lifetime_dist: Lifetime Distribution and Quantile Estimation for IGP...

In IGPFrailty: Inverse Gaussian Process Degradation Models with Frailty

Lifetime Distribution and Quantile Estimation for IGP Degradation Models

Description

Computes the implied lifetime cumulative distribution function (CDF), probability density function (PDF), survival function, and quantiles with asymptotic confidence intervals for a predefined failure threshold \rho.

Usage

lifetime_dist(
  object,
  threshold,
  times = NULL,
  probs = c(0.01, 0.05, 0.1, 0.5, 0.8),
  conf_level = 0.95
)

Arguments

object	An object of class "igp_fit".
threshold	Numeric failure threshold \rho > 0 at which a unit is declared failed.
times	Optional numeric vector of evaluation time points. If NULL, an automatic sequence spanning the lifetime domain is generated.
probs	Numeric vector of quantile probabilities. Default is c(0.01, 0.05, 0.1, 0.5, 0.8).
conf_level	Nominal confidence level for quantile asymptotic confidence intervals. Default is 0.95.

Details

In threshold degradation models, the soft failure time T is defined as:

T = \inf\{t \ge 0 : D(t) \ge \rho\}

The lifetime CDF F_T(t) is the probability that cumulative degradation exceeds \rho:

F_T(t) = P(D(t) \ge \rho) = 1 - F(\rho \mid g_\theta(t), \eta, \xi) = R(\rho \mid g_\theta(t), \eta, \xi)

For the **classical IGP** model:

F_{T,\text{IGP}}(t) = \Phi\left(-\sqrt{\frac{\eta}{\rho}}(\rho - g_\theta(t))\right) - \exp(2\eta g_\theta(t)) \Phi\left(-\sqrt{\frac{\eta}{\rho}}(\rho + g_\theta(t))\right)

For the **IGP-Gamma** model:

F_{T,\text{Gamma}}(t) = \frac{2 \xi^{-1/(2\xi)} (H_{\text{IGP}}(\rho))^{1/(2\xi)} K_{1/\xi}\left(2\sqrt{\frac{H_{\text{IGP}}(\rho)}{\xi}}\right)}{\Gamma(1/\xi)}

For the **IGP-IG** model:

F_{T,\text{IG}}(t) = \frac{\exp\left(-\frac{\sqrt{1 + 2\xi H_{\text{IGP}}(\rho)} - 1}{\xi}\right)}{\sqrt{1 + 2\xi H_{\text{IGP}}(\rho)}}

where H_{\text{IGP}}(\rho) = -\log R_{\text{IGP}}(\rho \mid g_\theta(t), \eta).

Quantiles t_p are computed by numerical root finding on F_T(t_p) = p. Asymptotic standard errors for \hat t_p are obtained via the Delta method using the estimated variance-covariance matrix of parameter estimates.

Value

An object of class "lifetime_dist" containing:

threshold	The specified failure threshold \rho.
quantiles	Data frame containing estimated quantiles Estimate, Std. Error, CI Lower, and CI Upper.
curve	Data frame with evaluation grid: t, CDF, PDF, Survival, and Hazard.
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, individual_frailty

Examples

data(laser)
fit_gam <- igp_fit(laser, time_col = "t", deg_col = "increase",
                   unit_col = "unit", frailty = "gamma")
lt <- lifetime_dist(fit_gam, threshold = 10, probs = c(0.01, 0.05, 0.1, 0.5, 0.8))
print(lt)

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