cramer_vonmises {cylcop}

Cramér-von-Mises criterion

Description

Calculate the Cramér-von-Mises criterion with a p-value (via parametric bootstrapping) to assess the goodness of fit of a parametric copula compared to the empirical copula of the data.

Usage

cramer_vonmises(
  copula,
  theta,
  x,
  n_bootstrap = 1000,
  parameters = NULL,
  optim.method = "L-BFGS-B",
  optim.control = list(maxit = 100)
)

Arguments

copula

R object of class 'cyl_copula' or 'Copula' (package 'copula'.

theta

numeric vector of angles (measurements of a circular variable) or "circular" component of pseudo-observations.

x

numeric vector of step lengths (measurements of a linear variable) or "linear" component of pseudo-observations.

n_bootstrap

integer number of bootstrap replicates. If n_bootstrap is smaller than 1, no p-value is calculated.

parameters

vector of character strings holding the names of the parameters to be optimized when using the bootstrap procedure. These can be any parameters in copula@parameters. Default is to optimize the first 2 parameters. parameters has no effect if copula is of class 'Copula' (package 'copula'

optim.method

character string, optimizer used in optim(), can be "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", or "Brent".

optim.control

list of additional controls passed to optim().

Details

The Cramér-von Misses criterion is calculated as the sum of the squared differences between the empirical copula and the parametric copula, copula, evaluated at the pseudo-observations obtained from theta and x. If the bootstrap procedure is used, a random sample is drawn from copula and converted to pseudo-observations. A new (set of) copula parameter(s) is then fit to those pseudo-observations using maximum likelihood (function cylcop::fit_cylcop_ml()).

Value

A list of length 2 containing the Cramér-von Mises criterion and the p-value.

References

Genest C, Rémillard B (2008). “Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models.” Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 44(6), 1096 – 1127. doi:10.1214/07-AIHP148.

Examples

set.seed(1234)
sample <- rcylcop(100,cyl_cubsec(0.1, 0.1))

opt_cop <- fit_cylcop_ml(copula = cyl_quadsec(),
  theta = sample[,1],
  x = sample[,2],
  parameters = "a",
  start = 0
)$copula
cramer_vonmises(opt_cop,
  theta = sample[,1],
  x = sample[,2],
  n_bootstrap=5)