Stochastic orders in time transformed exponential models with applications

Li, Xiaohu; Lin, Jianhui

doi:10.1016/j.insmatheco.2011.02.002

“…the involved densities being defined in (9), (22), and (23). Since X and Y describe the random lifetimes of two systems, M X,Y (s, t) measures the dependence between their remaining lifetimes at different ages s and t. See the analogy between (24) and the mutual information of…”

Section: Mutual Information For Residual Lifetimesmentioning

confidence: 99%

“…We recall that a pair of random lifetimes (X, Y ) is said to follow the time-transformed exponential (TTE) model if its joint survival function may be expressed in the following way: [4], [22], [24], [28], and [31]. Hereafter, we investigate the bivariate dynamic residual mutual information within the TTE model.…”

Section: Dynamic Mutual Information For the Time-transformed Exponentmentioning

confidence: 99%

See 1 more Smart Citation

On dynamic mutual information for bivariate lifetimes

Ahmadi

¹

,

Crescenzo

²

,

Longobardi

³

2015

Advances in Applied Probability

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We consider dynamic versions of the mutual information of lifetime distributions, with a focus on past lifetimes, residual lifetimes and mixed lifetimes evaluated at different instants. This allows us to study multicomponent systems, by measuring the dependence in conditional lifetimes of two components having possibly different ages. We provide some bounds, and investigate the mutual information of residual lifetimes within the time-transformed exponential model (under both the assumptions of unbounded and truncated lifetimes). Moreover, with reference to the order statistics of a random sample, we evaluate explicitly the mutual information between the minimum and the maximum, conditional on inspection at different times, and show that it is distribution-free in a special case. Finally, we develop a copula-based approach aiming to express the dynamic mutual information for past and residual bivariate lifetimes in an alternative way.

show abstract

“…We recall that a pair of random lifetimes (X, Y ) is said to follow the time-transformed exponential (TTE) model if its joint survival function may be expressed in the following way: [4], [22], [24], [28], and [31].…”

Section: Dynamic Mutual Information For Time-transformed Exponential mentioning

confidence: 99%

“…Under the given assumptions the densities in (9), (22), and (23) can still be expressed respectively as in (38) for 0 ≤ x ≤ R −1 1 (ω −R 2 (t))−s, as in (39) for 0 ≤ y ≤ R −1 2 (ω −R 1 (s))−t, and as in (40) for all nonnegative x, y such that R 1 (x + s) + R 2 (y + t) ≤ ω. Note that the assumption W ′ (ω) = 0 is essential to ascertain that the integral of f X,Y (x, y; s, t) is unity.…”

Section: Propositionmentioning

confidence: 99%

On dynamic mutual information for bivariate lifetimes

Ahmadi

¹

,

Crescenzo

²

,

Longobardi

³

2015

View full text Add to dashboard Cite

We consider dynamic versions of the mutual information of lifetime distributions, with a focus on past lifetimes, residual lifetimes and mixed lifetimes evaluated at different instants. This allows us to study multicomponent systems, by measuring the dependence in conditional lifetimes of two components having possibly different ages. We provide some bounds, and investigate the mutual information of residual lifetimes within the time-transformed exponential model (under both the assumptions of unbounded and truncated lifetimes). Moreover, with reference to the order statistics of a random sample, we evaluate explicitly the mutual information between the minimum and the maximum, conditional on inspection at different times, and show that it is distribution-free in a special case. Finally, we develop a copula-based approach aiming to express the dynamic mutual information for past and residual bivariate lifetimes in an alternative way.

show abstract