Methods for delays with an exponentially tilted primary

Analytical primary event censored CDFs for exponential, gamma and normal delay distributions with an exponentially tilted primary event window, the dexpgrowth() primary distribution with r equal to the tilt $\rho$. They honour use_numeric, and use the numerical method of pcens_cdf.default() when no closed form applies.

Usage

# S3 method for class 'pcens_pexp_dexpgrowth'
pcens_cdf(object, q, pwindow, use_numeric = FALSE)

# S3 method for class 'pcens_pgamma_dexpgrowth'
pcens_cdf(object, q, pwindow, use_numeric = FALSE)

# S3 method for class 'pcens_pnorm_dexpgrowth'
pcens_cdf(object, q, pwindow, use_numeric = FALSE)

Arguments

object: A primarycensored object as created by new_pcens().
q: Vector of quantiles
pwindow: Primary event window. Use pwindow = 0 for an exactly observed primary event, in which case the delay CDF is used directly.
use_numeric: Logical, if TRUE forces use of numeric integration even for distributions with analytical solutions. This is primarily useful for testing purposes or for settings where the analytical solution breaks down.

Value

Vector of computed primary event censored CDFs

Details

With window width $w$, delay CDF $F$ and $J(x) = \int e^{-\rho u} f(u) du$ up to $x$, the CDF at $q$ is $$ F_\rho(q) = F(q - w) + \frac{e^{\rho q} \{J(q) - J(q - w)\} - \{F(q) - F(q - w)\}}{e^{\rho w} - 1}. $$ Every term depends on one endpoint, $q$ or $q - w$, so each endpoint is evaluated once and shared between neighbouring q. Terms are on the log scale, and each difference is taken between the lower or the upper tail terms, whichever loses less precision.

The exponential and gamma forms need $\lambda + \rho > 0$ for rate $\lambda$, so that the tilted delay distribution exists. Otherwise the numerical method is used. The numerical method is less accurate in the lower tail of a gamma with shape below 1, with a relative error of about 1e-5 in R and 1e-4 in Stan.

The direct form cancels as $\rho \to 0$. With $G_k(t) = \int (t - u)^k f(u) du$ up to $t$, two series in $\rho$ truncated after the second order replace it.

For $|\rho| w$ below $10^{-2}$, or $10^{-3}$ for delays on the reals, and $|\rho| (|q| + w) < 0.1$, the uniform window limit with its corrections, $\{G_1(q) - G_1(q - w)\} / w + \rho \{G_2(q) - w G_1(q) - G_2(q - w) - w G_1(q - w)\} / (2 w) + \rho^2 \{G_3(q) - G_3(q - w) - 3 w [G_2(q) + G_2(q - w)] / 2 + w^2 [G_1(q) - G_1(q - w)] / 2\} / (6 w)$. Its relative error is below $(|\rho| w)^3 / 160$.
Otherwise, for delays on the non-negative reals with $q < w$ and $|\rho| q < 5 \times 10^{-3}$, $\rho \{G_1(q) + \rho G_2(q) / 2 + \rho^2 G_3(q) / 6\} / (e^{\rho w} - 1)$. Its relative error is below $(|\rho| q)^3 / 24$.

Both errors are below $10^{-8}$ at these limits.

The CDF agrees with numerical integration to a relative 1e-6 in the package tests.

Examples

# Exponential delay with a growing primary event process
pexp_obj <- new_pcens(
  pdist = pexp, dprimary = dexpgrowth,
  primary_args = list(r = 0.3), rate = 0.5
)
pcens_cdf(pexp_obj, q = c(1, 4, 8), pwindow = 2)
#> [1] 0.08649693 0.75597288 0.96697452

# Gamma delay with a declining process
pgamma_obj <- new_pcens(
  pdist = pgamma, dprimary = dexpgrowth,
  primary_args = list(r = -0.3), shape = 2, rate = 1
)
pcens_cdf(pgamma_obj, q = c(1, 4, 8), pwindow = 2)
#> [1] 0.06357351 0.79940359 0.99243183

# Normal delay, for example a difference of event times
pnorm_obj <- new_pcens(
  pdist = pnorm, dprimary = dexpgrowth,
  primary_args = list(r = 0.1), mean = 3, sd = 2
)
pcens_cdf(pnorm_obj, q = c(-1, 3, 8), pwindow = 2)
#> [1] 0.007780575 0.309880218 0.971725556

Usage

Arguments

Value

Details

See also

Examples