Comparison of Different Estimators of P [Y < X] for a Scaled Burr Type X Distribution

Abstract

In this article, we consider the estimation of P[Y < X], when Y and X are two independent scaled Burr Type X distribution having the same scale parameters. The maximum likelihood estimator and its asymptotic distribution is used to construct an asymptotic confidence interval of P[Y < X]. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator, and approximate Bayes estimators of P[Y < X] are discussed. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations. One data set has been analyzed for illustrative purposes.

Keywords:

• Asymptotic distributions
• Bayes estimator
• Bootstrap confidence intervals
• Credible intervals
• Maximum likelihood estimator
• Stress-strength model

Mathematics Subject Classification:

• 62F10
• 62E20
• 62F12
• 62F15

Acknowledgments

The authors would like to thank one referee for his/her very valuable comments.

Notes

The first rows represent the average biases and the corresponding MSEs are reported within brackets. The second, third, and fourth rows represent the average lengths and corresponding coverage percentages of the asymptotic, boot-p, and boot-t confidence intervals. The fifth row represents the average lengths and corresponding coverage percentages bases on the formula (4.1), simply putting λ = λˆ.

The first, second, third, fourth, and fifth rows represent the biases and the corresponding MSEs by MLEs, UMVUES, approximate Bayes (with respect to 0-1 loss function), approximate Bayes (Lindley's approximation), and the Bayes estimators (with respect to the squared error loss function) are reported within brackets

The first and second rows represent the confidence intervals based on (4.1) using the estimate of R as MLE or UMVUE. The third rows represent the average HPD intervals and the corresponding coverage percentages based on MCMC.

The first rows represent the average biases and the corresponding MSEs are reported within brackets. Second, third, and fourth rows represent the average lengths and the corresponding coverage percentages of the asymptotic, boot-p, and boot-t confidence intervals. The fifth row represents the average lengths and corresponding coverage percentages bases on the formula (4.1), simply putting λ = λˆ.

The first, second, third, fourth, and fifth rows represent the biases and the corresponding MSEs by MLEs, UMVUES, approximate Bayes (with respect to 0-1 loss function), approximate Bayes (Lindley's approximation), and the Bayes estimators (with respect to the squared error loss function) are reported within brackets.

The first and second rows represent the confidence intervals based on (4.1) using the estimate of R as MLE or UMVUE. The third row represents the average HPD intervals and the corresponding coverage percentages based on MCMC.