Mean Residual Life and Reversed Mean Residual Life

Usage

bd_mrl(t, a, b, c, k, rel.tol = 1e-10)

bd_rmrl(t, a, b, c, k, rel.tol = 1e-10)

Arguments

t: Vector of times.
a: Shape parameter (beta generator).
b: Shape parameter governing the tail.
c: Shape parameter (baseline).
k: Scale parameter (baseline).
rel.tol: Passed to [stats::integrate()].

Value

A numeric vector the length of `t`, or all `NA` when $b \le 1$.

Details

Following the definitions in the underlying dissertation, with $\Upsilon_1$ the upper and $M_1$ the lower incomplete first moment, $$m(t) = E(Z - t \mid Z > t) = \Upsilon_1(t)/S(t) - t,$$ $$\bar m(t) = E(t - Z \mid Z \le t) = t - M_1(t)/F(t).$$ The reversed form is also called the mean inactivity time. Both require the first moment to exist, hence $b > 1$.

No monotonicity is asserted. Ageing classification requires a separate argument and is not implied by these values; see [bd_hazard_shape()] for the hazard-rate counterpart.

Examples

bd_mrl(c(0.5, 1, 2, 5), a = 1.5, b = 3, c = 2, k = 1)
#> [1] 1.591853 1.664506 2.036508 3.443689
bd_rmrl(c(0.5, 1, 2, 5), a = 1.5, b = 3, c = 2, k = 1)
#> [1] 0.1641383 0.4004914 1.0316699 3.5385297