Analytic solutions for censored delay distributions
Source:vignettes/analytic-solutions.Rmd
analytic-solutions.Rmd1 Introduction
1.1 What are we going to do in this vignette
In this vignette, we’ll derive the analytic solutions for the primary censored delay distributions for a range of commonly used distributions.
- Examine the case of exponentially tilted primary event times
- Analyse the uniform primary event time case (\(r=0\))
- Explore general partial expectation and its role in solving equation (3.1)
- Provide specific analytic solutions for distributions where the partial expectation can be reduced to an analytic expression
2 Analytic solutions for exponentially tilted primary event times
In epidemiological analysis, it is common for primary events to occur at exponentially increasing or decreasing rates, for example, incidence of new infections in an epidemic. In this case, the distribution of the primary event time within its censoring window is biased by the exponential growth or decay[1] (i.e. for exponential growth, the event time will more likely to be closer to the end of the censoring window and vice versa for decay). If we assume a reference uniform distribution within a primary censoring window \([k, k + w_P)\) then the distribution of the primary event time within the censoring window is the exponential tilted uniform distribution:
\[ f_P(t) \propto \exp(r t) \mathbb{1}_{[k, k + w_P]}(t). \]
In this case, the distribution function for \(C_P\), that is the length of time left in the primary censor window after the primary event time, is given by:
\[ F_{C_P}(p; r) = { 1 - \exp(-r p) \over 1 - \exp(-r w_P)}, \qquad p \in [0, w_P]. \tag{2.1} \] Note that taking the limit \(\lim_r \rightarrow 0\) in equation (2.1) gives the uniform distribution function \(F_{C_P}(p, 0) = p / w_P\).
In the following, it is convenient to use a (linear) difference operator defined as:
\[ \Delta_{w}f(t) = f(t + w) - f(t). \]
3 Uniform primary event time (\(r=0\))
Applying a uniform primary event time distribution to equation 3.1 from “Why it works” gives:
\[ Q_{S_+}(t) = Q_T(t + w_P) + { 1 \over w_P} \int_0^{w_P} f_T(t+p) p~ dp. \]
This is analytically solvable whenever the upper distribution function of \(T\) is known and the mean of \(T\) is analytically solvable from its integral definition.
In each case considered below it is easier to change the integration variable:
\[ \begin{aligned} Q_{S_+}(t) &= Q_T(t + w_P) + { 1 \over w_P} \int_t^{t+w_P} f_T(z) (z-t)~ dz \\ &= Q_T(t + w_P) + { 1 \over w_P} \Big[ \int_t^{t+w_P} f_T(z) z~ dz - t \Delta_{w_P}F_T(t) \Big]. \end{aligned} \tag{3.1} \]
3.1 General partial expectation
Note that for any distribution with an analytically available distribution function \(F_T\) equation (3.1) can be solved so long as the partial expectation
\[ \int_t^{t+w_P} f_T(z) z~ dz \tag{3.2} \]
can be reduced to an analytic expression.
The insight here is that this will be possible for any distribution where the average of the distribution can be calculated analytically, which includes commonly used non-negative distributions such as the Gamma, Log-Normal and Weibull distributions.
3.2 General discrete censored delay distribution
First, we note that equation 3.3 from “Why it works” can be written using the difference operator: \(f_n = -\Delta_1 Q_{S_+}(n-1)\). We can insert this expression into equation (3.1) to give the discrete censored delay distribution for uniformly distributed primary event times:
\[ \begin{aligned} f_n &= \Delta_1\Big[(n-1) \Delta_1F_T(n-1)\Big] - \Delta_1Q_T(n) - \Delta_1\Big[ \int_{n-1}^n f_T(z) z ~dz \Big] \\ &= (n+1)F_T(n+1) + (n-1)F_T(n-1) - 2nF_T(n) - \Delta_1\Big[ \int_{n-1}^n f_T(z) z ~dz \Big]. \end{aligned} \tag{3.3} \]
We now consider some specific delay distributions.
3.3 Gamma distributed delay times
The Gamma distribution has the density function:
\[ f_T(z;k, \theta) = {1 \over \Gamma(k) \theta^k} z^{k-1} \exp(-z/\theta). \] Where \(\Gamma\) is the Gamma function.
The Gamma distribution has the distribution function:
\[ \begin{aligned} F_T(z;k, \theta) &= {\gamma(k, z/\theta) \over \Gamma(k)}, \qquad z\geq 0,\\ F_T(z;k, \theta) &= 0, \qquad z < 0. \end{aligned} \] Where \(\gamma\) is the lower incomplete gamma function.
3.3.1 Gamma partial expectation
We know that the full expectation of the Gamma distribution is \(\mathbb{E}[T] = k\theta\), which can be calculated as a standard integral. Doing the same integral for the partial expectation gives:
\[ \begin{aligned} \int_t^{t+w_P} f_T(z) z~ dz &= {1 \over \Gamma(k) \theta^k} \int_t^{t+w_P} \mathbb{1}(z \geq 0) z z^{k-1} \exp(-z/\theta)~dz \\ &= {\Gamma(k+1) \theta^{k+1} \over \Gamma(k) \theta^k} {1 \over \Gamma(k+1) \theta^{k+1}} \int_t^{t+w_P} \mathbb{1}(z \geq 0) z^{k} \exp(-z/\theta)~dz\\ &= k\theta \Delta_{w_P} F_T(t; k + 1, \theta). \end{aligned} \tag{3.4} \]
3.3.2 Survival function of \(S_{+}\) for Gamma distribution
By substituting equation (3.4) into equation (3.3) we can solve for the survival function of \(S_+\) in terms of analytically available functions:
\[ Q_{S_+}(t; k, \theta) = Q_T(t + w_P; k, \theta) + { 1 \over w_P} \big[ k \theta \Delta_{w_P}F_T(t; k+1, \theta) - t \Delta_{w_P}F_T(t; k, \theta) \big]. \tag{3.5} \]
3.3.3 CDF form of \(S_{+}\) for Gamma distribution
Rearranging equation (3.5) via \(F_{S_+} = 1 - Q_{S_+}\) gives the equivalent CDF-direct form
\[ F_{S_+}(d; k, \theta) = \frac{1}{w_P}\Big[ d\, F_T(d; k, \theta) - q\, F_T(q; k, \theta) - k\theta\big( F_T(d; k{+}1, \theta) - F_T(q; k{+}1, \theta) \big) \Big], \]
where \(q = \max(d - w_P, 0)\).
3.3.4 Gamma discrete censored delay distribution
By substituting (3.5) into (3.3) we get the discrete censored delay distribution in terms of analytically available functions: \[ \begin{aligned} f_n &= (n+1) F_T(n+1; k, \theta) + (n-1) F_T(n-1; k, \theta) - 2 n F_T(n; k, \theta) - k \theta \Delta_1^{(2)}F_T(n-1; k+1, \theta)\\ &= (n+1) F_T(n+1; k, \theta) + (n-1) F_T(n-1; k, \theta) - 2 n F_T(n; k, \theta) \\ &+ k \theta \Big( 2 F_T(n; k+1, \theta) - F_T(n-1; k+1, \theta) - F_T(n+1; k+1,\theta) \Big) \qquad n = 0, 1, \dots. \end{aligned} \]
Which was also found by Cori et al[2].
3.4 Log-Normal distribution
The Log-Normal distribution has the density function:
\[ \begin{aligned} f_T(z;\mu, \sigma) &= {1 \over z \sigma \sqrt{2\pi}} \exp\left( - {(\log(z) - \mu)^2 \over 2 \sigma^2} \right)\\ F_T(z;\mu, \sigma) &= 0, \qquad z < 0. \end{aligned} \] And distribution function:
\[ F_T(z;\mu, \sigma) = \Phi\left( {\log(z) - \mu \over \sigma} \right). \] Where \(\Phi\) is the standard normal distribution function.
3.4.1 Log-Normal partial expectation
We know that the full expectation of the Log-Normal distribution is \(\mathbb{E}[T] = e^{\mu + \frac{1}{2} \sigma^2}\), which can be calculated by integration with the integration substitution \(y = (\ln z - \mu) / \sigma\). This has transformation Jacobian:
\[ \frac{dz}{dy} = \sigma e^{\sigma y + \mu}. \]
Doing the same integral for the partial expectation, and using the same integration substitution gives:
\[ \begin{aligned} \int_t^{t+w_P} z~ f_T(z; \mu, \sigma) dz &= {1 \over \sigma \sqrt{2\pi}} \int_t^{t+w_P} \mathbb{1}(z \geq 0) \exp\left( - {(\log(z) - \mu)^2 \over 2 \sigma^2} \right) dz \\ &= {1 \over \sqrt{2\pi}} \int_{(\ln t - \mu)/\sigma}^{(\ln(t+w_P) - \mu)/\sigma} e^{\sigma y + \mu} e^{-y^2/2} dy\\ &= {e^{\mu + \frac{1}{2} \sigma^2} \over \sqrt{2 \pi} } \int_{(\ln t - \mu)/\sigma}^{(\ln(t+w_P) - \mu)/\sigma} e^{-(y- \sigma)^2/2} dy \\ &= e^{\mu + \frac{1}{2} \sigma^2} \Big[\Phi\Big({\ln(t+w_P) - \mu \over \sigma} - \sigma\Big) - \Phi\Big({\ln(t) - \mu \over \sigma} - \sigma\Big) \Big]\\ &= e^{\mu + \frac{1}{2} \sigma^2} \Delta_{w_P}F_T(t; \mu + \sigma^2, \sigma). \end{aligned} \tag{3.6} \]
3.4.2 Survival function of \(S_{+}\) for Log-Normal distribution
By substituting equation (3.6) into equation (3.3) we can solve for the survival function of \(S_+\) in terms of analytically available functions:
\[ Q_{S+}(t ;\mu, \sigma) = Q_T(t + w_P;\mu, \sigma) + { 1 \over w_P} \Big[ e^{\mu + \frac{1}{2} \sigma^2} \Delta_{w_P}F_T(t; \mu + \sigma^2, \sigma) - t\Delta_{w_P}F_T(t; \mu, \sigma) \Big] \]
3.4.3 CDF form of \(S_{+}\) for Log-Normal distribution
Applying \(F_{S_+} = 1 - Q_{S_+}\) to the preceding survival form gives the equivalent CDF-direct form
\[ F_{S_+}(d; \mu, \sigma) = \frac{1}{w_P}\Big[ d\, F_T(d; \mu, \sigma) - q\, F_T(q; \mu, \sigma) - e^{\mu + \sigma^2/2}\big( F_T(d; \mu{+}\sigma^2, \sigma) - F_T(q; \mu{+}\sigma^2, \sigma) \big) \Big], \]
where \(q = \max(d - w_P, 0)\).
3.4.4 Log-Normal discrete censored delay distribution
By substituting (3.6) into (3.3) we get the discrete censored delay distribution in terms of analytically available functions:
\[ \begin{aligned} f_n &= (n+1) F_T(n+1; \mu, \sigma) + (n-1) F_T(n-1; \mu, \sigma) - 2 n F_T(n; \mu, \sigma) \\ &- e^{\mu + \frac{1}{2} \sigma^2} \Delta_1^{(2)}F_T(n-1;\mu + \sigma^2, \sigma) \\ &= (n+1) F_T(n+1; \mu, \sigma) + (n-1) F_T(n-1; \mu, \sigma) - 2 n F_T(n; \mu, \sigma) \\ &+ e^{\mu + \frac{1}{2} \sigma^2} \Big( 2 F_T(n; \mu + \sigma^2, \sigma) - F_T(n+1; \mu + \sigma^2, \sigma) - F_T(n-1; \mu + \sigma^2, \sigma) \Big)\qquad n = 0, 1, \dots. \end{aligned} \]
3.5 Weibull distribution
The Weibull distribution has the density function:
\[ f_T(z;\lambda,k) = \begin{cases} \frac{k}{\lambda}\left(\frac{z}{\lambda}\right)^{k-1}e^{-(z/\lambda)^{k}}, & z\geq0 ,\\ 0, & z<0, \end{cases} \] And distribution function:
\[ F_T(z;\lambda,k))=\begin{cases}1 - e^{-(z/\lambda)^k}, & z\geq0,\\ 0, & z<0.\end{cases} \] Where \(\Phi\) is the standard normal distribution function.
3.5.1 Weibull partial expectation
We know that the full expectation of the Weibull distribution is \(\mathbb{E}[T] = \lambda \Gamma(1 + 1/k)\), which can be calculated by integration using the integration substitution \(y = (z / \lambda)^k\), which has transformation Jacobian:
\[ \frac{dz}{dy} = \frac{\lambda}{k}y^{1/k - 1}. \]
Doing the same integral for the partial expectation, and using the same integration substitution gives:
\[ \begin{aligned} \int_{t}^{t+w_P} z~ f_T(z; \lambda,k) dz &= \int_t^{t+w_P} \mathbb{1}(z \geq 0) \frac{kz}{\lambda}\left(\frac{z}{\lambda}\right)^{k-1}e^{-(z/\lambda)^{k}} dz \\ &= k\int_t^{t+w_P} \mathbb{1}(z \geq 0) \left(\frac{z}{\lambda}\right)^{k}e^{-(z/\lambda)^{k}} dz \\ &= \lambda k \int_{(t / \lambda)^k}^{((t + w_P) / \lambda)^k} \mathbb{1}(y \geq 0) y y^{1/k - 1} e^{-y} dy \\ &= \lambda\int_{(t / \lambda)^k}^{((t + w_P) / \lambda)^k} \mathbb{1}(y \geq 0) y^{1/k} e^{-y} dy\\ &= \lambda \Delta_{w_P} g(t; \lambda,k) \end{aligned} \tag{3.7} \]
Where
\[ g(t; \lambda, k) = \gamma\left(1 + 1/k, \left({t\vee 0 \over \lambda}\right)^k\right) = \frac{1}{k}\gamma\left(1/k, \left({t\vee 0 \over \lambda}\right)^k\right) - \frac{t}{\lambda}\exp\left(-\left({t\vee 0 \over \lambda}\right)^k\right) \] is a reparametrisation of the lower incomplete gamma function. Note that the \(\vee\) operator \(t \vee 0 = \text{max}(0, t)\) comes into the expression due to \(\mathbb{1}(y \geq 0)\) term in the integrand.
3.6 Survival function of \(S_{+}\) for Weibull distribution
By substituting equation (3.7) into equation (3.3) we can solve for the survival function of \(S_+\) in terms of analytically available functions:
\[ Q_{S+}(t ;\lambda,k) = Q_T(t + w_P; \lambda,k) + { 1 \over w_P} \Big[ \lambda \Delta_{w_P} g(t; \lambda,k) - t\Delta_{w_P}F_T(t; \lambda,k)\Big]. \]
3.6.1 CDF form of \(S_{+}\) for Weibull distribution
Applying \(F_{S_+} = 1 - Q_{S_+}\) to the preceding survival form gives the equivalent CDF-direct form
\[ F_{S_+}(d; \lambda, k) = \frac{1}{w_P}\Big[ d\, F_T(d; \lambda, k) - q\, F_T(q; \lambda, k) - \lambda\big( g(d; \lambda, k) - g(q; \lambda, k) \big) \Big], \]
where \(q = \max(d - w_P, 0)\) and \(g\) is as defined in equation (3.7).
3.6.2 Weibull discrete censored delay distribution
By substituting (3.7) into (3.3) we get the discrete censored delay distribution in terms of analytically available functions:
\[ \begin{aligned} f_n &= (n+1)F_T(n+1) + (n-1)F_T(n-1) - 2nF_T(n) - \Delta_1\Big[ \int_{n-1}^n f_T(z) z ~dz \Big] \\ &= (n+1)F_T(n+1) + (n-1)F_T(n-1) - 2nF_T(n) - \lambda \Delta_1^{(2)} g(n-1; \lambda,k) \\ &= (n+1)F_T(n+1) + (n-1)F_T(n-1) - 2nF_T(n) \\ &+ \lambda [2 g(n; \lambda,k) - g(n+1; \lambda,k) - g(n-1; \lambda,k)] \qquad n = 0, 1, \dots. \end{aligned} \]
Which was also found by Cori et al[2].
3.7 Generalised gamma distribution
The generalised gamma distribution, in the parameterisation of Stacy[3] as implemented by flexsurv::pgengamma.orig(), has the density function:
\[ f_T(z; a, \theta, k) = \begin{cases} \frac{a}{\Gamma(k)\theta}\left(\frac{z}{\theta}\right)^{a k-1}e^{-(z/\theta)^{a}}, & z\geq0 ,\\ 0, & z<0, \end{cases} \] And distribution function:
\[ F_T(z; a, \theta, k) =\begin{cases} {\gamma\left(k, (z/\theta)^a\right) \over \Gamma(k)}, & z\geq0,\\ 0, & z<0.\end{cases} \] Where \(\gamma\) is the lower incomplete gamma function. The Gamma distribution is the special case \(a = 1\) and the Weibull distribution is the special case \(k = 1\).
3.7.1 Generalised gamma partial expectation
Using the integration substitution \(y = (z / \theta)^a\) as for the Weibull distribution, the full expectation is \(\mathbb{E}[T] = \theta \Gamma(k + 1/a) / \Gamma(k)\). Doing the same integral for the partial expectation gives:
\[ \begin{aligned} \int_{t}^{t+w_P} z~ f_T(z; a, \theta, k) dz &= \frac{\theta}{\Gamma(k)} \int_{(t / \theta)^a}^{((t + w_P) / \theta)^a} \mathbb{1}(y \geq 0) y^{k + 1/a - 1} e^{-y} dy\\ &= \theta \frac{\Gamma(k + 1/a)}{\Gamma(k)} \Delta_{w_P} F_T(t; a, \theta, k + 1/a). \end{aligned} \tag{3.8} \]
That is, the partial expectation distribution is the generalised gamma distribution with \(k\) replaced by \(k + 1/a\). For \(a = 1\) this recovers equation (3.4) and for \(k = 1\) it recovers equation (3.7).
3.7.2 Survival function of \(S_{+}\) for generalised gamma distribution
By substituting equation (3.8) into equation (3.1) we can solve for the survival function of \(S_+\) in terms of analytically available functions:
\[ Q_{S_+}(t; a, \theta, k) = Q_T(t + w_P; a, \theta, k) + { 1 \over w_P} \Big[ \theta \frac{\Gamma(k + 1/a)}{\Gamma(k)} \Delta_{w_P}F_T(t; a, \theta, k + 1/a) - t \Delta_{w_P}F_T(t; a, \theta, k) \Big]. \tag{3.9} \]
3.7.3 CDF form of \(S_{+}\) for generalised gamma distribution
Rearranging equation (3.9) via \(F_{S_+} = 1 - Q_{S_+}\) gives the equivalent CDF-direct form
\[ F_{S_+}(d; a, \theta, k) = \frac{1}{w_P}\Big[ d\, F_T(d; a, \theta, k) - q\, F_T(q; a, \theta, k) - \theta \frac{\Gamma(k + 1/a)}{\Gamma(k)} \big( F_T(d; a, \theta, k + 1/a) - F_T(q; a, \theta, k + 1/a) \big) \Big], \]
where \(q = \max(d - w_P, 0)\).
This is the form implemented in pcens_cdf() and in the Stan function primarycensored_gengamma_uniform_lcdf().
The discrete censored delay distribution follows from equation (3.3) exactly as for the Gamma distribution, with \(k\theta F_T(\cdot; k + 1, \theta)\) replaced by \(\theta \frac{\Gamma(k + 1/a)}{\Gamma(k)} F_T(\cdot; a, \theta, k + 1/a)\).
3.8 Antiderivative form
Equation (3.1) is the integral of the delay CDF over the primary event window. For a uniform primary event time \(F_{S_+}(d) = \frac{1}{w_P} \int_q^d F_T(u)~ du\), with \(q = \max(d - w_P, 0)\) for delays with non-negative support and \(q = d - w_P\) for delays with support on the reals. So for any antiderivative \(G\) of \(F_T\)
\[ F_{S_+}(d) = \frac{G(d) - G(q)}{w_P}. \tag{3.10} \]
Integrating by parts gives \(G(t) = t F_T(t) - \int_{-\infty}^t z f_T(z)~ dz\), which is the form used for the Gamma, Log-Normal, Weibull and generalised gamma distributions above.
Each of these needs only the delay CDF and the lower partial expectation at \(d\) and at \(q\).
The two terms of \(G\) are each evaluated once per endpoint, so the Stan functions primarycensored_uniform_terms() and primarycensored_uniform_lcdf_from_terms() reuse them across neighbouring delays.
Where \(G\) has a simple positive closed form it is used directly, which avoids differences of terms that can be negative.
The difference \(G(d) - G(q)\) loses precision when \(d\) is much larger than \(w_P\), with a relative error of about \(10^{-16} d / (w_P F_{S_+}(d))\).
3.9 Exponential distribution
The exponential distribution with rate \(\lambda\) has \(F_T(t) = 1 - e^{-\lambda t}\) for \(t \geq 0\). An antiderivative that is zero at \(t = 0\) is
\[ G(t) = t - \frac{1}{\lambda} + \frac{e^{-\lambda t}}{\lambda}, \qquad t \geq 0, \]
with \(G(t) = 0\) for \(t < 0\). Equation (3.10) then gives, for \(d \geq w_P\),
\[ F_{S_+}(d; \lambda) = 1 - \frac{e^{-\lambda (d - w_P)} \left(1 - e^{-\lambda w_P}\right)}{\lambda w_P}. \]
This is the Gamma solution with shape 1. It is implemented directly so that it does not need the incomplete gamma function.
3.10 Normal distribution
The normal distribution has support on the reals, so the primary event window \([d - w_P, d]\) is not clipped at zero. With \(z = (t - \mu) / \sigma\), the partial expectation is \(\int_{-\infty}^t z' f_T(z')~ dz' = \mu \Phi(z) - \sigma \phi(z)\). This can be negative, so a log scale evaluation cannot hold it as a single term. Substituting it into \(G(t) = t F_T(t) - \int_{-\infty}^t z' f_T(z')~ dz'\) gives the positive antiderivative
\[ G(t; \mu, \sigma) = \sigma \left( z \Phi(z) + \phi(z) \right) = \mathbb{E}\left[ (t - T)^+ \right]. \]
In the lower tail \(z \Phi(z)\) and \(\phi(z)\) nearly cancel. For \(z < -10\) the asymptotic series \(\sigma \phi(z) z^{-2} \sum_{n \geq 0} (-1)^n (2n + 1)!! z^{-2n}\) is used instead, with 20 terms.
3.11 Chi-square and beta distributions
The chi-square distribution with \(\nu\) degrees of freedom is the Gamma distribution with shape \(\nu / 2\) and scale 2, so it uses the Gamma solution.
The beta distribution has support on \([0, 1]\) and mean \(a / (a + b)\). The partial expectation is \(\int_0^t z f_T(z)~ dz = \frac{a}{a + b} F_T(t; a + 1, b)\) for \(t \leq 1\), and the full mean for \(t \geq 1\) where \(F_T = 1\). So the terms \(t F_T(t; a, b)\) and \(\frac{a}{a + b} F_T(t; a + 1, b)\) replace \(F_T\) with 1 above \(t = 1\).
3.12 Inverse gamma family
The inverse gamma distribution with shape \(\alpha\) and scale \(\beta\) has mean \(\beta / (\alpha - 1)\) for \(\alpha > 1\). Its partial expectation is an inverse gamma distribution function with shape \(\alpha - 1\)
\[ \int_0^t z f_T(z; \alpha, \beta)~ dz = \frac{\beta}{\alpha - 1} F_T(t; \alpha - 1, \beta). \]
The solution is then equation (3.10) with \(G(t) = t F_T(t; \alpha, \beta) - \frac{\beta}{\alpha - 1} F_T(t; \alpha - 1, \beta)\).
For \(\alpha \leq 1\) the mean is infinite and this form does not apply, so the numerical path is used.
The inverse chi-square distribution with \(\nu\) degrees of freedom is the inverse gamma with shape \(\nu / 2\) and scale \(1 / 2\).
The scaled inverse chi-square distribution with scale \(s\) is the inverse gamma with shape \(\nu / 2\) and scale \(\nu s^2 / 2\).
Both need \(\nu > 2\) and otherwise use the numerical path.
Base R has none of these distributions, so in R pcens_cdf() uses this solution for extraDistr::pinvgamma() and extraDistr::pinvchisq(), which need the extraDistr package.
3.13 Pareto distribution
The Pareto distribution with minimum \(y_{\min}\) and shape \(\alpha\) has \(F_T(t) = 1 - (y_{\min} / t)^\alpha\) for \(t \geq y_{\min}\) and 0 below. An antiderivative that is zero at \(y_{\min}\) is
\[ G(t) = (t - y_{\min}) - y_{\min} \frac{1 - (y_{\min} / t)^{\alpha - 1}}{\alpha - 1}, \qquad t \geq y_{\min}, \]
which is \((t - y_{\min}) - y_{\min} \log(t / y_{\min})\) for \(\alpha = 1\).
Unlike the forms that use the mean, this holds for every \(\alpha > 0\) because only the integral of \(F_T\) over the window enters.
Base R has no Pareto distribution, so in R pcens_cdf() uses this solution for extraDistr::ppareto(), which needs the extraDistr package.
4 Learning more
- For more mathematical background on the analytic solutions see the
vignette("why-it-works"). - For a more introductory explanation of the primary event censored distribution see the
vignette("primarycensored").