individual_frailty: Individual Frailty Estimation for Degradation Units

In IGPFrailty: Inverse Gaussian Process Degradation Models with Frailty

Individual Frailty Estimation for Degradation Units

Description

Computes empirical Bayes / posterior individual frailty estimates E[z_i \mid \text{Data}_i] and posterior variances \text{Var}(z_i \mid \text{Data}_i) for each experimental unit.

Usage

individual_frailty(object, conf_level = 0.95)

Arguments

object	An object of class "igp_fit" fitted with frailty ("gamma" or "ig").
conf_level	Nominal confidence level for posterior frailty intervals. Default is 0.95.

Details

Under the **IGP-Gamma** model, the posterior distribution of the frailty variable given the degradation path is:

\pi(z_i \mid \mathbf{y}_i) \propto z_i^{1/\xi - n_i - 1} \exp\left(-\frac{z_i}{\xi} - \frac{S_{Hi}}{z_i}\right)

which corresponds to a Generalized Inverse Gaussian (GIG) distribution.

Under the **IGP-IG** model, the posterior distribution is:

\pi(z_i \mid \mathbf{y}_i) \propto z_i^{-1.5 - n_i} \exp\left(-\frac{(z_i - 1)^2}{2\xi z_i} - \frac{S_{Hi}}{z_i}\right)

also belonging to the GIG family.

Posterior moments are evaluated analytically using modified Bessel functions of the second kind K_\nu(\cdot). Units with higher \hat z_i exhibit higher degradation rates and greater failure proneness.

Value

An object of class "individual_frailty" containing:

estimates	Data frame with columns: unit, n_obs, cum_deg, E_z (posterior mean), Var_z, SD_z, CI_Lower, CI_Upper, and fragility_rank.
frailty_type	Type of frailty distribution.
most_fragile	Subset of top fragile units.
least_fragile	Subset of least fragile units.
conf_level	Nominal confidence level.

References

Morita, L. H. M., Tomazella, V. L. D., Balakrishnan, N., Ramos, P. L., Ferreira, P. H., & Louzada, F. (2021). Inverse Gaussian process model with frailty term in reliability analysis. Quality and Reliability Engineering International, 37(2), 763-784. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/qre.2762")}.

See Also

igp_fit, lifetime_dist

Examples

data(laser)
fit_gam <- igp_fit(laser, time_col = "t", deg_col = "increase",
                   unit_col = "unit", frailty = "gamma")
frail_est <- individual_frailty(fit_gam)
print(frail_est)

IGPFrailty documentation built on Aug. 25, 2026, 9:08 a.m.