Abstract
The profile likelihood of the reliability parameter θ = P(X < Y) or of the ratio of means, when X and Y are independent exponential random variables, has a simple analytical expression and is a powerful tool for making inferences. Inferences about θ can be given in terms of likelihood-confidence intervals with a simple algebraic structure even for small and unequal samples. The case of right censored data can also be handled in a simple way. This is in marked contrast with the complicated expressions that depend on cumbersome numerical calculations of multidimensional integrals required to obtain asymptotic confidence intervals that have been traditionally presented in scientific literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc B 26: 211–252
Chao A (1982) On comparing estimators of P(Y < X) in the exponential case. IEEE Trans Reliab 31: 389–392
Cramer E, Kamps U (1997) A note on UMVUE of Pr (X < Y) in the exponential case. Commun Stat Theory Methods 26: 1051–1055
Díaz-Trujillo A, Contreras J, Medina A, Silveyra-León G, Antaramian A, Quirarte G, Prado-Alcalá R (2009) Enhanced inhibitory avoidance learning prevents the long-term memory-impairing effects of cycloheximide, a protein synthesis inhibitor. Neurobiol Learn Mem 91: 310–314
Enis P, Geisser S (1971) Estimation of the probability that Y > X. J Am Stat Assoc 66: 162–168
Ismail R, Jeyaratnan S, Panchapakesan S (1986) Estimation of Pr[X > Y] for gamma distributions. J Stat Comput Simul 26: 253–267
Jiang L, Wong ACM (2008) A note on inference for P(X < Y) for right truncated exponentially distributed data. Stat Pap 49: 637–651
Kalbfleisch JG (1985) Probability and statistical inference. Springer, New York
Kelley GD, Kelley JA, Schucany WR (1976) Efficient estimation of P(Y < X) in the exponential case. Technometrics 18: 359–360
Kotz S, Lumelskii Y, Pensky M (2003) The stress–strength model and its generalizations: theory and applications. World Scientific, River Edge
Krzanowski WJ, Hand DJ (2009) ROC curves for continuous data. CRC Press, Boca Raton
Kundu D, Gupta RD (2005) Estimation of P[Y < X] for generalized exponential distribution. Metrika 61: 291–308
Lawless JF (2003) Statistical models and methods for lifetime data. Wiley, Hoboken
Montoya JA, Díaz-Francés E, Sprott DA (2009) On a criticism of the profile likelihood function. Stat Pap 50: 195–202
Saracoglu B, Kinaci I, Kundu D (2011) On estimation of R = P(Y < X) for exponential distribution under progressive type-II censoring. J Stat Comput Simul. doi:10.1080/00949655.2010.551772
Sathe YS, Shah SP (1981) On estimating P(X > Y) for the exponential distribution. Commun Stat Theory Methods 10: 39–47
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Sprott DA (2000) Statistical inference in science. Springer, New York
Sullivan Pepe M (2000) An interpretation for the ROC curve and inference using GLM procedures. Biometrics 56: 352–359
Tong H (1974) A note on the estimation of P(Y < X) in the exponential case. Technometrics 16: 625
Tong H (1975) Errata: a note on the estimation of P(Y < X) in the exponential case. Technometrics 17: 395
Ventura L, Racugno W (2011) Recent advances on Bayesian inference for P(Y < X). Bayesian Anal 6: 1–18
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Díaz-Francés, E., Montoya, J.A. The simplicity of likelihood based inferences for P(X < Y) and for the ratio of means in the exponential model. Stat Papers 54, 499–522 (2013). https://doi.org/10.1007/s00362-012-0446-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-012-0446-1
Keywords
- Comparison of exponential distributions
- Exponential right censored data
- Exponential stress–strength models
- Profile likelihood of reliability parameter
- ROC curves