Probabilistic forecasts take the form of probability distributions over the set of possible outcomes.
When the outcome is real-valued (\(Y \in \mathbb{R}\)), probabilistic forecasts often come in the
form of parametric distributions. The scoringrules package provides the functionality to calculate
the CRPS and Log score for a range of parametric distributions. In this document, we list the
distributions that are available, along with their probability density/mass function (pdf/pmf),
cumulative distribution function (cdf), and an analytical formula for their CRPS. The Log score
is simply the negative logarithm of the pdf/pmf, when this exists. The analytical
formulae for the CRPS are all taken from Appendix A of the scoringRules R package (Jordan et al., 2017).
Discrete outcomes
Binomial distribution
The binomial distribution has two parameters that describe the number of trials, \(n = 0, 1, \dots, \)
and the probability of a success, \(p \in [0, 1]\). Its pmf, cdf, and CRPS are
\[\begin{split}
f_{n, p}(x) =
\begin{cases}
{n \choose x} p^{x} (1 - p)^{n - x}, & x = 0, 1, \dots, n, \\
0, & \text{otherwise,}
\end{cases}
\end{split}\]
\[\begin{split}
F_{n, p}(x) =
\begin{cases}
I(n - \lfloor x \rfloor, \lfloor x \rfloor + 1, 1 - p), & x \ge 0, \\
0, & x < 0,
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{n, p}, y) = 2 \sum_{x = 0}^{n} f_{n,p}(x) (1\{y < x\}
- F_{n,p}(x) + f_{n,p}(x)/2) (x - y),
\]
where \(I(a, b, x)\) is the regularised incomplete beta function.
Hypergeometric distribution
The hypergeometric distribution has three parameters that describe the number of objects with
the relevant feature, \(m = 0, 1, \dots, \) the number of objects without this feature, \(n = 0, 1, \dots \),
and the size of the sample to be drawn without replacement from these objects, \(k = 0, 1, \dots, m + n\).
Its pmf, cdf, and CRPS are
\[\begin{split}
f_{m, n, k}(x) =
\begin{cases}
\frac{{m \choose x} {n \choose k-x}}{m + n \choose k}, & x = \max{0, k - n}, \dots, \min{k, m}, \\
0, & \text{otherwise,}
\end{cases}
\end{split}\]
\[\begin{split}
F_{m, n, k}(x) =
\begin{cases}
\sum_{i = 0}^{\lfloor x \rfloor} f_{m, n, k}(i), & x \ge 0, \\
0, & x < 0,
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{m, n, k}, y) = 2 \sum_{x = 0}^{n} f_{m,n,k}(x) (1\{y < x\}
- F_{m,n,k}(x) + f_{m,n,k}(x)/2) (x - y).
\]
Negative binomial distribution
The negative binomial distribution has two parameters that describe the number of successes,
\(n > 0\), and the probability of a success, \(p \in [0, 1]\). Its pmf, cdf, and CRPS are
\[\begin{split}
f_{n, p}(x) =
\begin{cases}
\frac{\Gamma(x + n)}{\Gamma(n) x!} p^{n} (1 - p)^{x} , & x = 0, 1, 2, \dots, \\
0, & \text{otherwise,}
\end{cases}
\end{split}\]
\[\begin{split}
F_{n, p}(x) =
\begin{cases}
I(n, \lfloor x + 1 \rfloor, p), & x \ge 0, \\
0, & x < 0,
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{n, p}, y) = y (2 F_{n, p}(y) - 1) - \frac{n(1 - p)}{p^{2}} \left( p (2 F_{n+1, p}(y - 1) - 1) +
_{2} F_{1} \left( n + 1, \frac{1}{2}; 2; -\frac{4(1 - p)}{p^{2}} \right) \right),
\]
where \(I(a, b, x)\) is the regularised incomplete beta function, and \(_2 F_{1}(a, b; c; x)\) is the hypergeometric function.
Poisson distribution
The Poisson distribution has one mean parameter, \(\lambda > 0\). Its pmf, cdf, and CRPS are
\[\begin{split}
f_{\lambda}(x) =
\begin{cases}
\frac{\lambda^{x}}{x!}e^{-\lambda}, & x = 0, 1, 2, \dots, \\
0, & \text{otherwise,}
\end{cases}
\end{split}\]
\[\begin{split}
F_{\lambda}(x) =
\begin{cases}
\frac{\Gamma_{u}(\lfloor x + 1 \rfloor, \lambda)}{\Gamma(\lfloor x + 1 \rfloor)}, & x \ge 0, \\
0, & x < 0,
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{\lambda}, y) = (y - \lambda) (2F_{\lambda}(y) - 1)
+ 2 \lambda f_{\lambda}(\lfloor y \rfloor )
- \lambda e^{-2 \lambda} (I_{0} (2 \lambda) + I_{1} (2 \lambda)),
\]
where \(\Gamma_{u}(a, x)\) is the upper incomplete gamma function, and \(I_{m}(x)\) is the
modified Bessel function of the first kind.
Continuous real-valued outcomes
Laplace distribution
The Laplace distribution has one location parameter, \(\mu \in \mathbb{R}\), and one scale parameter \(\sigma > 0\).
Its pdf, cdf, and CRPS are
\[
f_{\mu, \sigma}(x) = \frac{1}{2\sigma} \exp \left( - \frac{|x - \mu|}{\sigma} \right)
\]
\[\begin{split}
F_{\mu, \sigma}(x) =
\begin{cases}
\frac{1}{2} \exp \left( \frac{x - \mu}{\sigma} \right), & x \le \mu, \\
1 - \frac{1}{2} \exp \left( - \frac{x - \mu}{\sigma} \right), & x \ge \mu, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1}, y) &= |y| + \exp(-|y|) - \frac{3}{4}, \\
\mathrm{CRPS}(F_{\mu, \sigma}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1}, \frac{y - \mu}{\sigma} \right).
\end{aligned}
\end{split}\]
Logistic distribution
The Logistic distribution has one location parameter, \(\mu \in \mathbb{R}\), and one scale parameter, \(\sigma > 0\).
Its pdf, cdf, and CRPS are
\[
f_{\mu, \sigma}(x) = \frac{\exp \left( - \frac{x - \mu}{\sigma} \right)}{\sigma \left( 1 + \exp \left( - \frac{x - \mu}{\sigma} \right) \right)^{2}},
\]
\[
F_{\mu, \sigma}(x) = \frac{1}{1 + \exp \left( - \frac{x - \mu}{\sigma} \right)},
\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1}, y) &= y - 2 \log \left( F_{0, 1}(y) \right) - 1, \\
\mathrm{CRPS}(F_{\mu, \sigma}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1}, \frac{y - \mu}{\sigma} \right).
\end{aligned}
\end{split}\]
Normal distribution
The normal distribution has one location parameter, \(\mu \in \mathbb{R}\), and one scale parameter, \(\sigma > 0\).
Its pdf, cdf, and CRPS are
\[
f_{\mu, \sigma}(x) = \frac{1}{\sigma} \phi \left( \frac{x - \mu}{\sigma} \right) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left( - \frac{(x - \mu)^{2}}{2\sigma^{2}} \right),
\]
\[
F_{\mu, \sigma}(x) = \Phi \left( \frac{x - \mu}{\sigma} \right) = \frac{1}{2} \left[ 1 + \text{erf} \left( \frac{x - \mu}{\sigma \sqrt{2}} \right) \right],
\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1}, y) &= y (2\Phi(y) - 1) + 2 \phi(y) - \frac{1}{\sqrt{\pi}}, \\
\mathrm{CRPS}(F_{\mu, \sigma}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1}, \frac{y - \mu}{\sigma} \right),
\end{aligned}
\end{split}\]
where \(\phi\) and \(\Phi\) represent the standard normal pdf and cdf, respectively, and
\(\text{erf}\) is the error function.
Mixture of normal distributions
A mixture of normal distributions has \(M\) location parameters, \(\mu_{1}, \dots, \mu_{M} \in \mathbb{R}\),
\(M\) scale parameters, \(\sigma_{1}, \dots, \sigma_{M} > 0\), and \(M\) weight parameters, \(w_{1}, \dots, w_{M} \ge 0\)
that sum to one. Its pdf, cdf, and CRPS are
\[
f(x) = \sum_{m=1}^{M} \frac{w_{m}}{\sigma_{m}} \phi \left( \frac{x - \mu_{m}}{\sigma_{m}} \right),
\]
\[
F(x) = \sum_{m=1}^{M} w_{m} \Phi \left( \frac{x - \mu_{m}}{\sigma_{m}} \right),
\]
\[
\mathrm{CRPS}(F, y) = \sum_{m=1}^{M} w_{m} A(y - \mu_{m}, \sigma_{m}^{2}) - \frac{1}{2} \sum_{m=1}^{M} \sum_{k=1}^{M} w_{m} w_{k} A(\mu_{m} - \mu_{k}, \sigma_{m}^{2} + \sigma_{k}^{2}),
\]
where \(A(\mu, \sigma^{2}) = \mu (2 \Phi(\frac{\mu}{\sigma}) - 1) + 2\sigma \phi(\frac{\mu}{\sigma})\),
with \(\phi\) and \(\Phi\) the standard normal pdf and cdf, respectively.
Student’s t distribution
The (generalised) Student’s t distribution has one degrees of freedom parameter, \(\nu > 0\),
one location parameter, \(\mu \in \mathbb{R}\), and one scale parameter, \(\sigma > 0\).
Its pdf, cdf, and CRPS are
\[
f_{\nu, \mu, \sigma}(x) = \frac{1}{\sigma \sqrt{\nu} B(\frac{1}{2}, \frac{\nu}{2})} \left( 1 + \frac{(x - \mu)^{2}}{\sigma^{2} \nu} \right)^{- \frac{\nu + 1}{2}},
\]
\[
F_{\nu, \mu, \sigma}(x) = \frac{1}{2} + \left( \frac{x - \mu}{\sigma} \right) \frac{_{2} F_{1} (\frac{1}{2}, \frac{\nu + 1}{2} ; \frac{3}{2} ; - \frac{(x - \mu)^{2}}{\sigma^{2} \nu})}{ \sqrt{\nu} B(\frac{1}{2}, \frac{\nu}{2}) },
\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{\nu, 0, 1}, y) &= y (2F_{\nu, 0, 1}(y) - 1) + 2 f_{\nu, 0, 1}(y) \left( \frac{\nu + y^{2}}{\nu - 1} \right) - \frac{2 \sqrt{\nu}}{\nu - 1} \frac{B(\frac{1}{2}, \nu - \frac{1}{2})}{B(\frac{1}{2}, \frac{\nu}{2})^{2}}, \\
\mathrm{CRPS}(F_{\nu, \mu, \sigma}, y) &= \sigma \mathrm{CRPS} \left( F_{\nu, 0, 1}, \frac{y - \mu}{\sigma} \right).
\end{aligned}
\end{split}\]
Two-piece exponential distribution
The two-piece exponential distribution has one location parameter, \(\mu \in \mathbb{R}\), and
two scale parameters, \(\sigma_{1}, \sigma_{2} > 0\). Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma_{1}, \sigma_{2}}(x) =
\begin{cases}
\frac{1}{\sigma_{1} + \sigma_{2}} \exp \left( - \frac{\mu - x}{\sigma_{1}} \right), & x \le \mu, \\
\frac{1}{\sigma_{1} + \sigma_{2}} \exp \left( - \frac{x - \mu}{\sigma_{2}} \right), & x \ge \mu, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma_{1}, \sigma_{2}}(x) =
\begin{cases}
\frac{\sigma_{1}}{\sigma_{1} + \sigma_{2}} \exp \left( - \frac{\mu - x}{\sigma_{1}} \right), & x \le \mu, \\
1 - \frac{\sigma_{2}}{\sigma_{1} + \sigma_{2}} \exp \left( - \frac{x - \mu}{\sigma_{2}} \right), & x \ge \mu, \\
\end{cases}
\end{split}\]
\[\begin{split}
\mathrm{CRPS}(F_{\mu, \sigma_{1}, \sigma_{2}}, y) =
\begin{cases}
|y - \mu| + \frac{2 \sigma_{1}^{2}}{\sigma_{1} + \sigma_{2}} \left[ \exp \left( - \frac{|y - \mu|}{\sigma_{1}} \right) - 1 \right] + \frac{\sigma_{1}^{3} + \sigma_{2}^{3}}{2(\sigma_{1} + \sigma_{2})^{2}}, & y \le \mu, \\
|y - \mu| + \frac{2 \sigma_{2}^{2}}{\sigma_{1} + \sigma_{2}} \left[ \exp \left( - \frac{|y - \mu|}{\sigma_{2}} \right) - 1 \right] + \frac{\sigma_{1}^{3} + \sigma_{2}^{3}}{2(\sigma_{1} + \sigma_{2})^{2}}, & y \ge \mu.
\end{cases}
\end{split}\]
Two-piece normal distribution
The two-piece normal distribution has one location parameter, \(\mu \in \mathbb{R}\), and
two scale parameters, \(\sigma_{1}, \sigma_{2} > 0\). Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma_{1}, \sigma_{2}}(x) =
\begin{cases}
\frac{2}{\sigma_{1} + \sigma_{2}} \phi \left( \frac{x - \mu}{\sigma_{1}} \right), & x \le \mu, \\
\frac{2}{\sigma_{1} + \sigma_{2}} \phi \left( \frac{x - \mu}{\sigma_{2}} \right), & x \ge \mu, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma_{1}, \sigma_{2}}(x) =
\begin{cases}
\frac{2\sigma_{1}}{\sigma_{1} + \sigma_{2}} \Phi \left( \frac{x - \mu}{\sigma_{1}} \right), & x \le \mu, \\
\frac{\sigma_{1} - \sigma_{2}}{\sigma_{1} + \sigma_{2}} + \frac{2\sigma_{2}}{\sigma_{1} + \sigma_{2}} \Phi \left( \frac{x - \mu}{\sigma_{2}} \right), & x \ge \mu, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{\mu, \sigma_{1}, \sigma_{2}}, y) = & \sigma_{1} \mathrm{CRPS} \left( F_{-\infty, 0}^{0, \sigma_{2}/(\sigma_{1} + \sigma_{2})}, \min \left(0, \frac{y - \mu}{\sigma_{1}} \right) \right) \\
& + \sigma_{2} \mathrm{CRPS} \left( F^{\infty, 0}_{0, \sigma_{1}/(\sigma_{1} + \sigma_{2})}, \max \left(0, \frac{y - \mu}{\sigma_{2}} \right) \right),
\end{aligned}
\end{split}\]
where \(\phi\) and \(\Phi\) the standard normal pdf and cdf, respectively, and \(F_{l, L}^{u, U}\)
is the cdf of the truncated and censored normal distribution.
Continuous positive outcomes
Exponential distribution
The exponential distribution has one rate parameter, \(\lambda > 0\). Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\lambda}(x) =
\begin{cases}
\lambda e^{-\lambda x}, & x \ge 0, \\
0, & x < 0, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\lambda}(x) =
\begin{cases}
1 - e^{-\lambda x}, & x \ge 0, \\
0, & x < 0, \\
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{\lambda}, y) = |y| - \frac{2 F_{\lambda}(y)}{\lambda} + \frac{1}{2\lambda}.
\]
Gamma distribution
The gamma distribution has one shape parameter, \(\alpha > 0\), and one rate parameter, \(\beta > 0\). Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\alpha, \beta}(x) =
\begin{cases}
\frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, & x \ge 0, \\
0, & x < 0, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\alpha, \beta}(x) =
\begin{cases}
\frac{\Gamma_{l}(\alpha, \beta x)}{\Gamma(\alpha)} , & x \ge 0, \\
0, & x < 0, \\
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{\alpha, \beta}, y) = y (2 F_{\alpha, \beta}(y) - 1) - \frac{\alpha}{\beta} (2 F_{\alpha + 1, \beta}(y) - 1) - \frac{1}{\beta B(\frac{1}{2}, \alpha)},
\]
where \(\Gamma_{l}\) is the lower incomplete gamma function, and \(B\) is the beta function.
Log-Laplace distribution
The log-Laplace distribution has one location parameter, \(\mu \in \mathbb{R}\), and one scale parameter, \(\sigma > 0\).
Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma}(x) =
\begin{cases}
\frac{1}{2 \sigma x} \exp \left( - \frac{| \log x - \mu |}{\sigma} \right), & x > 0, \\
0, & x \le 0, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma}(x) =
\begin{cases}
0, & x \le 0, \\
\frac{1}{2} \exp \left( \frac{\log x - \mu}{\sigma} \right) , & 0 < \log x < \mu, \\
1 - \frac{1}{2} \exp \left( - \frac{\log x - \mu}{\sigma} \right), & \log x \ge \mu, \\
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{\mu, \sigma}, y) = y (2 F_{\mu, \sigma}(y) - 1) + e^{\mu} \left( \frac{\sigma}{4 - \sigma^{2}} + A(y) \right),
\]
where
\[\begin{split}
A(x) =
\begin{cases}
\frac{1}{1 + \sigma} \left( 1 - (2 F_{\mu, \sigma}(x))^{1 + \sigma} \right), & \log x < \mu, \\
-\frac{1}{1 - \sigma} \left( 1 - (2 (1 - F_{\mu, \sigma}(x)))^{1 - \sigma} \right), & \log x \ge \mu. \\
\end{cases}
\end{split}\]
Log-logistic distribution
The log-logistic distribution has one location parameter, \(\mu \in \mathbb{R}\), and one scale parameter, \(\sigma > 0\).
Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma}(x) =
\begin{cases}
\frac{\exp \left(\frac{\log x - \mu}{\sigma} \right)}{\sigma x \left(1 + \exp \left(\frac{\log x - \mu}{\sigma} \right) \right)^{2}}, & x > 0, \\
0, & x \le 0, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma}(x) =
\begin{cases}
\left(1 + \exp \left( - \frac{\log x - \mu}{\sigma} \right) \right)^{-1}, & x > 0, \\
0, & x \le 0, \\
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{\mu, \sigma}, y) = y (2 F_{\mu, \sigma}(y) - 1) - e^{\mu} B(1 + \sigma, 1 - \sigma) ( 2I(1 + \sigma, 1 - \sigma, F_{\mu, \sigma}(y)) + \sigma - 1 ),
\]
where \(B\) is the beta function, and \(I\) is the regularised incomplete beta function.
Log-normal distribution
The log-normal distribution has one location parameter, \(\mu \in \mathbb{R}\), and one scale parameter, \(\sigma > 0\).
Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma}(x) =
\begin{cases}
\frac{1}{\sigma x} \phi \left( \frac{\log x - \mu}{\sigma} \right) = \frac{1}{\sigma x \sqrt{2 \pi}} \exp \left( - \frac{(\log x - \mu)^{2}}{2\sigma^{2}} \right), & x > 0, \\
0, & x \le 0, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma}(x) =
\begin{cases}
\Phi \left( \frac{\log x - \mu}{\sigma} \right) = \frac{1}{2} \left[ 1 + \text{erf} \left( \frac{\log x - \mu}{\sigma \sqrt{2}} \right) \right], & x > 0, \\
0, & x \le 0, \\
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{\mu, \sigma}, y) = y (2 F_{\mu, \sigma}(y) - 1) - 2 \exp \left( \mu + \frac{\sigma^{2}}{2} \right) \left( \Phi \left( \frac{\log y - \mu - \sigma^{2}}{\sigma} \right) + \Phi \left( \frac{\sigma}{\sqrt{2}} \right) - 1 \right),
\]
where \(\phi\) and \(\Phi\) represent the standard normal pdf and cdf, respectively, and \(\text{erf}\) is the error function.
Bounded or censored outcomes
Beta distribution
The (generalised) beta distribution has two shape parameters, \(\alpha, \beta > 0\), and
lower and upper bound parameters \(l,u \in \mathbb{R}\), such that \(l < u\). Its pdf, cdf, and CRPS are
\[\begin{split}
f_{\alpha, \beta}^{l, u}(x) =
\begin{cases}
\frac{1}{B(\alpha, \beta)} \left( \frac{x - l}{u - l} \right)^{\alpha - 1} \left(\frac{u - x}{u - l} \right)^{\beta - 1} , & l \le x \le u, \\
0, & x < l \quad \text{or} \quad x > u, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\alpha, \beta}^{l, u}(x) =
\begin{cases}
0, & x < l, \\
I \left( \alpha, \beta, \frac{x - l}{u - l} \right), & l \le x \le u, \\
1, & x > u, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{\alpha, \beta}^{0, 1}, y) &= y (2F_{\alpha, \beta}^{0, 1}(y) - 1) + \frac{\alpha}{\alpha + \beta} \left( 1 - 2 F_{\alpha + 1, \beta}^{0, 1}(y) - \frac{2B(2 \alpha, 2 \beta)}{\alpha B(\alpha, \beta)^{2}} \right), \\
\mathrm{CRPS}(F_{\alpha, \beta}^{l, u}, y) &= (u - l) \mathrm{CRPS} \left( F_{\alpha, \beta}^{0, 1}, \frac{y - l}{u - l} \right),
\end{aligned}
\end{split}\]
where \(B\) is the beta function, and \(I\) is the regularised incomplete beta function.
Continuous uniform distribution
The continuous uniform distribution has lower and upper bound parameters \(l,u \in \mathbb{R}\), \(l < u\),
and point mass parameters \(L, U \ge 0\), \(L + U < 1\) that assign mass to the boundary points.
Its pdf+pmf, cdf, and CRPS are
\[\begin{split}
f_{l, u}^{L, U}(x) =
\begin{cases}
0, & x < l \quad \text{or} \quad x > u, \\
L, & x = l, \\
U, & x = u, \\
\frac{1 - L - U}{u - l}, & l < x < u, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{l, u}^{L, U}(x) =
\begin{cases}
0, & x < l, \\
L + (1 - L - U) \frac{x - l}{u - l}, & l \le x \le u, \\
1, & x > u, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1}^{L, U}, y) &= | y - F(y) | + F(y)^{2}(1 - L - U) - F(y)(1 - 2L) + \frac{(1 - L - U)^{2}}{3} + (1 - L)U, \\
\mathrm{CRPS}(F_{l, u}^{L, U}, y) &= (u - l) \mathrm{CRPS} \left( F_{0, 1}^{L, U}, \frac{y - l}{u - l} \right),
\end{aligned}
\end{split}\]
where \(F = F_{0, 1}^{0, 0}\) is the standard uniform distribution function.
Exponential distribution with point mass
The exponential distribution can be generalised to different supports, and to include a
point mass at the boundary. This generalised exponential distribution has one location parameter, \(\mu \in \mathbb{R}\),
one scale parameter \(\sigma > 0\), and one point mass parameter \(M \in [0, 1]\). Its pdf+pmf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma, M}(x) =
\begin{cases}
0, & x < \mu, \\
M, & x = \mu, \\
\frac{1 - M}{\sigma} \exp \left( - \frac{x - \mu}{\sigma} \right), & x > \mu, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, M}(x) =
\begin{cases}
0, & x < \mu, \\
M + (1 - M) \left(1 - \exp \left( - \frac{x - \mu}{\sigma} \right) \right), & x \ge \mu, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1, M}, y) &= |y| - 2(1 - M)F(y) + \frac{(1 - M)^{2}}{2}, \\
\mathrm{CRPS}(F_{\mu, \sigma, M}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1, M}, \frac{y - \mu}{\sigma} \right),
\end{aligned}
\end{split}\]
where \(F = F_{0, 1, 0}\) is the standard exponential distribution function.
Generalised extreme value (GEV) distribution
The generalised extreme value (GEV) distribution has one location parameter, \(\mu \in \mathbb{R}\),
one scale parameter, \(\sigma > 0\), and one shape parameter, \(\xi \in \mathbb{R}\). The GEV distribution can be divided
into three types, depending on the value of \(\xi\). The pdf, cdf, and CRPS for each type are:
If \(\xi = 0\),
\[
f_{\mu, \sigma, \xi}(x) = \frac{1}{\sigma} \exp \left( - \frac{x - \mu}{\sigma} \right) \exp \left( - \exp \left( - \frac{x - \mu}{\sigma} \right) \right),
\]
\[
F_{\mu, \sigma, \xi}(x) = \exp \left( - \exp \left( - \frac{x - \mu}{\sigma} \right) \right),
\]
\[
\mathrm{CRPS}(F_{0, 1, \xi}, y) = - y - 2 \text{Ei}(\log F_{0, 1, \xi}(y)) + \gamma - \log 2,
\]
where \(\text{Ei}\) is the exponential integral, and \(\gamma\) is the Euler-Mascheroni constant.
If \(\xi > 0\),
\[\begin{split}
f_{\mu, \sigma, \xi}(x) =
\begin{cases}
0, & x < \mu - \frac{\sigma}{\xi}, \\
\frac{1}{\sigma} \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-(\xi + 1)/\xi} \exp \left( - \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-1/\xi} \right), & x \ge \mu - \frac{\sigma}{\xi},
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, \xi}(x) =
\begin{cases}
0, & x < \mu - \frac{\sigma}{\xi}, \\
\exp \left( - \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-1/\xi} \right), & x \ge \mu - \frac{\sigma}{\xi},
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{0, 1, \xi}, y) = y (2 F_{0, 1, \xi}(y) - 1) - 2 G_{\xi} (y) - \frac{1 - (2 - 2^{\xi}) \Gamma(1 - \xi)}{\xi},
\]
where
\[\begin{split}
G_{\xi}(x) =
\begin{cases}
0, & x \le -\frac{1}{\xi}, \\
-\frac{F_{0, 1, \xi}(x)}{\xi} + \frac{\Gamma_{u}(1 - \xi, - \log F_{0, 1, \xi}(x))}{\xi}, & x > -\frac{1}{\xi},
\end{cases}
\end{split}\]
with \(\Gamma\) the gamma function, and \(\Gamma_{u}\) the upper incomplete gamma function.
If \(\xi < 0\),
\[\begin{split}
f_{\mu, \sigma, \xi}(x) =
\begin{cases}
\frac{1}{\sigma} \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-(\xi + 1)/\xi} \exp \left( - \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-1/\xi} \right), & x \le \mu - \frac{\sigma}{\xi}, \\
0, & x > \mu - \frac{\sigma}{\xi},
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, \xi}(x) =
\begin{cases}
\exp \left( - \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-1/\xi} \right), & x \le \mu - \frac{\sigma}{\xi}, \\
1, & x > \mu - \frac{\sigma}{\xi},
\end{cases}
\end{split}\]
\[
\mathrm{CRPS}(F_{0, 1, \xi}, y) = y (2 F_{0, 1, \xi}(y) - 1) - 2 G_{\xi} (y) - \frac{1 - (2 - 2^{\xi}) \Gamma(1 - \xi)}{\xi},
\]
where
\[\begin{split}
G_{\xi}(x) =
\begin{cases}
-\frac{F_{0, 1, \xi}(x)}{\xi} + \frac{\Gamma_{u}(1 - \xi, - \log F_{0, 1, \xi}(x))}{\xi}, & x < -\frac{1}{\xi}, \\
-\frac{1}{\xi} + \frac{\Gamma(1 - \xi)}{\xi}, & x \ge -\frac{1}{\xi},
\end{cases}
\end{split}\]
with \(\Gamma\) the gamma function, and \(\Gamma_{u}\) the upper incomplete gamma function.
For all \(\xi\), we have that
\[
\mathrm{CRPS}(F_{\mu, \sigma, \xi}, y) = \sigma \mathrm{CRPS} \left( F_{0, 1, \xi}, \frac{y - \mu}{\sigma} \right).
\]
Generalised Pareto distribution (GPD) with point mass
The generalised Pareto distribution (GPD) has one location parameter, \(\mu \in \mathbb{R}\),
one scale parameter, \(\sigma > 0\), one shape parameter, \(\xi \in \mathbb{R}\), and one point mass
parameter at the lower boundary, \(M \in [0, 1]\). Its pdf+pmf, cdf, and CRPS are:
If \(\xi = 0\),
\[\begin{split}
f_{\mu, \sigma, \xi, M}(x) =
\begin{cases}
0, & x < \mu, \\
M, & x = \mu, \\
\frac{1 - M}{\sigma} \exp \left( - \frac{x - \mu}{\sigma} \right), & x > \mu,
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, \xi, M}(x) =
\begin{cases}
0, & x < \mu, \\
M + (1 - M) \left( 1 - \exp \left( - \frac{x - \mu}{\sigma} \right) \right), & x \ge \mu.
\end{cases}
\end{split}\]
If \(\xi > 0\),
\[\begin{split}
f_{\mu, \sigma, \xi, M}(x) =
\begin{cases}
0, & x < \mu, \\
M, & x = \mu, \\
\frac{1 - M}{\sigma} \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{- (\xi + 1) / \xi}, & x > \mu,
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, \xi, M}(x) =
\begin{cases}
0, & x < \mu, \\
M + (1 - M) \left( 1 - \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{- 1 / \xi} \right), & x \ge \mu.
\end{cases}
\end{split}\]
If \(\xi < 0\),
\[\begin{split}
f_{\mu, \sigma, \xi, M}(x) =
\begin{cases}
M, & x = \mu, \\
\frac{1 - M}{\sigma} \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{- (\xi + 1) / \xi}, & \mu < x \le \mu - \frac{\sigma}{\xi}, \\
0, & \text{otherwise},
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, \xi, M}(x) =
\begin{cases}
0, & x < \mu, \\
M + (1 - M) \left( 1 - \left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{- 1 / \xi} \right), & \mu < x \le \mu - \frac{\sigma}{\xi}, \\
1, & x > \mu - \frac{\sigma}{\xi}.
\end{cases}
\end{split}\]
For all \(\xi\),
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1, \xi, M}, y) &= |y| - \frac{2(1 - M)}{1 - \xi} \left( 1 - (1 - F_{0, 1, \xi, 0}(y))^{1 - \xi} \right) + \frac{(1 - M)^{2}}{2 - \xi}, \\
\mathrm{CRPS}(F_{\mu, \sigma, \xi, M}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1, \xi, M}, \frac{y - \mu}{\sigma} \right).
\end{aligned}
\end{split}\]
Generalised truncated or censored logistic distribution
The logistic distribution can be generalised to account for truncation, censoring, and to
allow point masses at the boundary points of its support. This generalised logistic distribution
has one location parameter, \(\mu \in \mathbb{R}\), one scale parameter \(\sigma > 0\),
lower and upper bound parameters \(l,u \in \mathbb{R}\), \(l < u\), and point mass parameters
\(L, U \ge 0\), \(L + U < 1\), that assign mass to the boundary points \(l\) and \(u\) respectively.
Its pdf+pmf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma, l, L}^{u, U}(x) =
\begin{cases}
0, & x < l, \\
L, & x = l, \\
(1 - L - U) f_{\mu, \sigma}(x), & l < x < u, \\
U, & x = u, \\
0, & x > u, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, l, L}^{u, U}(x) =
\begin{cases}
0, & x < l, \\
L + (1 - L - U) \frac{F_{\mu, \sigma}(x) - F_{\mu, \sigma}(l)}{F_{\mu, \sigma}(u) - F_{\mu, \sigma}(l)}, & l \le x < u, \\
1, & x \ge u, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1, l, L}^{u, U}, y) &= |y - z| + uU^{2} - lL^{2} - \left( \frac{1 - L - U}{F(u) - F(l)} \right) z \left( \frac{(1 - 2L)F(u) + (1 - 2U)F(l)}{1 - L - U} \right) - \left( \frac{1 - L - U}{F(u) - F(l)} \right) (2 \log F(-z) - 2G(u)U - 2G(l)L) - \left( \frac{1 - L - U}{F(u) - F(l)} \right)^{2} (H(u) - H(l)) , \\
\mathrm{CRPS}(F_{\mu, \sigma, l, L}^{u, U}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1, (l - \mu)/\sigma, L}^{(u - \mu)/\sigma, U}, \frac{y - \mu}{\sigma} \right),
\end{aligned}
\end{split}\]
where \(f_{\mu, \sigma}\) and \(F_{\mu, \sigma}\) denote the pdf and cdf of the logistic distribution
with location parameter \(\mu\) and scale parameter \(\sigma\), \(F = F_{0, 1}\) is the cdf of the
standard logistic distribution, and with
\[\begin{split}
\begin{aligned}
G(x) &= xF(x) + \log F(-x), \\
H(x) &= F(x) - xF(x)^{2} + (1 - 2F(x)) \log F(-x), \\
z &= \max \{l, \min \{y, u\}\}.
\end{aligned}
\end{split}\]
Generalised truncated or censored normal distribution
The normal distribution can similarly be generalised to account for truncation, censoring, and to
allow point masses at the boundary points of its support. This generalised normal distribution
has one location parameter, \(\mu \in \mathbb{R}\), one scale parameter \(\sigma > 0\),
lower and upper bound parameters \(l,u \in \mathbb{R}\), \(l < u\), and point mass parameters
\(L, U \ge 0\), \(L + U < 1\), that assign mass to the boundary points \(l\) and \(u\) respectively.
Its pdf+pmf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma, l, L}^{u, U}(x) =
\begin{cases}
0, & x < l, \\
L, & x = l, \\
(1 - L - U) f_{\mu, \sigma}(x), & l < x < u, \\
U, & x = u, \\
0, & x > u, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, l, L}^{u, U}(x) =
\begin{cases}
0, & x < l, \\
L + (1 - L - U) \frac{F_{\mu, \sigma}(x) - F_{\mu, \sigma}(l)}{F_{\mu, \sigma}(u) - F_{\mu, \sigma}(l)}, & l \le x < u, \\
1, & x \ge u, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1, l, L}^{u, U}, y) &= |y - z| + uU^{2} - lL^{2} + \left( \frac{1 - L - U}{\Phi(u) - \Phi(l)} \right) z \left( 2 \Phi(z) - \frac{(1 - 2L)\Phi(u) + (1 - 2U)\Phi(l)}{1 - L - U} \right) + \left( \frac{1 - L - U}{\Phi(u) - \Phi(l)} \right) (2 \phi(z) - 2 \phi(u)U - 2 \phi(l)L) - \left( \frac{1 - L - U}{\Phi(u) - \Phi(l)} \right)^{2} \left( \frac{1}{\sqrt{\pi}} \right) \left(\Phi \left( u\sqrt{2} \right) - \Phi \left( l\sqrt{2} \right) \right), \\
\mathrm{CRPS}(F_{\mu, \sigma, l, L}^{u, U}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1, (l - \mu)/\sigma, L}^{(u - \mu)/\sigma, U}, \frac{y - \mu}{\sigma} \right),
\end{aligned}
\end{split}\]
where \(f_{\mu, \sigma}\) and \(F_{\mu, \sigma}\) denote the pdf and cdf of the normal distribution
with location parameter \(\mu\) and scale parameter \(\sigma\), and \(\phi = f_{0, 1}\) and \(\Phi = F_{0, 1}\)
are the pdf and cdf of the standard normal distribution.
Generalised truncated or censored Student’s t distribution
The Student’s t distribution can also be generalised to account for truncation, censoring, and to
allow point masses at the boundary points of its support. This generalised t distribution
has one location parameter, \(\mu \in \mathbb{R}\), one scale parameter \(\sigma > 0\),
one degrees of freedom parameter, \(\nu > 0\), lower and upper bound parameters \(l,u \in \mathbb{R}\),
\(l < u\), and point mass parameters \(L, U \ge 0\), \(L + U < 1\), that assign mass to the boundary points \(l\) and \(u\) respectively.
Its pdf+pmf, cdf, and CRPS are
\[\begin{split}
f_{\mu, \sigma, \nu, l, L}^{u, U}(x) =
\begin{cases}
0, & x < l, \\
L, & x = l, \\
(1 - L - U) f_{\mu, \sigma, \nu}(x), & l < x < u, \\
U, & x = u, \\
0, & x > u, \\
\end{cases}
\end{split}\]
\[\begin{split}
F_{\mu, \sigma, \nu, l, L}^{u, U}(x) =
\begin{cases}
0, & x < l, \\
L + (1 - L - U) \frac{F_{\mu, \sigma, \nu}(x) - F_{\mu, \sigma, \nu}(l)}{F_{\mu, \sigma, \nu}(u) - F_{\mu, \sigma, \nu}(l)}, & l \le x < u, \\
1, & x \ge u, \\
\end{cases}
\end{split}\]
\[\begin{split}
\begin{aligned}
\mathrm{CRPS}(F_{0, 1, \nu, l, L}^{u, U}, y) &= |y - z| + uU^{2} - lL^{2} + \left( \frac{1 - L - U}{F_{\nu}(u) - F_{\nu}(l)} \right) z \left( 2 F_{\nu}(z) - \frac{(1 - 2L)F_{\nu}(u) + (1 - 2U)F_{\nu}(l)}{1 - L - U} \right) - \left( \frac{1 - L - U}{F_{\nu}(u) - F_{\nu}(l)} \right) (2 G_{\nu}(z) - 2 G_{\nu}(u)U - 2 G_{\nu}(l)L) - \left( \frac{1 - L - U}{F_{\nu}(u) - F_{\nu}(l)} \right)^{2} \bar{B}_{\nu} (H_{\nu}(u) - H_{\nu}(l)), \\
\mathrm{CRPS}(F_{\mu, \sigma, \nu, l, L}^{u, U}, y) &= \sigma \mathrm{CRPS} \left( F_{0, 1, \nu, (l - \mu)/\sigma, L}^{(u - \mu)/\sigma, U}, \frac{y - \mu}{\sigma} \right),
\end{aligned}
\end{split}\]
where \(f_{\mu, \sigma, \nu}\) and \(F_{\mu, \sigma, \nu}\) denote the pdf and cdf of the Student’s t distribution
with location parameter \(\mu\), scale parameter \(\sigma\), and degrees of freedom parameter \(\nu\);
\(f_{\nu} = f_{0, 1, \nu}\) and \(F_{\nu} = F_{0, 1, \nu}\) are the pdf and cdf of the standard t distribution with \(\nu\)
degrees of freedom; and
\[\begin{split}
\begin{aligned}
G_{\nu}(x) &= - \left( \frac{\nu + x^{2}}{\nu - 1} \right) f_{\nu}(x), \\
H_{\nu}(x) &= \frac{1}{2} + \frac{1}{2} \text{sgn}(x) I \left(\frac{1}{2}, \nu - \frac{1}{2}, \frac{x^{2}}{\nu + x^{2}} \right), \\
\bar{B}_{\nu} &= \left( \frac{2 \sqrt{\nu}}{\nu - 1} \right) \frac{B \left(\frac{1}{2}, \nu - \frac{1}{2} \right)}{B(\frac{1}{2}, \frac{\nu}{2})}, \\
z &= \max \{l, \min \{y, u\}\},
\end{aligned}
\end{split}\]
where \(I(a, b, x)\) is the regularised incomplete beta function, and \(B\) is the beta function.
