Description Usage Arguments Details Value Author(s) References Examples
Functions for testing lowdiscrepancy sequences.
1 2 3 4  sum_of_squares(u)
sobol_g(u, copula = indepCopula(dim = ncol(u)), alpha = 1:ncol(u), ...)
exceedance(x, q, p = 0.99, method = c("indicator", "individual.given.sum.exceeds",
"sum.given.sum.exceeds"))

u 
(n, d)matrix containing n
ddimensional realizations (of a potential quasirandom number
generator). For 
copula 

alpha 
vector of parameters of Sobol's g test function. 
... 
additional arguments passed to the underlying

x 
(n, d)matrix containing n ddimensional realizations. 
q 

p 
If

method 

For examples see the demo man_test_functions
.
See ES_np(<matrix>)
from qrmtools for another test function.
sum_of_squares()
returns an nvector
(numeric(n)
) with the rowwise computed scaled sum
of squares (theoretically integrating to 1).
sobol_g()
returns an nvector (numeric(n)
)
with the rowwise computed Sobol' g functions.
exceedance()
's return value depends on method
:
returns indicators whether,
componentwise, x
exceeds the threshold determined by q
.
returns all rows of x
whose sum exceeds the threshold determined by q
.
returns the row sums of those
rows of x
whose sum exceeds the threshold determined by q
.
Marius Hofert and Christiane Lemieux
Radovic, I., Sobol', I. M. and Tichy, R. F. (1996). QuasiMonte Carlo methods for numerical integration: Comparison of different low discrepancy sequences. Monte Carlo Methods and Applications 2(1), 1–14.
Faure, H., Lemieux, C. (2009). Generalized Halton Sequences in 2008: A Comparative Study. ACMTOMACS 19(4), Article 15.
Owen, A. B. (2003). The dimension distribution and quadrature test functions. Stat. Sinica 13, 1–17.
Sobol', I. M. and Asotsky, D. I. (2003). One more experiment on estimating highdimensional integrals by quasiMonte Carlo methods. Math. Comput. Simul. 62, 255–263.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  ## Generate some (here: copula, pseudorandom) data
library(copula)
set.seed(271)
cop < claytonCopula(iTau(claytonCopula(), tau = 0.5)) # Clayton copula
U < rCopula(1000, copula = cop)
## Compute sum of squares test function
mean(sum_of_squares(U)) # estimate of E(3(sum_{j=1}^d U_j^2)/d)
## Compute the Sobol' g test function
if(packageVersion("copula") >= "0.99920")
mean(sobol_g(U)) # estimate of E(<Sobol's g function>)
## Compute an exceedance probability
X < qnorm(U)
mean(exceedance(X, q = qnorm(0.99))) # fixed threshold q
mean(exceedance(X, p = 0.99)) # empirically estimated marginal pquantiles as thresholds
## Compute 99% expected shortfall for the sum
mean(exceedance(X, p = 0.99, method = "sum.given.sum.exceeds"))
## Or use ES_np(X, level = 0.99) from 'qrmtools'

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