Hostname: page-component-6bf8c574d5-9nwgx Total loading time: 0 Render date: 2025-02-21T21:23:46.031Z Has data issue: false hasContentIssue false
  • English
  • Français

RELIABILITY STUDIES OF BIVARIATE BIRNBAUM–SAUNDERS DISTRIBUTION

Published online by Cambridge University Press:  20 January 2015

Ramesh C. Gupta
Ramesh C. Gupta*
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, Maine 04469-5752, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the bivariate Birnbaum–Saunders (BVBS) distribution from a reliability point of view. The monotonicity of the hazard rates of the univariate as well as the conditional distributions is discussed. Clayton's association measure is obtained in terms of the hazard gradient and its value in the case of the BVBS distribution is derived. The probability distributions, in the case of series and parallel systems, are derived and the monotonicity of the failure rate, in the case of series system, is discussed.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 2 , April 2015 , pp. 265 - 276
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Balakrishnan, N., Gupta, R.C., Kundu, D., Leiva, V. & Sanhueza, A. (2010). On some mixture models based on the Birnbaum–Saunders distribution and associated inference. Journal of Statistical Planning and Inference 141: 2175–2190.CrossRefGoogle Scholar
2.Barlow, R.E. & Proschan, F. (1975). Statistical Theory and Life Testing. New York: Holt, Rinehart and Winston Inc.Google Scholar
3.Barlow, R.E., Marshall, A.W. & Proschan, F. (1963). Properties of probability distributions with monotone hazard rate. Annals of Mathematical Statistics 34: 375–389.Google Scholar
4.Birnbaum, Z.W. & Saunders, S.C. (1969). A new family of life distributions. Journal of Applied Probability 6: 319–327.Google Scholar
5.Clayton, D.G. (1978). A model for association in bivariate life tables and its applications in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65: 141–151.CrossRefGoogle Scholar
6.Desmond, A.F. (1985). Stochastic models of failure in random environments. Canadian Journal of Statistics 13: 171–183.CrossRefGoogle Scholar
7.Elandt-Johnson, R.C. & Johnson, N.L. (1980). Survival Models and Data Analysis. New York: John Wiley and Sons.Google Scholar
8.Gaynor, J.J., Feuer, E.J., Tan, C.C., Wu, D.H., Little, C.R., Straus, D.J., Clarkson, B.D. & Brennan, M.F. (1993). On the use of cause-specific failure and conditional failure probabilities: example from clinical oncology data. Journal of the American Statistical Association 88(422): 400–409.CrossRefGoogle Scholar
9.Glaser, R.E. (1980). Bath tub and related failure rate characterizations. Journal of the American Statistical Association 75: 667–672.Google Scholar
10.Gupta, P.L. & Gupta, R.C. (1997). On the multivariate normal hazard. Journal of Multivariate Analysis 62(1): 64–73.Google Scholar
11.Gupta, P.L. & Gupta, R.C. (2001). Failure rate of the minimum and maximum of a multivariate normal distribution. Metrika 53: 39–49.Google Scholar
12.Gupta, R.C. (1979). Some counter examples in the competing risk analysis. Communications in Statistics A 8(15): 1535–1540.CrossRefGoogle Scholar
13.Gupta, R.C. (1987). On the monotonic properties of the residual variance and their applications in reliability. Journal of Statistical Planning and Inference 16: 329–335.Google Scholar
14.Gupta, R.C. & Akman, O. (1995). On the reliability studies of weighted inverse Gaussian distribution. Journal of Statistical Planning and Inference 48: 69–83.CrossRefGoogle Scholar
15.Gupta, R.C. & Akman, O. (1997). Estimation of critical points in the mixture Inverse Gaussian mode l. Statistical Papers 38: 445–452.Google Scholar
16.Gupta, R.C. & Arnold, B. (2014). Preservation of failure rate function shape in weighted distributions. Under review.Google Scholar
17.Gupta, R.C. & Warren, R. (2001). Determination of change points of non-monotonic failure rates. Communications in Statistics, 30: 1903–1920.CrossRefGoogle Scholar
18.Hanley, J.A. & Parnes, M.N. (1983). Nonparametric estimation of mutivariate distribution in the presence of censoring. Biometrika 39: 129–139.CrossRefGoogle Scholar
19.Holland, P.W. & Wang, Y.J. (1987). Dependence function for continuous bivariate densities. Communications in Statistics: Theory and Methods 16: 863–876.Google Scholar
20.Johnson, N.L. & Kotz, S. (1975). A vector valued multivariate hazard rate. Journal of Multivariate Analysis 5: 53–66.Google Scholar
21.Johnson, N.L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, vol. 2, 2nd ed.New York: John Wiley & Sons.Google Scholar
22.Kundu, D., Kannan, N. & Balakrishnan, N. (2008). On the hazard function of Birnbaum–Saunders distribution and associated inference. Computational Analysis and Data Analysis 52: 2692–2702.Google Scholar
23.Kundu, D., Balakrishnan, N. & Jamalizadeh, A. (2010). Bivariate Birnbaum–Saunders distribution and associated inference. Journal of multivariate Analysis 101: 113–125.Google Scholar
24.Manatunga, A.K. & Oakes, D. (1996). A measure of association for bivariate frailty distributions. Journal of Multivariate Analysis 56: 60–74.CrossRefGoogle Scholar
25.Marshall, A.W. & Olkin, I. (2007). Life Distributions. New York: Springer-Verlag.Google Scholar
26.McGill, J.I. (1992). The multivariate hazard gradient and moments of the truncated multinormal distribution. Communications in Statistics, Theory and Methods 21(11): 3053–3060.Google Scholar
27.Ng, H.K.T., Kundu, D. & Balakrishnan, N. (2003). Modified momemt estimation for the two parameter Birnbaum–Saunders distribution. Computational Statistics and Data Analysis 43: 283–298.Google Scholar
28.Ng, H.K.T., Kundu, D. & Balakrishnan, N. (2006). Point and interval estimations for the two parameter Birnbaum–Saunders distribution based on type II censored samples. Computational Statistics and Data Analysis 50: 3222–3242.CrossRefGoogle Scholar
29.Oakes, D. (1989). Bivariate survival models induced by fralities. Journal of the American Statistical Association 84: 487–493.Google Scholar
30.Shaked, M. (1977). A family of concepts of dependence for bivariate distributions. Journal of the American Statistical Association 72: 642–650.Google Scholar