Stochastic comparisons for time transformed exponential models

Abstract

a b s t r a c tDifferent sufficient conditions for stochastic comparisons between random vectors have been described in the literature. In particular, conditions for the comparison of random vectors having the same copula, i.e., the same dependence structure, may be found in Müller and Scarsini (2001). Here we provide conditions for the comparison, in the usual stochastic order sense and in other weaker stochastic orders, of two time transformed exponential bivariate lifetimes having different copulas. Some examples of applications are provided too.

Stochastic orders in time transformed exponential models with applications

“…Comparison of such expectations are conducted based on some stochastic orders of the involved univariate survival functions in the models, and also the upper orthant-convex order between two bivariate random risks in TTE models is built. This corrects Theorem 2.3 of Mulero et al (2010) and invalidates some results there. Some applications in actuarial science are presented as well.…”

“…In particular, the upper orthant-convex order between two vectors in TTE models is built as well. This result corrects Theorem 2.3 of Mulero et al (2010), which was proved in an incorrect way and misled some invalid results. At last, Section 3 presents some applications of the main results in actuarial science.…”

“…(ii) On the other hand, Theorem 2.3 of Mulero et al (2010) also claims in the bracket statement that E[g 1 (X 1 )g 2 (X 2 )] ≤ E[g 1 (Y 1 )g 2 (Y 2 )] still holds if R is convex and one of two increasing functions g 1 , g 2 is concave.…”

“…For the case with g(x 1 , x 2 ) = g 1 (x 1 ) · g 2 (x 2 ), Theorem 2.3 in Mulero et al (2010) obtains a better result in the sense that g 1 and g 2 are not necessarily absolute continuous. However, the proof is incorrect, and just as is demonstrated by some counterexamples in the sequel, this misleads the statement in brackets as well as Theorem 2.4 there, which is actually invalid.…”

On dynamic mutual information for bivariate lifetimes

“…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].…”