Estimation of Constant Stress Partially Accelerated Life Test for Fréchet Distribution with Type-I Censoring

Rehman Ahmad Khan Sherwani,Wali Khan Mashwani,Abdullah Ali H Ahmadini,Dr Dilbag Singh,Miftahuddin Miftahuddin,Shokrya S Alshqaq,Mohammed Elgarhy,Faiza Razaq,Farrukh Jamal,Sanaa Al-Marzouki,Kamran Abbas

doi:10.1155/2021/9957944

Abstract

Modern reliability engineering accelerated life tests (ALT) and partially accelerated life tests (PALT) are widely used to obtain the timely information on the reliability of objects, products, elements, and materials as well as to save time and cost. The ALTs or PALTs are useful in determining the failed manners of the items at routine conditions by using the information of the data generated from the experiment. PALT is the most sensible method to be used for estimating both ordinary and ALTs. In this research, constant stress PALT design for the Fréchet distribution with type-I censoring has been investigated due to a wide applicability of the Fréchet distribution in engineering problems especially in hydrology. The distribution parameters and acceleration factor are obtained by using the maximum likelihood method. Fisher's information matrix is used to develop the asymptotic confidence interval estimates of the model parameters. A simulation study is conducted to illustrate the statistical properties of the parameters and the confidence intervals by using the R software. The results indicated that the constant stress PALT plan works well. Moreover, a numerical example is given to exemplify the performance of the proposed methods.

Highlights

Constant stress partially accelerated life tests (PALT) design for the Frechet distribution with type-I censoring has been investigated due to a wide applicability of the Frechet distribution in engineering problems especially in hydrology. e distribution parameters and acceleration factor are obtained by using the maximum likelihood method
A simulation study is conducted to illustrate the statistical properties of the parameters and the confidence intervals by using the R software. e results indicated that the constant stress PALT plan works well
We examine that the results support theoretical findings of constant stress PALT (CSPALT) for Frechet distribution

Summary

Model Description e following notations are used for model description:

X (vi) ti observed lifetime of ith object at usual use condition (vii) xj observed lifetime of jth unit at accelerated condition (viii) t(1) ≤ . . . ≤ t(nu) ≤ τ ordered failure time at use condition (ix) x(1) ≤ . . . ≤ x(na) ≤ τ ordered failure time at accelerated condition. ≤ t(nu) ≤ τ ordered failure time at use condition (ix) x(1) ≤ . ≤ x(na) ≤ τ ordered failure time at accelerated condition. (i) Under normal-use condition, lifespan of an object supports Frechet distribution. N(1 − r) of objects allocated to the normal condition are independently and identically distributed (i.i.d) random variables. Let δui and δaj be the indicator functions under the use and accelerated conditions, respectively, such that. Α δaj i 1 θ θ θ θ j 1 θ θ θ βτ − α δaj ×􏼢1 − exp􏼠− 􏼠 􏼡 􏼡􏼣. Let nu and na be the numbers of items that failed at let cu and ca be the numbers of items censored at normal and normal and accelerated conditions, respectively. The asymptotic variance of the MLE can be attained by the inverse of the observed Fisher information matrix: AVar(α􏽢) ACov(α􏽢θ􏽢) ACov(α􏽢β􏽢).

Simulation Study

Numerical Example

Conclusion

Recommendation