Characterizations of Arnold and Strauss’ and related bivariate exponential models

• Published in: Journal of multivariate analysis 2007-08, Vol.98 (7), p.1494-1507
• Main Authors: Kotz, Samuel, Navarro, Jorge, Ruiz, Jose M.
• Format: Article
• Language: English
• Subjects: Applications; Conditional moment; Distribution theory; Exact sciences and technology; Exponential model; Hazard gradient function; Hazard gradient function Mean residual life Conditional moment Exponential model; Mathematical models; Mathematics; Mean residual life; Multivariate analysis; Probability and statistics; Probability theory and stochastic processes; Reliability, life testing, quality control; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistics; Studies
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2006.09.002

Characterizations of probability distributions is a topic of great popularity in applied probability and reliability literature for over last 30 years. Beside the intrinsic mathematical interest (often related to functional equations) the results in this area are helpful for probabilistic and statistical modelling, especially in engineering and biostatistical problems. A substantial number of characterizations has been devoted to a legion of variants of exponential distributions. The main reliability measures associated with a random vector X are the conditional moment function defined by m φ ( x ) = E ( φ ( X ) | X ⩾ x ) (which is equivalent to the mean residual life function e ( x ) = m φ ( x ) - x when φ ( x ) = x ) and the hazard gradient function h ( x ) = - ∇ log R ( x ) , where R ( x ) is the reliability (survival) function, R ( x ) = Pr ( X ⩾ x ) , and ∇ is the operator ∇ = ( ∂ ∂ x 1 , ∂ ∂ x 2 , … , ∂ ∂ x n ) . In this paper we study the consequences of a linear relationship between the hazard gradient and the conditional moment functions for continuous bivariate and multivariate distributions. We obtain a general characterization result which is the applied to characterize Arnold and Strauss’ bivariate exponential distribution and some related models.