wolfCOPtest_data_smlsam: Small Sample Components for Schweizer-Wolff Sigma Test

In copBasic: General Bivariate Copula Theory and Many Utility Functions

Small Sample Components for Schweizer–Wolff Sigma Test

Description

Data for the test components for small samples. Empirical sampling distributions of Schweizer–Wolff Sigma (\hat\sigma_\mathbf{C}, wolfCOP) for n \in 3, \cdots, 40 of the \mathrm{NULL} : \mathbf{C} = \mathbf{\Pi} (Independence copula, P), where \sigma_\mathbf{C} is defined under wolfCOP, using simulation sizes of 900,000 for each n. So, total simulation size was 34,200,000. Continuing, this data set is a concatenated table of empirical distributions formed from the unique number of \hat\sigma_\mathbf{C} (discrete variables in small samples), which large-simulation attainable up to \pm4.26 standard deviations, were computed and tabulated in the wolfCOPtest_data_smlsam table. Algorithms for the small-sample components are located at inst/make_wolfCOPtest/smlsam/make_wolfCOPtestP_smlsam.R, and this script is the origin of the n = 40 in wolfCOPtest source code.

For the smallest of sample sizes, the empirical distribution is a step function, and for each plateau, a nudge using .Machine$double.eps has been used to make approx() for sensible in wolfCOPtest. In the Examples herein, the jaggedness of the empirical distribution can readily be seen and visually smoothing by about n > 10. Finally, the wolfCOPtest function internally loads the wolfCOPtest_data_smlsam tables, and most users will not need to access this dataset themselves unless for curiosity.

Usage

data(wolfCOPtest_data_smlsam)

Format

An R data.frame titled wolfCOPtest_data_smlsam with

n

Sample size;

probs

Nonexceedance probability of empirical distribution of \hat\sigma_\mathbf{C} (wolfCOP). Note, quantile(..., type=4) was used for computations because of structural similarity to sample \hat\sigma_\mathbf{C} as implemented in copBasic;

wolfemp

Value for \hat\sigma_\mathbf{C} (wolfCOP);

See Also

wolfCOPtest

Examples

data(wolfCOPtest_data_smlsam)

## Not run: 
  # Demonstrate the empirical distributions of Schweizer-Wolff against a parametric
  # distribution model. We see quite quickly that the distribution model mimics
  # the empirical distribution at quite small sample sizes.
  data("wolfCOPtest_data_smlsam")
  ff <- pnorm(seq(-3, 3, by=0.1))
  plot(c(-3, 3), c(0,1), type="n", xlab="Standard normal variate",
       ylab="Schweizer-Wolff Sigma given C(u,v)=PI")
  ns <- c(3, 5, 10, 20, 40)
  for(n in ns) {
    sata <- wolfCOPtest_data_smlsam[wolfCOPtest_data_smlsam$n == n,]
    lines(qnorm(sata$probs), sata$wolfemp, col="red", lwd=2)
  }
  ns <- c(3, 4, 5, 7, 10, 15, 20, 30, 40, 50, 70, 100, 130, 200, 500, 1000, 5000)
  for(n in ns) {
    wt   <- wolfCOPtest(0, n) # next isolate the distribution type
    # If copy and paste from R help, then one needs to add backslashes to avoid
    # unrecognized escape errors from R in the next line of code.
    type <- gsub("\)$", "", gsub("p.value\(dist_", "", names(wt$p.value)[1]))
    para <- lmomco::vec2par(wt$distpara_by_lmoms, type=type) # type == pe3
    qq   <- 1/(1 + exp(-lmomco::qlmomco(ff, para)))
    lines(qnorm(ff), qq, col="blue")
    text(0, 1/(1 + exp(-lmomco::qlmomco(0.5, para))), n, cex=0.8, font=2)
  }
  par(lheight=0.8)
  txt <- c(paste0(lmomco::prettydist(type), " model by estimated L-moments\n",
                  "of Sigma distribution for select sample sizes"),
           paste0("Empirical distribution for select sample sizes\n",
                  "in wolfCOPtest_data_smlsam data object"))
  legend("topleft", txt, box.lty=0, inset=0.01, cex=0.8, adj=c(0, 0.8),
         y.intersp=1.6, lwd=c(1,2), col=c("blue", "red"), bg=NA)
  par(lheight=1) #
## End(Not run)

## Not run: 
  W <- seq(0, 1, by=0.01) # Span all Schweizer-Wolff Sigma, when the first and second
  # arguments to wolfCOPtest are scalars, these are treated as sigma and sample size.
  for(i in 3:40) { # samples sizes 3 to 40 in wolfCOPtest_data_smlsam.
    Pd <- sapply(W, function(w) wolfCOPtest(w, i, usepade=FALSE)$p.value[1])
    Pe <- sapply(W, function(w) wolfCOPtest(w, i, usepade=FALSE)$p.value[2])
    Pp <- sapply(W, function(w) wolfCOPtest(w, i, usepade=TRUE )$p.value[1])
    # p-value by the generalized normal (parametric) distribution
    plot( W, Pd, type="l", col="grey70", lty=2, lwd=7, main=paste0("n=", i),
         xlab="Schweizer-Wolff Sigma", ylab="Exceedance probability (p-value)")
    points(W, Pp, col="skyblue2", lwd=1) # large sample p-values
    # p-value derived from wolfCOP_data_smlsam tables:
    lines(W, Pe, col="green4",    lwd=4) # small sample p-values
  } #
## End(Not run)

copBasic documentation built on July 23, 2026, 1:07 a.m.