating techniques, based on (WLS) and (RB) methods. c-Comparison: Following the discovery of reliability function estimators, which were used to assess the precision of estimate techniques, the method that produces the lowest mean squares error (MSE) result is considered the best-fitting one.

Results

Conclusion

By observing the tables, it is shown that the Bayes (RB) method gave the lowest comparison of (MLS) than the weighted least squares method (WLS), and therefore the Bayes method is better than the weighted least squares method to calculate the reliability of the contamination Weibull distribution at a contamination rate of 15% and 30%, the values of estimating the contamination Weibull distribution's reliability function with the exponential function with a contamination rate of 15% and 30% for both methods and all sizes.

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