# 1 Introduction

The continuous ranked probability score can be given in multiple equivalent forms, \begin{align*} \label{eq:kernel}\mathrm{CRPS}(F,y) &= \mathbb{E}_F|Y-y| - \frac{1}{2}\mathbb{E}_F|Y-Y'| \\ &= \int_{-\infty}^y F(x)^2 \, \mathrm{d} x + \int_y^{\infty} \left(1 - F(x)\right)^2 \, \mathrm{d} x \\ &= 2\int_0^{F(y)} \alpha\left(y - F^{-1}(\alpha)\right) \, \mathrm{d} \alpha + 2\int_{F(y)}^1 (1 - \alpha)\left(F^{-1}(\alpha) - y\right) \, \mathrm{d} \alpha, \end{align*}

where the first is the kernel representation, followed by the threshold decomposition, and lastly the quantile decomposition. The threshold decomposition corresponds to the integral of the Brier score over all event thresholds, while the quantile decomposition is the integral of the quantile score over all probabilities.

# 2 Distributions for variables on the real line

## 2.1 Laplace distribution ("lapl")

CRPS formula: \begin{align*} \mathrm{CRPS}(F, y) &= |y| + \exp(-|y|) - \frac{3}{4}, \\ \mathrm{CRPS}(F_{\mu, \sigma}, y) &= \sigma\, \mathrm{CRPS}\left(F, \tfrac{y - \mu}{\sigma}\right). \end{align*} Laplace distribution: \begin{align*} F(x) &= \begin{cases} \frac{1}{2} \exp(x), & x < 0,\\ 1 - \frac{1}{2} \exp(-x), & x \geq 0, \end{cases} \\ F_{\mu, \sigma}(x) &= F\left(\tfrac{x - \mu}{\sigma}\right). \end{align*}

Parameters:

Name Domain
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$

## 2.2 Logistic distribution ("logis")

CRPS formula: \begin{align*} \mathrm{CRPS}(F, y) &= y - 2\log(F(y)) - 1, \\ \mathrm{CRPS}(F_{\mu, \sigma}, y) &= \sigma\, \mathrm{CRPS}\left(F, \tfrac{y - \mu}{\sigma} \right). \end{align*} Logistic distribution: \begin{align*} F(x) &= \frac{1}{1 + \exp(-x)}, \\ F_{\mu, \sigma}(x) &= F\left(\tfrac{x - \mu}{\sigma}\right). \end{align*}

Parameters:

Name Domain
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$

## 2.3 Normal distribution ("norm")

CRPS formula: \begin{align*} \mathrm{CRPS}(F, y) &= y\left(2F(y)-1\right) + 2f(y) - \frac{1}{\sqrt{\pi}}, \\ \mathrm{CRPS}(F_{\mu, \sigma}, y) &= \sigma\, \mathrm{CRPS}\left(F, \tfrac{y - \mu}{\sigma} \right). \end{align*} Normal distribution: \begin{align*} f(x) &= \tfrac{1}{\sqrt{2\pi}}\exp\left(-\tfrac{x^2}{2}\right), \\ F(x) &= \int_{-\infty}^x f(t)\, \mathrm{d} x, \\ F_{\mu, \sigma}(x) &= F\left(\tfrac{x - \mu}{\sigma}\right). \end{align*}

Parameters:

Name Domain
"mean" $$\mu \in \mathbb{R}$$
"sd" $$\sigma > 0$$

Notes:

• Gneiting et al. (2005) is the original source of this CRPS formula.

## 2.4 Mixture of normal distributions ("mixnorm")

CRPS formula: \begin{align*} \mathrm{CRPS}(F,y) &= \sum_{i=1}^M \omega_i A\left(y-\mu_i,\sigma_i^2\right) - \frac{1}{2} \sum_{i=1}^M\sum_{j=1}^{M}\omega_i \omega_j A\left(\mu_i-\mu_j,\sigma_i^2+\sigma_j^2\right). \end{align*} Finite mixture of normal distributions: \begin{align*} F(x) &= \sum_{i=1}^M \omega_i \Phi\left(\tfrac{x-\mu_i}{\sigma_i}\right), \\ A\left(\mu,\sigma^2\right) &= \mu\left(2\Phi\left(\tfrac{\mu}{\sigma}\right) -1 \right) + 2\sigma \varphi\left(\tfrac{\mu}{\sigma}\right), \\ \varphi(x) &= \tfrac{1}{\sqrt{2\pi}}\exp\left(-\tfrac{x^2}{2}\right), \\ \Phi(x) &= \int_{-\infty}^x \varphi(t)\, \mathrm{d} x. \end{align*}

Parameters:

Name Domain
"m" $$\mu_1, \ldots, \mu_M \in \mathbb{R}$$
"s" $$\sigma_1, \ldots, \sigma_M > 0$$
"w" $$\omega_1, \ldots, \omega_M > 0$$, $$\omega_1 + \ldots + \omega_M = 1$$

Notes:

• Grimit et al. (2006) is the original source of this CRPS formula.
• Computation time increases quadratically in the number of mixture components. For $$M$$ in the order of several thousand, numerical integration to machine precision may be faster than using the formula above.

## 2.5 Student’s $$t$$-distribution ("t")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_\nu, y) &= y\Big(2F_\nu(y) - 1\Big) + 2f_\nu(y) \left(\frac{\nu + y^2}{\nu - 1}\right) - \frac{2\sqrt{\nu}}{\nu - 1}\frac{B(\tfrac{1}{2}, \nu - \tfrac{1}{2})}{B(\tfrac{1}{2}, \tfrac{\nu}{2})^2}, \\ \mathrm{CRPS}(F_{\nu, \mu, \sigma}, y) &= \sigma\, \mathrm{CRPS}\left(F_\nu, \tfrac{y - \mu}{\sigma} \right). \end{align*} Student’s $$t$$-distribution: \begin{align*} f_\nu(x) &= \frac{1}{\sqrt{\nu}B(\tfrac{1}{2},\tfrac{\nu}{2})}\left(1+\frac{x^2}{\nu}\right)^{-\tfrac{\nu+1}{2}}, \\ F_\nu(x) &= \frac{1}{2} + \frac{x\ {}_2F_1(\tfrac{1}{2},\tfrac{\nu+1}{2};\tfrac{3}{2};-\tfrac{x^2}{\nu})}{\sqrt{\nu} B(\tfrac{1}{2},\tfrac{\nu}{2})}, \\ F_{\nu, \mu, \sigma}(x) &= F_\nu\left(\tfrac{x - \mu}{\sigma}\right). \end{align*}

Parameters:

Name Domain
"df" $$\nu > 1$$ (for $$\nu \in (0, 1]$$ the CRPS is undefined)
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$

Mathematical functions:

Symbol Name
$${}_2F_1$$ hypergeometric function
$$B$$ beta function

## 2.6 Two-piece exponential distribution ("2pexp")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{\sigma_1, \sigma_2}, y) &= \begin{cases} \left\lvert y \right\rvert + \frac{2\sigma_1^2}{\sigma_1 + \sigma_2}\exp\left(-\left\lvert \frac{y}{\sigma_1}\right\rvert \right) - \frac{ 2\sigma_1^2}{\sigma_1 + \sigma_2} + \frac{\sigma_1^3 + \sigma_2^3}{2(\sigma_1 + \sigma_2)^2}, & y < 0, \\ \left\lvert y \right\rvert + \frac{2\sigma_2^2}{\sigma_1 + \sigma_2}\exp\left(-\left\lvert \frac{y}{\sigma_2}\right\rvert \right) - \frac{ 2\sigma_2^2}{\sigma_1 + \sigma_2} + \frac{\sigma_1^3 + \sigma_2^3}{2(\sigma_1 + \sigma_2)^2}, & y \ge 0, \end{cases} \\ \mathrm{CRPS}(F_{\mu, \sigma_1, \sigma_2}, y) &= \mathrm{CRPS}(F_{\sigma_1, \sigma_2}, y - \mu). \end{align*} Two-piece exponential distribution: \begin{align*} F_{\sigma_1, \sigma_2}(x) &= \begin{cases} \frac{\sigma_1}{\sigma_1 + \sigma_2}\exp\left(\frac{x}{\sigma_1}\right), & x < 0, \\ 1 - \frac{\sigma_2}{\sigma_1 + \sigma_2}\exp\left(-\frac{x}{\sigma_2}\right), & x \ge 0,\end{cases} \\ F_{\mu, \sigma_1, \sigma_2}(x) &= F_{\sigma_1, \sigma_2}(x - \mu). \end{align*}

Parameters:

Name Domain
"location" $$\mu \in \mathbb{R}$$
"scale1" $$\sigma_1 > 0$$
"scale2" $$\sigma_2 > 0$$

## 2.7 Two-piece normal distribution ("2pnorm")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{\sigma_1, \sigma_2}, y) &= \sigma_1\, \mathrm{CRPS}\left(F_{-\infty, 0}^{0, \sigma_2/(\sigma_1 + \sigma_2)}, \tfrac{\min(0, y)}{\sigma_1}\right) \\ &\quad + \sigma_2\, \mathrm{CRPS}\left(F_{0, \sigma_1/(\sigma_1 + \sigma_2)}^{\infty, 0}, \tfrac{\max(0, y)}{\sigma_2}\right), \\ \mathrm{CRPS}(F_{\mu, \sigma_1, \sigma_2}, y) &= \mathrm{CRPS}(F_{\sigma_1, \sigma_2}, y - \mu), \end{align*}

with $$F_{l, L}^{u, U}$$ as in Generalized truncated/censored normal distribution.

Two-piece normal distribution: \begin{align*} F_{\sigma_1,\sigma_2}(x) &= \begin{cases} \frac{2\sigma_1}{\sigma_1+\sigma_2}\Phi\left(\frac{x}{\sigma_1}\right), & x < 0,\\ \frac{\sigma_1-\sigma_2}{\sigma_1+\sigma_2} + \frac{2\sigma_2}{\sigma_1+\sigma_2} \Phi\left(\frac{x}{\sigma_2}\right), & x \ge 0, \end{cases} \\ F_{\mu, \sigma_1, \sigma_2}(x) &= F_{\sigma_1, \sigma_2}(x - \mu), \\ \Phi(x) &= \int_{-\infty}^x \varphi(t)\, \mathrm{d} x, \\ \varphi(x) &= \tfrac{1}{\sqrt{2\pi}}\exp\left(-\tfrac{x^2}{2}\right). \end{align*}

Parameters:

Name Domain
"location" $$\mu \in \mathbb{R}$$
"scale1" $$\sigma_1 > 0$$
"scale2" $$\sigma_2 > 0$$

Notes:

• Gneiting and Thorarinsdottir (2010) give an explicit CRPS formula.
• The left tail follows a right-censored normal distribution, censored at 0 with mass , with location parameter 0 and scale parameter $$\sigma_1$$. The right tail follows a left-censored normal distribution, censored at 0 with mass , with location parameter 0 and scale parameter $$\sigma_2$$.
• The same construction can be used to implement two-piece families based on the $$t$$, logistic, exponential (see section Two-piece exponential), and generalized Pareto distributions.

# 3 Distributions for non-negative variables

## 3.1 Exponential distribution ("exp")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_\lambda, y) &= |y| - \frac{2F_\lambda(y)}{\lambda} + \frac{1}{2\lambda}. \end{align*} Exponential distribution: \begin{align*} F_\lambda(x) = \begin{cases} 1 - \exp(-\lambda x), & x \ge 0, \\ 0, & x < 0.\end{cases} \end{align*}

Parameters:

Name Domain
"rate" $$\lambda > 0$$

## 3.2 Gamma distribution ("gamma")

CRPS formula: $\mathrm{CRPS}(F_{\alpha,\beta},y) = y\left(2F_{\alpha,\beta}(y)-1\right) - \frac{\alpha}{\beta}\left(2F_{\alpha + 1, \beta}(y) -1\right) - \frac{1}{\beta B\left(\tfrac{1}{2},\alpha\right)}$ Gamma distribution: \begin{align*} F_{\alpha,\beta}(x) &= \begin{cases}\frac{\Gamma_l(\alpha,\beta x)}{\Gamma(\alpha)}, & x \geq 0, \\ 0, & x < 0. \end{cases} \end{align*}

Parameters:

Name Domain
"shape" $$\alpha > 0$$
"rate" $$\beta > 0$$

Mathematical functions:

Symbol Name
$$B$$ beta function
$$\Gamma$$ gamma function
$$\Gamma_l$$ lower incomplete gamma function

Notes:

• Möller and Scheuerer (2015) is the original source of this CRPS formula.

## 3.3 Log-Laplace distribution ("llapl")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{\mu, \sigma}, y) &= y\Big(2F_{\mu, \sigma}(y) - 1\Big) + \exp(\mu) \left(\tfrac{\sigma}{4 - \sigma^2} + A(y) \right) \end{align*} Log-Laplace distribution: \begin{align*} F_{\mu, \sigma}(x) &= \begin{cases} 0, & x \leq 0,\\ \frac{1}{2}\exp\left(\frac{\log x - \mu}{\sigma}\right), & 0 < x < \exp(\mu), \\ 1 - \frac{1}{2}\exp\left(-\frac{\log x - \mu}{\sigma}\right), & x \geq \exp(\mu), \end{cases} \\ A(x) &= \begin{cases} \frac{1}{1 + \sigma}\left(1-\left(2F_{\mu, \sigma}(x)\right)^{1+\sigma}\right), & x < \exp(\mu), \\ -\frac{1}{1-\sigma}\left(1-\left(2(1-F_{\mu, \sigma}(x))\right)^{1-\sigma}\right), & y \geq \exp(\mu). \end{cases} \end{align*}

Parameters:

Name Domain
"locationlog" $$\mu \in \mathbb{R}$$
"scalelog" $$\sigma \in (0, 1)$$ (for $$\sigma \ge 1$$ the CRPS is undefined)

## 3.4 Log-logistic distribution ("llogis")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{\mu, \sigma}, y) &= y\left(2F_{\mu, \sigma}(y) - 1\right) - 2\exp(\mu) B(F_{\mu, \sigma}(y); 1 + \sigma, 1 - \sigma) \\ &\quad + \exp(\mu)(1 - \sigma)B(1 + \sigma, 1 - \sigma) \end{align*} Log-logistic distribution: \begin{align*} F_{\mu, \sigma}(x) &= \begin{cases} 0, & x \leq 0, \\ \left(1 + \exp\left(-\tfrac{\log x - \mu}{\sigma}\right)\right)^{-1}, & x > 0, \end{cases} \end{align*}

Parameters:

Name Domain
"locationlog" $$\mu \in \mathbb{R}$$
"scalelog" $$\sigma \in (0, 1)$$ (for $$\sigma \ge 1$$ the CRPS is undefined)

Mathematical functions:

Symbol Name
$$B(\cdot, \cdot)$$ beta function
$$B(x; \cdot, \cdot)$$ incomplete beta function

Notes:

• Taillardat et al. (2016) give an alternative CRPS formula.

## 3.5 Log-normal distribution ("lnorm")

\begin{align*} \mathrm{CRPS}(F_{\mu,\sigma},y) &= y\left(2F_{\mu, \sigma}(y) - 1\right) - 2 \exp(\mu+\sigma^2/2)\left(\Phi\left(\tfrac{\log y -\mu - \sigma^2}{\sigma}\right) + \Phi\left(\tfrac{\sigma}{\sqrt{2}}\right) - 1\right) \end{align*} Log-normal distribution: \begin{align*} F_{\mu,\sigma}(x) &= \begin{cases} 0, & x\leq 0, \\ \Phi\left(\tfrac{\log x - \mu}{\sigma}\right), & x > 0, \end{cases} \\ \varphi(x) &= \tfrac{1}{\sqrt{2\pi}}\exp\left(-\tfrac{x^2}{2}\right), \\ \Phi(x) &= \int_{-\infty}^x \varphi(t)\, \mathrm{d} x. \end{align*}

Parameters:

Name Domain
"locationlog" $$\mu \in \mathbb{R}$$
"scalelog" $$\sigma > 0$$

Notes:

• Baran and Lerch (2015) is the original source of this CRPS formula.

# 4 Distributions with flexible support and/or point masses

## 4.1 Beta distribution ("beta")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{\alpha, \beta}, y) &= y(2F_{\alpha, \beta}(y) - 1) + \frac{\alpha}{\alpha + \beta} \left(1 - 2F_{\alpha + 1, \beta}(y) - \frac{2B(2\alpha, 2\beta)}{\alpha B(\alpha, \beta)^2} \right), \\ \mathrm{CRPS}(F_{l, \alpha, \beta}^{u}, y) &= (u - l)\, \mathrm{CRPS}\left(F_{\alpha, \beta}, \tfrac{y - l}{u - l} \right). \end{align*} Beta distribution: \begin{align*} F_{\alpha, \beta}(x) &= \begin{cases} 0 & x < 0\\ I(x; \alpha, \beta) & 0 \leq x < 1\\ 1 & x \geq 1 \end{cases}, \\ F_{l, \alpha, \beta}^{u}(x) &= F_{\alpha, \beta}\left(\tfrac{x - l}{u - l}\right). \end{align*}

Parameters:

Name Domain
"shape1" $$\alpha > 0$$
"shape2" $$\beta > 0$$
"lower" $$l \in \mathbb{R}$$, $$l < u$$
"upper" $$u \in \mathbb{R}$$, $$l < u$$

Mathematical functions:

Symbol Name
$$B$$ beta function
$$I$$ regularized incomplete beta function

Notes:

• Taillardat et al. (2016) give an equivalent expression.

## 4.2 Continuous uniform ("unif")

CRPS formula: \begin{align*} \mathrm{CRPS}(F, y) &= |y - z| + z^2 - z + \frac{1}{3}, \\ \text{where}\quad z &= \begin{cases} 0, & y < 0, \\ y, & 0 \le y < 1, \\ 1, & y \ge 1. \end{cases} \end{align*} Continuous uniform: \begin{align*} F(x) &= \begin{cases} 0, & x < 0,\\ x, & 0 \leq x < 1,\\ 1, & x > 1. \end{cases} \end{align*}

### 4.2.1 Generalized continuous uniform ("unif")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{L}^{U}, y) &= |y - z| + z^2(1 - L - U) - z(1 - 2L) \\ &\quad + \frac{(1 - L - U)^2}{3} + (1 - L)U, \\ \text{where}\quad z &= \begin{cases} 0, & y < 0, \\ y, & 0 \le y < 1, \\ 1, & y \ge 1, \end{cases} \\ \mathrm{CRPS}(F_{l, L}^{u, U}, y) &= (u - l)\, \mathrm{CRPS}\left(F_{L}^{U}, \tfrac{y - l}{u - l} \right). \end{align*} Continuous uniform with point masses and support transformation: \begin{align*} F_{L}^{U}(x) &= \begin{cases} 0, & x < 0, \\ L + (1 - L - U) x, & 0 \leq x < 1,\\ 1, & x \geq 1, \end{cases} \\ F_{l, L}^{u, U}(x) &= F_{L}^{U}\left(\tfrac{x - l}{u - l}\right). \end{align*}

Parameters:

Name Domain
"min" $$l \in \mathbb{R}$$, $$l < u$$
"max" $$u \in \mathbb{R}$$, $$l < u$$
"lmass" $$L \ge 0$$, $$L + U < 1$$
"umass" $$U \ge 0$$, $$L + U < 1$$

## 4.3 Exponential distribution with point mass ("expM")

\begin{align*} \mathrm{CRPS}(F_{M}, y) &= |y| - 2 (1 - M)F(y) + \frac{(1 - M)^2}{2}, \\ \mathrm{CRPS}(F_{M, \mu, \sigma}, y) &= \sigma\, \mathrm{CRPS}\left(F_M, \tfrac{y - \mu}{\sigma} \right). \end{align*} Exponential distribution with point mass: \begin{align*} F_M(x) &= \begin{cases} M + (1 - M)F(x), & x \ge 0, \\ 0, & x < 0, \end{cases} \\ F(x) &= \begin{cases} 1 - \exp(-x), & x \ge 0, \\ 0, & x < 0, \end{cases} \\ F_{M, \mu, \sigma}(x) &= F_M\left(\tfrac{x - \mu}{\sigma}\right). \end{align*}

Parameters:

Name Domain
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$
"mass" $$M \in [0, 1]$$

## 4.4 Generalized extreme value distribution ("gev")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{\xi, \mu, \sigma}, y) &= \sigma\, \mathrm{CRPS}\left(F_{\xi}, \tfrac{y - \mu}{\sigma} \right), \\ \text{For \xi = 0:}\quad \mathrm{CRPS}(F_{\xi},y) &= - y - 2 \mathrm{Ei}(\log F_{\xi}(y)) + C - \log 2, \\ \text{For \xi \neq 0:} \quad \mathrm{CRPS}(F_\xi, y) &= y\left(2F_\xi(y) - 1\right) - 2G_\xi(y) - \frac{1 - \left(2 - 2^\xi\right)\Gamma(1 - \xi)}{\xi}. \end{align*} Generalized extreme value distribution: \begin{align*} F_{\xi, \mu, \sigma}(x) &= F_{\xi}\left(\tfrac{x - \mu}{\sigma}\right), \\ \text{for \xi = 0:}\quad F_{\xi}(x) &= \exp\left(-\exp(-x)\right) \\ \text{for \xi > 0:}\quad F_{\xi}(x) &= \begin{cases} 0, & x \le -\frac{1}{\xi}, \\ \exp\left(-(1+\xi x)^{-1/\xi}\right), & x > -\frac{1}{\xi}, \end{cases}\\ G_{\xi}(x) &= \begin{cases} 0, & x \le -\frac{1}{\xi}, \\ -\frac{F_\xi(x)}{\xi} + \frac{\Gamma_u(1 - \xi, -\log F_\xi(x))}{\xi}, & x > -\frac{1}{\xi}, \end{cases} \\ \text{for \xi < 0:}\quad F_{\xi}(x) &= \begin{cases} \exp\left(-(1+\xi x)^{-1/\xi}\right), & x < -\frac{1}{\xi}, \\ 1, & x \ge -\frac{1}{\xi}, \end{cases} \\ G_\xi(x) &= \begin{cases} -\frac{F_\xi(x)}{\xi} + \frac{\Gamma_u(1 - \xi, -\log F_\xi(x))}{\xi}, & x < -\frac{1}{\xi}, \\ -\frac{1}{\xi} + \frac{\Gamma(1 - \xi)}{\xi}, & x \ge -\frac{1}{\xi}. \end{cases} \end{align*}

Parameters:

Name Domain
"shape" $$\xi < 1$$ (for $$\xi \ge 1$$ the CRPS is undefined)
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$

Mathematical constants and functions:

Symbol Name
$$C$$ Euler-Mascheroni constant
$$\mathrm{Ei}$$ exponential integral
$$\Gamma$$ gamma function
$$\Gamma_u$$ upper incomplete gamma function

Notes:

• Friederichs and Thorarinsdottir (2012) is the original source of this formula.

## 4.5 Generalized Pareto distribution with point mass ("gpd")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{M, \xi}, y) &= |y| - \frac{2(1 - M)}{1 - \xi}\left(1 - \left(1 - F_\xi(y)\right)^{1 - \xi}\right) + \frac{(1 - M)^2}{2 - \xi}, \\ \mathrm{CRPS}(F_{M, \xi, \mu, \sigma}, y) &= \sigma\, \mathrm{CRPS}\left(F_{M, \xi}, \tfrac{y - \mu}{\sigma} \right). \end{align*} Generalized Pareto distribution with point mass: \begin{align*} F_{M, \xi}(x) &= \begin{cases} M + (1 - M)F_\xi(x), & x \ge 0, \\ 0, & x < 0, \end{cases} \\ F_{M, \xi, \mu, \sigma}(x) &= F_{M, \xi}\left(\tfrac{x - \mu}{\sigma}\right) \\ \text{for \xi > 0:}\quad F_\xi(x) &= \begin{cases} 0, & x < 0, \\ 1 - (1 + \xi x)^{-1/\xi}, & x \ge 0, \end{cases} \\ \text{for \xi < 0:}\quad F_\xi(x) &= \begin{cases} 0, & x < 0, \\ 1 - (1 + \xi x)^{-1/\xi}, & 0 \le x < |\xi|^{-1}, \\ 1, & x \ge |\xi|^{-1}, \end{cases} \\ \text{for \xi = 0:}\quad F_\xi(x) &= \begin{cases} 0, & x < 0, \\ 1 - \exp(-x), & x \ge 0. \end{cases} \end{align*}

Parameters:

Name Domain
"shape" $$\xi < 1$$ (for $$\xi \ge 1$$ the CRPS is undefined)
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$
"mass" $$M \in [0, 1]$$

Notes:

• Friederichs and Thorarinsdottir (2012) give CRPS formulas for the generalized Pareto distribution without a point mass.

## 4.6 Generalized truncated/censored logistic distribution ("gtclogis")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l, L}^{u, U}, y\right) &= |y - z| + uU^2 - lL^2 \\ &\quad - \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) \\ &\quad - \left(\frac{1 - L - U}{F(u) - F(l)}\right)\left(2\log F(-z) - 2G(u)U - 2G(l)L\right) \\ &\quad - \left(\frac{1 - L - U}{F(u) - F(l)}\right)^2 (H(u) - H(l)), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u, \end{cases} \\ \mathrm{CRPS}(F_{l, L, \mu, \sigma}^{u, U}, y) &= \sigma\, \mathrm{CRPS}\left(F_{(l - \mu)/\sigma, L}^{(u - \mu)/\sigma, U}, \tfrac{y - \mu}{\sigma} \right). \end{align*} Generalized truncated/censored logistic distribution: \begin{align*} F_{l, L}^{u, U}(x) &= \begin{cases} 0, & x < l, \\ \frac{1 - L - U}{F(u) - F(l)} F(z) - \frac{1 - L - U}{F(u) - F(l)} F(l) + L, & l \leq x < u, \\ 1, & x \geq u, \end{cases} \\ F_{l, L, \mu, \sigma}^{u, U}(x) &= F_{(l - \mu)/\sigma, L}^{(u - \mu)/\sigma, U}\left(\tfrac{x - \mu}{\sigma}\right), \\ F(x) &= \frac{1}{1 + \exp(-x)}, \\ G(x) &= xF(x) + \log F(-x), \\ H(x) &= F(x) - xF(x)^2 + (1 - 2F(x))\log F(-x). \end{align*}

Parameters:

Name Domain
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$
"lower" $$l \in \mathbb{R}$$, $$l < u$$
"upper" $$u \in \mathbb{R}$$, $$l < u$$
"lmass" $$L \ge 0$$, $$L + U < 1$$
"umass" $$U \ge 0$$, $$L + U < 1$$

### 4.6.1 Censored logistic distribution ("clogis")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l}^{u}, y\right) &= |y - z| + z + \log \left(\frac{F(-l)F(u)}{F(z)^2}\right) - F(u) + F(l), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u. \end{cases} \end{align*} Censored logistic distribution: \begin{align*} F_{l}^{u}(x) = \begin{cases} 0, & x < l, \\ F(x), & l \leq x < u, \\ 1, & x \geq u, \end{cases} \end{align*}

where all other symbols are as given in Generalized truncated/censored logistic distribution.

Notes:

• Taillardat et al. (2016) give a CRPS formula for a left-censored logistic distribution with boundary 0.

### 4.6.2 Truncated logistic distribution ("tlogis")

\begin{align*} \mathrm{CRPS}\left(F_{l}^{u}, y\right) &= |y - z| - \frac{zF(u) + zF(l) + 2\log F(-z)}{F(u) - F(l)} - \frac{H(u) - H(l)}{(F(u) - F(l))^2}, \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u. \end{cases} \end{align*} Truncated logistic distribution: \begin{align*} F_{l}^{u}(x) = \begin{cases} 0, & x < l, \\ \frac{F(x) - F(l)}{F(u) - F(l)}, & l \leq x < u, \\ 1, & x \geq u, \end{cases} \end{align*}

where all other symbols are as given in Generalized truncated/censored logistic distribution.

Notes:

• Möller and Scheuerer (2015) give a CRPS formula for a left-truncated logistic distribution with boundary 0.

## 4.7 Generalized truncated/censored normal distribution ("gtcnorm")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l, L}^{u, U}, y\right) &= |y - z| + uU^2 - lL^2 \\ &\quad + \left(\frac{1 - L - U}{F(u) - F(l)}\right)z\left(2F(z) - \frac{(1 - 2L)F(u) + (1 - 2U)F(l)}{1 - L - U}\right) \\ &\quad + \left(\frac{1 - L - U}{F(u) - F(l)}\right)\left(2f(z) - 2f(u)U - 2f(l)L\right) \\ &\quad - \left(\frac{1 - L - U}{F(u) - F(l)}\right)^2 \left(\frac{1}{\sqrt{\pi}}\right) \left(F\left(u\sqrt{2}\right) - F\left(l\sqrt{2}\right)\right), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u, \end{cases} \\ \mathrm{CRPS}(F_{l, L, \mu, \sigma}^{u, U}, y) &= \sigma\, \mathrm{CRPS}\left(F_{(l - \mu)/\sigma, L}^{(u - \mu)/\sigma, U}, \tfrac{y - \mu}{\sigma} \right). \end{align*} Generalized truncated/censored normal distribution: \begin{align*} F_{l, L}^{u, U}(x) &= \begin{cases} 0, & x < l, \\ \frac{1 - L - U}{F(u) - F(l)} F(z) - \frac{1 - L - U}{F(u) - F(l)} F(l) + L, & l \leq x < u, \\ 1, & x \geq u, \end{cases} \\ F_{l, L, \mu, \sigma}^{u, U}(x) &= F_{(l - \mu)/\sigma, L}^{(u - \mu)/\sigma, U}\left(\tfrac{x - \mu}{\sigma}\right), \\ F(x) &= \int_{-\infty}^x f(t)\, \mathrm{d}t, \\ f(x) &= \frac{1}{\sqrt{2\pi}}\exp(-x^2/2). \end{align*}

Parameters:

Name Domain
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$
"lower" $$l \in \mathbb{R}$$, $$l < u$$
"upper" $$u \in \mathbb{R}$$, $$l < u$$
"lmass" $$L \ge 0$$, $$L + U < 1$$
"umass" $$U \ge 0$$, $$L + U < 1$$

### 4.7.1 Censored normal distribution ("cnorm")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l}^{u}, y\right) &= |y - z| + uF(-u)^2 - lF(l)^2 \\ & \quad + z(2F(z) - 1) \\ & \quad + 2f(z) - 2f(u)F(-u) - 2f(l)F(l) \\ & \quad - \left(\frac{1}{\sqrt{\pi}}\right) \left(F\left(u\sqrt{2}\right) - F\left(l\sqrt{2}\right)\right), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u. \end{cases} \end{align*} Censored normal distribution: \begin{align*} F_{l}^{u}(x) = \begin{cases} 0, & x < l, \\ F(x), & l \leq x < u, \\ 1, & x \geq u, \end{cases} \end{align*}

where all other symbols are as given in Generalized truncated/censored normal distribution.

Notes:

• Thorarinsdottir and Gneiting (2010) give a CRPS formula for a left-censored normal distribution with boundary 0.

### 4.7.2 Truncated normal distribution ("tnorm")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l}^{u}, y\right) &= |y - z| \\ & \quad + \left(\frac{1}{F(u) - F(l)}\right) z\left(2F(z) - F(u) - F(l)\right) \\ & \quad + \left(\frac{1}{F(u) - F(l)}\right) 2f(z) \\ & \quad - \left(\frac{1}{F(u) - F(l)}\right)^2 \left(\frac{1}{\sqrt{\pi}}\right) \left(F\left(u\sqrt{2}\right) - F\left(l\sqrt{2}\right)\right), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u. \end{cases} \end{align*} Truncated normal distribution: \begin{align*} F_{l}^{u}(x) = \begin{cases} 0, & x < l, \\ \frac{F(x) - F(l)}{F(u) - F(l)}, & l \leq x < u, \\ 1, & x \geq u, \end{cases} \end{align*}

where all other symbols are as given in Generalized truncated/censored normal distribution.

Notes:

• Gneiting et al. (2006) give a CRPS formula for a left-truncated normal distribution with boundary 0.

## 4.8 Generalized truncated/censored Student’s $$t$$-distribution ("gtct")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l, L, \nu}^{u, U}, y\right) &= |y - z| + uU^2 - lL^2 \\ &\quad + \left(\frac{1 - L - U}{F_\nu(u) - F_\nu(l)}\right) z\left(2F_\nu(z) - \frac{(1 - 2L)F_\nu(u) + (1 - 2U)F_\nu(l)}{1 - L - U}\right) \\ &\quad - \left(\frac{1 - L - U}{F_\nu(u) - F_\nu(l)}\right)\left(2G_\nu(z) - 2G_\nu(u)U - 2G_\nu(l)L\right) \\ &\quad - \left(\frac{1 - L - U}{F_\nu(u) - F_\nu(l)}\right)^2 \bar{B}_\nu \left(H_\nu(u) - H_\nu(l)\right), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u, \end{cases} \\ \mathrm{CRPS}(F_{l, L, \nu, \mu, \sigma}^{u, U}, y) &= \sigma\, \mathrm{CRPS}\left(F_{(l - \mu)/\sigma, L, \nu}^{(u - \mu)/\sigma, U}, \tfrac{y - \mu}{\sigma} \right). \end{align*} Generalized truncated/censored Student’s $$t$$-distribution: \begin{align*} F_{l, L, \nu}^{u, U}(x) &= \begin{cases} 0, & x < l, \\ \frac{1 - L - U}{F(u) - F(l)} F(z) - \frac{1 - L - U}{F(u) - F(l)} F(l) + L, & l \leq x < u, \\ 1, & x \geq u, \end{cases} \\ F_{l, L, \nu, \mu, \sigma}^{u, U}(x) &= F_{\tfrac{l - \mu}{\sigma}, L, \nu}^{\tfrac{u - \mu}{\sigma}, U}\left(\tfrac{x - \mu}{\sigma}\right), \\ f_\nu(x) &= \frac{1}{\sqrt{\nu}B\left(\tfrac{1}{2}, \tfrac{\nu}{2}\right)}\left(1 + \frac{x^2}{\nu}\right)^{-(\nu + 1)/2}, \\ F_\nu(x) &= \frac{1}{2} + \frac{x\ {}_2F_1\left(\tfrac{1}{2},\tfrac{\nu+1}{2};\tfrac{3}{2};-\tfrac{x^2}{\nu}\right)}{\sqrt{\nu} B\left(\tfrac{1}{2},\tfrac{\nu}{2}\right)}, \\ G_\nu(x) &= -\left(\frac{\nu + x^2}{\nu - 1}\right) f_\nu(x), \\ H_\nu(x) &= \frac{1}{2} + \frac{1}{2}\, \mathrm{sgn}(x)\, I \left(\tfrac{x^2}{\nu + x^2}; \tfrac{1}{2}, \nu - \tfrac{1}{2}\right), \\ \bar{B}_\nu &= \left(\frac{2\sqrt{\nu}}{\nu - 1}\right)\frac{B\left(\tfrac{1}{2}, \nu - \tfrac{1}{2}\right)}{B\left(\tfrac{1}{2}, \tfrac{\nu}{2}\right)^2}. \end{align*}

Parameters:

Name Domain
"df" $$\nu > 0$$ (for $$\nu \le 1$$ the CRPS is undefined)
"location" $$\mu \in \mathbb{R}$$
"scale" $$\sigma > 0$$
"lower" $$l \in \mathbb{R}$$, $$l < u$$
"upper" $$u \in \mathbb{R}$$, $$l < u$$
"lmass" $$L \ge 0$$, $$L + U < 1$$
"umass" $$U \ge 0$$, $$L + U < 1$$

Mathematical functions:

Symbol Name
$${}_2F_1$$ hypergeometric function
$$B$$ beta function
$$I$$ regularized incomplete beta function
$$\mathrm{sgn}$$ sign function

See also: Student’s $$t$$-distribution

### 4.8.1 Censored $$t$$-distribution ("ct")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l, \nu}^{u}, y\right) &= |y - z| + uF_\nu(-u)^2 - lF_\nu(l)^2 \\ & \quad + z(2F_\nu(z) - 1) \\ & \quad - 2G_\nu(z) + 2G_\nu(u)F_\nu(-u) + 2G_\nu(l)F_\nu(l) \\ & \quad - \bar{B}_\nu \left(H_\nu(u) - H_\nu(l) \right), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u. \end{cases} \end{align*} Censored Student’s $$t$$-distribution: \begin{align*} F_{l, \nu}^{u}(x) &= \begin{cases} 0, & x < l, \\ F_\nu(x), & l \leq x < u, \\ 1, & x \geq u, \end{cases} \end{align*}

where all other symbols are as given in Generalized truncated/censored Student’s $$t$$-distribution.

### 4.8.2 Truncated $$t$$-distribution ("tt")

CRPS formula: \begin{align*} \mathrm{CRPS}\left(F_{l, \nu}^{u}, y\right) &= |y - z| \\ & \quad + \left(\frac{1}{F_\nu(u) - F_\nu(l)}\right) z\left(2F_\nu(z) - F_\nu(u) - F_\nu(l)\right) \\ & \quad - \left(\frac{1}{F_\nu(u) - F_\nu(l)}\right) 2G_\nu(z) \\ & \quad - \left(\frac{1}{F_\nu(u) - F_\nu(l)}\right)^2 \bar{B}_\nu \left(H_\nu(u) - H_\nu(l)\right), \\ \text{where}\quad z&= \begin{cases} l, & y < l, \\ y, & l \le y < u, \\ u, & y \ge u. \end{cases} \end{align*} Truncated Student’s $$t$$-distribution: \begin{align*} F_{l, \nu}^{u}(x) &= \begin{cases} 0, & x < l, \\ \frac{F_\nu(x) - F_\nu(l)}{F_\nu(u) - F_\nu(l)}, & l \leq x < u, \\ 1, & x \geq u, \end{cases} \end{align*}

where all other symbols are as given in Generalized truncated/censored Student’s $$t$$-distribution.

# 5 Distributions for discrete variables

## 5.1 Negative binomial distribution ("nbinom")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_{n, p}, y) &= y\left(2F_{n, p}(y) - 1\right) \\ &\quad - \frac{n(1-p)}{p^2}\left(p\left(2F_{n+1,p}(y-1) - 1\right) + {\ }_2F_1\left(n+1, \tfrac{1}{2}; 2; -\tfrac{4(1-p)}{p^2}\right)\right) \end{align*} Negative binomial distribution: \begin{align*} F_{n, p}(x) &= \begin{cases} I\left(p; n, \lfloor x+1\rfloor\right), & x \geq 0, \\ 0, & x < 0, \end{cases} \\ f_{n, p}(x) &= \begin{cases}\frac{\Gamma(x+n)}{\Gamma(n) x!} p^n (1-p)^x, & x = 0, 1, 2, \ldots, \\ 0, & \text{otherwise}. \end{cases} \end{align*}

Parameters:

Name Domain
"size" $$n > 0$$
"prob" $$p \in (0, 1]$$

Mathematical functions:

Symbol Name
$$\lfloor\cdot\rfloor$$ floor function
$${}_2F_1$$ hypergeometric function
$$I$$ regularized incomplete beta function

Notes:

• Wei and Held (2014) is the original source of this CRPS formula.

## 5.2 Poisson distribution ("pois")

CRPS formula: \begin{align*} \mathrm{CRPS}(F_\lambda, y) &= (y - \lambda) \left(2F_\lambda(y) - 1\right) + 2\lambda f_\lambda\left(\lfloor y\rfloor\right) - \lambda \exp(-2\lambda)\left(I_0(2\lambda) + I_1(2\lambda)\right) \end{align*} Poisson distribution: \begin{align*} F_\lambda(x) &= \begin{cases} \frac{\Gamma_u(\lfloor x+1\rfloor, \lambda)}{\Gamma(\lfloor x+1 \rfloor)}, & x \geq 0,\\ 0, & x < 0, \end{cases}\\ f_\lambda(x) &= \begin{cases}\frac{\lambda^x}{x!}e^{-\lambda}, & x = 0, 1, 2, \ldots, \\ 0, & \text{otherwise}, \end{cases} \end{align*}

Parameter:

Name Domain
"lambda" $$\lambda > 0$$

Mathematical functions:

Symbol Name
$$\lfloor \cdot \rfloor$$ floor function
$$\Gamma$$ gamma function
$$\Gamma_u$$ upper incomplete gamma function
$$I_m$$ modified Bessel function of the first kind

Notes:

• Wei and Held (2014) is the original source of this CRPS formula.