### Journals Information

**
Mathematics and Statistics Vol. 9(4), pp. 527 - 534 DOI: 10.13189/ms.2021.090412 Reprint (PDF) (238Kb) **

## Inference on P[Y < X] for Geometric Extreme Exponential Distribution

**Reza Pakyari ^{*}**

Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar

**ABSTRACT**

Geometric Extreme Exponential Distribution (GEE) is one of the statistical models that can be useful in fitting and describing lifetime data. In this paper, the problem of estimation of the reliability R = P(Y < X) when X and Y are independent GEE random variables with common scale parameter but different shape parameters has been considered. The probability R = P(Y < X) is also known as stress-strength reliability parameter and demonstrates the case where a component has stress X and is subjected to strength Y. The reliability R = P(Y < X) has applications in engineering, finance and biomedical sciences. We present the maximum likelihood estimator of R and study its asymptotic behavior. We first study the asymptotic distribution of the maximum likelihood estimators of the GEE parameters. We prove that the maximum likelihood estimators and so the reliability R have asymptotic normal distribution. A bootstrap confidence interval for R is also presented. Monte Carlo simulations are performed to assess he performance of the proposed estimation method and validity of the confidence interval. We found that the performance of the maximum likelihood estimator and also the bootstrap confidence interval is satisfactory even for small sample sizes. Analysis of a dataset has been given for illustrative purposes.

**KEYWORDS**

Asymptotic Distribution, Bootstrap Confidence Interval, Geometric Extreme Exponential Distribution, Maximum Likelihood Estimation, Stress-strength Model

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Reza Pakyari , "Inference on P[Y < X] for Geometric Extreme Exponential Distribution," Mathematics and Statistics, Vol. 9, No. 4, pp. 527 - 534, 2021. DOI: 10.13189/ms.2021.090412.

(b). APA Format:

Reza Pakyari (2021). Inference on P[Y < X] for Geometric Extreme Exponential Distribution. Mathematics and Statistics, 9(4), 527 - 534. DOI: 10.13189/ms.2021.090412.