scores_t: Calculating scores for Student's \(t\)-distribution

Description

These functions calculate scores (CRPS, logarithmic score) and their gradient and Hessian with respect to the parameters of a location-scale transformed Student's \(t\)-distribution. Furthermore, the censoring transformation and the truncation transformation may be introduced on top of the location-scale transformed \(t\)-distribution.

Usage

## score functions
crps_t(y, df, location = 0, scale = 1)
crps_ct(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf)
crps_tt(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf)
crps_gtct(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf, lmass = 0, umass = 0)
logs_t(y, df, location = 0, scale = 1)
logs_tt(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf)
dss_t(y, df, location = 0, scale = 1)
## gradient (location, scale) functions
gradcrps_t(y, df, location = 0, scale = 1)
gradcrps_ct(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf)
gradcrps_tt(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf)
## Hessian (location, scale) functions
hesscrps_t(y, df, location = 0, scale = 1)
hesscrps_ct(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf)
hesscrps_tt(y, df, location = 0, scale = 1, lower = -Inf, upper = Inf)

Value

For the CRPS functions: a vector of score values.

For the gradient and Hessian functions: a matrix with column names corresponding to the respective partial derivatives.

Arguments

y: vector of observations.
df: vector of degrees of freedom.
location: vector of location parameters.
scale: vector of scale paramters.
lower, upper: lower and upper truncation/censoring bounds.
lmass, umass: vectors of point masses in lower and upper respectively.