The LRMoE.jl package supports a collection of distributions commonly used for modelling insurance claim frequency and severity.
Discrete Distributions (Frequency Modelling)
Binomial
- Paramater:
list(n = n, p = p), where n>0 and 0<=p<=1.
- PMF \[
P(X = k) = {n \choose k}p^k(1-p)^{n-k}, \quad \text{ for } k = 0,1,2, \ldots, n.
\]
- See also: Binomial Distribution (Wikipedia)
Negative Binomial
- Parameter:
list(n = n, p = p), where n>0 and 0<=p<=1.
- PMF: \[
P(X = k) = \frac{\Gamma(k+r)}{k! \Gamma(r)} p^r (1 - p)^k, \quad \text{for } k = 0,1,2,\ldots.
\]
- See also: Negative Binomial Distribution (Wolfram)
Poisson
- Parameter:
list(lambda = lambda), where lambda>0.
- PMF: \[
P(X = k) = \frac{\lambda^k}{k!} e^{-\lambda}, \quad \text{ for } k = 0,1,2,\ldots.
\]
- See also: Poisson Distribution (Wikipedia)
Gamma Count
- Parameter:
list(m = m, s = s), where m>0 and s>0.
- PMF: \[
P(X = k) = G(m k, s T) - G(m (k+1), s T), \quad \text{ for } k = 0,1,2, \ldots, n.
\] with \[
G(m k, s T) = \frac{1}{\Gamma(mk)} \int^{sT}_{0} u^{mk - 1} e^{-u} du
\] See also: Gamma Count Distribution (Arxiv)
Continuous Distributions (Severity Modelling)
Burr
- Parameter:
list(shape1 = k, shape2 = c, scale = lambda), where k>0, c>0 and lambda>0.
- PDF: \[
f(x; k, c, \lambda) = \frac{kc}{\lambda} \left( \frac{x}{\lambda} \right)^{c-1} \left( 1+ \left( \frac{x}{\lambda} \right)^{c} \right)^{-k-1},
\quad x \geq 0
\]
- See also: Burr Distribution (Mathworks, implemented in this package), Burr Distribution (Wikipedia, with
lambda = 1)
Gamma
- Parameter:
list(k = k, theta = theta), where k>0 and theta>0.
- PDF: \[
f(x; k, \theta) = \frac{x^{k-1} e^{-x/\theta}}{\Gamma(k) \theta^k},
\quad x > 0
\]
- See also: Gamma Distribution (Wikipedia), shape-scale parameterization
Inverse Gaussian
- Parameter:
list(mean = mu, shape = lambda), where mu>0 and lambda>0.
- PDF: PDF: \[
f(x; \mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}
\exp\left(\frac{-\lambda(x-\mu)^2}{2\mu^2x}\right),
\quad x > 0
\]
- See also: Inverse Gaussian Distribution (Wikipedia)
Lognormal
- Parameter:
list(meanlog = mu, sdlog = sigma), where sigma>0.
- PDF: PDF: \[
f(x; \mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}
\exp\left(\frac{-\lambda(x-\mu)^2}{2\mu^2x}\right),
\quad x > 0
\]
- See also: Inverse Gaussian Distribution (Wikipedia)
Weibull
- Parameter:
list(shape = k, scale = theta), where k>0 and theta>0.
- PDF: \[
f(x; k, \theta) = \frac{k}{\theta} \left( \frac{x}{\theta} \right)^{k-1} e^{-(x/\theta)^k},
\quad x \geq 0
\]
- See also: Weibull Distribution (Wikipedia)
Zero Inflation
Zero inflation is supported for all discrete and continuous experts. They can be constructed by adding zi in front of an expert function, with an additional parameter p_zero for modelling a probability mass at zero. Zero-inflated experts are used in the same way as their non-zero-inflated counterpart. For example, the parameters for a zero-inflated Poisson expert zipoisson are given by list(p_zero = p0, lambda = lambda).
Adding Customized Expert Functions
See here.