Abstract
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 are the best estimator with percentage (46%) ,(42%) respectively compared with other estimators.
Highlights
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)
Evans and Ragab. 1983 (3) present a Bayes that estimates the shape parameter (α) and the reliability function based on type-II censored samples
The maximum likelihood estimator is widely used in practice largely because of its conceptual simplicity
Summary
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). 1983 (3) present a Bayes that estimates the shape parameter (α) and the reliability function based on type-II censored samples. 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). (1) , i.e. y be r.vweibull(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 βr−1 e−λβ , β > 0, ɼ(r). Where, r is the shape parameter and λ is the scale parameter since y be r.vweibull(α, β) and one of its parameter β be r.v Gamma(r,λ) , y has a compound density function is (7)
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