Modelling lifetimes with bivariate Schur-constant equilibrium distributions from renewal theory
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Modelling lifetimes with bivariate Schur-constant equilibrium distributions from renewal theory N. Unnikrishnan Nair · P. G. Sankaran
Received: 30 August 2013 / Accepted: 8 May 2014 © Sapienza Università di Roma 2014
Abstract In the present work we study the asymptotic distribution of the age and residual life in a renewal process. The bivariate distribution so derived is Schur-constant with marginal distributions as equilibrium discuss the reliability properties and the copula. The bivariate ageing properties and some stochastic orders connecting them are explored. It is shown that various time dependent measures of association and dependence concepts can be inferred from the ageing properties of the baseline distribution. Keywords Bivariate equilibrium distribution · Schur-constancy · Archimedean copula · Bivariate ageing · Dependence concepts
1 Introduction The concept of univariate equilibrium distribution originated from renewal theory as the asymptotic distribution of the age or residual life in a renewal process [12]. If X is a nonnegative random variable representing lifetime of a component with finite mean μ = E(X ) ¯ and survival function F(x) = P(X > x), the corresponding equilibrium distribution is specified by the survival function F¯1 (x) =
∞ ¯ F(t) dt. μ
(1)
x
¯ The relationship between F(x) and F¯1 (x) was exploited in reliability to establish many results such as characterizations [15,24], new concepts of ageing [2,13], stochastic orderings [35] etc. A review of the results on the topic and additional references are available in [17].
N. U. Nair · P. G. Sankaran (B) Department of Statistics, Cochin University of Science and Technology, Cochin 682 022, India e-mail: [email protected]; [email protected] N. U. Nair e-mail: [email protected]
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N. U. Nair, P. G. Sankaran
There have been some attempts in literature to extend the concept of univariate equilibrium distribution to higher dimensions. These are reviewed separately in Sect. 2. In the present work, we focus attention on the properties of the asymptotic joint distribution of age and residual life represented by the survival function. 1 G¯ 1 (x, y) = μ
∞
¯ F(t)dt
(2)
x+y
arising from renewal theory [14]. The distribution (2) is a model of chance variations in the residual life of a device in relation to its age and therefore it has direct relevance to lifelength studies. Since the marginal survival functions of (2) are F¯1 (x) and F¯1 (y) defined in (1), we can consider (2) as an exchangeable bivariate equilibrium distribution. Sankaran et al. [34] have characterized the bivariate exponential, Lomax and Dirichlet distributions using some identities connecting the bivariate hazard rate and mean residual life of (2). An alternative definition of the mean residual life was proposed by [25] as the mean value of the conditional distribution of residual life given the age from the joint distribution of (2). A new class of life distributions and comparison of the members therein through a stochastic order were also
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