A Comparison Between Two Shape Parameters Estimators for (Burr-XII) Distribution

This paper deals with defining Burr-XII, and how to obtain its p.d.f., and CDF, since this distribution is one of failure distribution which is compound distribution from two failure models which are Gamma model and weibull model. Some equipment may have many important parts and the probability distributions representing which may be of different types, so found that Burr by its different compound formulas is the best model to be studied, and estimated its parameter to compute the mean time to failure rate. Here Burr-XII rather than other models is consider because it is used to model a wide variety of phenomena including crop prices, household income, option market price distributions, risk and travel time. It has two shape-parameters (α, r) and one scale parameter (λ) which is considered known. So, this paper defines the p.d.f. and CDF and derives its Moments formula about origin, and also derive the Moments estimators of two shapes parameters (α, r) in addition to maximum likelihood estimators as well as percentile estimators, the scale parameter (λ) is not estimated (as it is considered known). The comparison between three methods is done through simulation procedure taking different sample size (n=30, 60, 90) and different sets of initial values for (α, r, λ).It is observed that the moment estimators ?̂?momand ?̂?mom are the best estimator with percentage (46%) ,(42%) respectively compared with other estimators. Key word: Burr-XII failure model, Maximum likelihood estimator, Moments estimator, Percentile estimator. Introduction Twelve different methods of cumulative distribution functions are presented by Burr on the data of the lifetime modeling or the data of the survival (1). It is worthy to mention that there are two types from these twelve methods mentioned above which are considered as the most important methods due to their application in the study of biological, industrial, reliability and life testing, and several industrial and economic experiments, these types are Burr Type XII and Burr Type X (2). The Burr type XII distribution became a vital research area for many authors and many studies. Evans and Ragab. 1983 (3) present a Bayes that estimates the shape parameter (α) and the reliability function based on type-II censored samples. Saracoglu et al. 2013 (4) progressive type-II right censored samples are used to obtain the maximum likelihood, weighted least squares ,ordinary least squares, and best linear unbiased estimators for the shape parameter α . According to Abuzaid 2015 (5),middle-censoring is considered as a modern general scheme of censoring and studying the analysis of middle-censored data with Burr-XII distribution which is considered one of the most popular and flexible distributions for modeling stochastic events and lifetime for many products. Nasser and et al.2016(6) introduced an adaptive type-II progressive hybrid censoring scheme that is used to obtain the maximum likelihood and Bayesian estimation for the unknown parameters of the Burr type XII distribution and Bayes estimates of the unknown parameters The objective of this paper is to estimate two parameters (α, r) ,where scale parameter (λ) is Open Access Baghdad Science Journal P-ISSN: 2078-8665 2020, 17(3) Supplement (September):973-979 E-ISSN: 2411-7986 974 considered known of the Burr XII distribution by the three different types of estimators Moments, Maximum likelihood as well as percentile estimators . The paper is presented as follows: Section 2, gives an introduction about (Burr-XII), finding this p.d.f. and its cumulative CDF, and discuss the Moments, Maximum likelihood as well as percentile estimators for the two parameters (α, r) ,the scale parameter (λ) is not estimated (considered known). Section 3 focuses on the results and compares between three methods through simulation procedure. Section 4 covers some conclusions from the results. Theoretical Aspect The compound p.d.f. of Burr-XII distribution can be obtained by compounding (p.d.f. of Gamma)distribution with (p.d.f. of weibull) distribution. The formula for the probability density function of the Weibull distribution is fY(y) = β α y e α , y>0, α, β >0 ... (1) , i.e. y be r.v ͠ weibull(α, β) where, α is the shape parameter and β is the scale parameter and, the formula for the probability density function of the gamma distribution is f(β) = λr ɼ(r) β e , β > 0, λ, r > 0 ...(2) , i.e. β be r.v ͠ Gamma(r,λ) where, r is the shape parameter and λ is the scale parameter since y be r.v ͠ weibull(α, β) and one of its parameter β be r.v ͠ Gamma(r,λ) then , y has a compound density function is (7) f(y) = ∫ f(y|β). f(β)


Introduction
Twelve different methods of cumulative distribution functions are presented by Burr on the data of the lifetime modeling or the data of the survival (1). It is worthy to mention that there are two types from these twelve methods mentioned above which are considered as the most important methods due to their application in the study of biological, industrial, reliability and life testing, and several industrial and economic experiments, these types are Burr Type XII and Burr Type X (2). The Burr type XII distribution became a vital research area for many authors and many studies. Evans and Ragab. 1983 (3) present a Bayes that estimates the shape parameter (α) and the reliability function based on type-II censored samples. Saracoglu et al. 2013 (4) progressive type-II right censored samples are used to obtain the maximum likelihood, weighted least squares ,ordinary least squares, and best linear unbiased estimators for the shape parameter α . According to Abuzaid 2015 (5),middle-censoring is considered as a modern general scheme of censoring and studying the analysis of middle-censored data with Burr-XII distribution which is considered one of the most popular and flexible distributions for modeling stochastic events and lifetime for many products. Nasser and et al.2016 (6) introduced an adaptive type-II progressive hybrid censoring scheme that is used to obtain the maximum likelihood and Bayesian estimation for the unknown parameters of the Burr type XII distribution and Bayes estimates of the unknown parameters The objective of this paper is to estimate two parameters (α, r) ,where scale parameter (λ) is where, α is the shape parameter and β is the scale parameter and, the formula for the probability density function of the gamma distribution is , i.e. β be r.v ͠ Gamma(r,λ) where, r is the shape parameter and λ is the scale parameter since be r.v ͠ weibull(α, β) and one of its parameter β be r.v ͠ Gamma(r,λ) then , has a compound density function is (7) Where, ( |β) is a conditional density function depending on the parameter β ∴f(y | β, r, λ) = After some steps, (4) is (Burr-XII) distribution with (λ) is scale parameter and (r, α) are shape parameters.

Moments derivation
The (mth) moments formula about origin is Then: and this gives (8) And from equations: According to given values of equation (9) can be solved to obtain ̂ and ̂

Maximum Likelihood Estimator
Maximum likelihood estimation (MLE) is a procedure of finding the value of one or more parameters of a statistical model given observations, by finding the parameter values that maximize the likelihood of making the observations give the parameters. The maximum likelihood estimator is widely used in practice largely because of its conceptual simplicity. Now, let y1, y2, …. yn be a r.s. from p.d.f in equation (4), then: Since the scale parameter (λ) known is considered so , do not find and (̂) may be found from mean time to failure according to given values α , r.

Percentile estimators
The estimation by this method is obtained from minimizing the total sum squares of difference between cumulative distribution function and its non-Parametric estimator: ̂( | , , ) = +1 , then

Simulation Procedure
In this section, Monte Carlo simulation results have been conducted to examine and compare the performance of three Methods (moment ,maximum likelihood and Percentile) for the unknown shape parameters(α, r) considering scale parameter λ constant respecting to their MSE values with different cases and different sample sizes n=30,60,90.For a given values of(r, α, λ) ,generated a random sample ,say as Burr-XII distribution through the adoption inverse transformation method The results of the simulation study are summarized and tabulated in Table 1, Table 2    after applying the best estimators of (r, α) (λ is known) and also estimating E( 2 ), where these are necessary to obtain estimated (variance), which is important for finding confidence internal of estimators.