pslt: Population CDF/PDF/quantile function of a two-stage...

In stagelife: Random Number Generation and Inferences for Two-Stage Life-Testing Data

Population CDF/PDF/quantile function of a two-stage step-stress model

Description

pslt(), dslt() and qslt() implement the full stage life testing design, in which every unit switches from the baseline to the elevated stress at a common change-point tau1. Combined with the cumulative-exposure equivalent age slt_equiv_age(), the two branches (pre-/post-tau1) are pieced together into a single, continuous population CDF.

Usage

pslt(t, family, theta0, theta1, tau1)

dslt(x, family, theta0, theta1, tau1)

qslt(p, family, theta0, theta1, tau1)

Arguments

t, x	Numeric vector of times at which to evaluate the CDF/PDF.
family	A family key (see slt_families()) or family object.
theta0	Numeric c(lambda0, alpha0), baseline-stage parameters.
theta1	Numeric c(lambda1, alpha1), elevated-stage parameters.
tau1	Scalar change-point time (same argument as tau in slt_equiv_age()).
p	Numeric vector of probabilities at which to evaluate the quantile function.

Details

For a single unit switching from stage 0 to stage 1 at the fixed time tau1 (every unit switches – the classical simple-step-stress design, matching prop_transfer = 1 in rslt()), the CEM construction (see slt_equiv_age()) gives a population cdf

F(t;\theta_0,\theta_1,\tau_1) = \begin{cases} F_0(t;\theta_0), & 0 < t \le \tau_1, \\ F_1(t-\tau_1+\nu_1;\ \theta_1), & t > \tau_1, \end{cases}

where \nu_1 = \nu(\tau_1;\theta_0,\theta_1) is the stage-1 equivalent age (slt_equiv_age()) at the moment of the switch. Differentiating – a constant shift of the argument by \tau_1-\nu_1, not a general change of variables, so no extra Jacobian factor arises – gives the matching population pdf

f(t;\theta_0,\theta_1,\tau_1) = \begin{cases} f_0(t;\theta_0), & 0 < t \le \tau_1, \\ f_1(t-\tau_1+\nu_1;\ \theta_1), & t > \tau_1. \end{cases}

F is continuous at \tau_1 by construction of \nu_1 (F_1(\tau_1-\tau_1+\nu_1;\theta_1) = F_1(\nu_1;\theta_1) = F_0(\tau_1;\theta_0), verified as an automatic consequence of the equal-reliability definition of \nu_1, not a separate assumption) – but F is generally not differentiable at \tau_1 (the hazard rate jumps there), an intrinsic, well-known feature of the cumulative exposure model, not an artefact of this implementation. qslt() inverts F directly (analytically, not numerically): for p \le F_0(\tau_1;\theta_0) it returns F_0^{-1}(p;\theta_0); otherwise it returns \tau_1-\nu_1+ F_1^{-1}(p;\theta_1), using each stage's own quantile function (itself closed-form for every family in slt_families()).

Value

A numeric vector the same length as t/x/p.

See Also

slt_equiv_age() for nu1; loglik_slt() for the estimation log-likelihood; rslt() for simulating data under the actual random-withdrawal design; fit_slt() for maximum likelihood estimation.

Examples

pslt(c(1, 6, 10), "weibull", c(0.1, 1.5), c(0.3, 1.5), tau1 = 5)

stagelife documentation built on Nov. 3, 2026, 5:08 p.m.