EvuCOP {copBasic}

Expected value of V given U

Description

Compute the expected value of V given a U (the X direction) through the conditional distribution function F(X) using the appropriate partial derivative of a copula (\mathbf{C}(u,v)) with respect to U. The inversion of the partial derivative is the conditional quantile function. Basic principles provide the expectation for a x \ge 0 is

E[X] = \int_0^\infty xf(x)\mathrm{d}x = \int_0^\infty \bigl(1-F_x(X)\bigr)\mathrm{d}x\mbox{,}

which for the setting here becomes

E[V \mid U = u] = \int_0^1 \bigl(1 - \frac{\delta}{\delta u} \mathbf{C}(u,v)\bigr)\mathrm{d}v\mbox{.}

This function solves the integral using the derCOP function. Verification study is provided in the Note section.

Usage

EvuCOP(u=seq(0.01, 0.99, by=0.01), cop=NULL, para=NULL, asuv=FALSE, nsim=1E5,
    subdivisions=100L, rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)

Arguments

u

Nonexceedance probability u in the X direction;

cop

A copula function with vectorization as in asCOP;

para

Vector of parameters or other data structures, if needed, to pass to the copula;

asuv

Return a data frame of the U and V;

nsim

Number of simulations for Monte Carlo integration when the numerical integration fails (see Note in EvuCOP);

subdivisions

Argument of same name passed to integrate();

rel.tol

Argument of same name passed to integrate();

abs.tol

Argument of same name passed to integrate(); and

...

Additional arguments to pass to derCOP.

Value

Value(s) for the expectation are returned.

Note

For the PSP copula with no parameters, compute the the median and mean V given U=0.4, respectively:

  U <- 0.4; n <- 1E4
  med.regressCOP(u=U, cop=PSP)                            # V = 0.4912752
  set.seed(1)
  median(replicate(n, derCOPinv(cop=PSP, U, runif(1)) ) ) # V = 0.4876440
  mean(  replicate(n, derCOPinv(cop=PSP, U, runif(1)) ) ) # V = 0.5049729

It is seen in the above that the median V given U is very close to the mean, but is not equal. Using the derivative inversion within med.regressCOP the median is about 0.491 and then using large-sample simulation, about 0.491 too is computed. This confirms the median and long standing proven use of derCOP (conditional distribution function) and derCOPinv (conditional quantile function) within the package. The expectation (mean) by simulation provides the anchor point to check implementation of EuvCOP(). The mean for V given U is about 0.505. Continuing, the core logic of EvuCOP() is to use numerical integration of the conditional distribution function (the partial derivative) and not bother for speed purposes to use the inversion of the partial derivative:

  integrate(function(v)                      1-derCOP(   cop=PSP, U, v),
            lower=0, upper=1) # 0.5047805 with absolute error < 1.4e-11

  integrate(function(v) sapply(v, function(t)  derCOPinv(cop=PSP, U, t)),
            lower=0, upper=1) # 0.5047862 with absolute error < 7.2e-05

The two integrals match, which functions as a confirmation of the (1-F) term in the mathematical definition. Finally, the two integrals match the simulation results. The expectation or mean V \mid U=0.4 for the PSP copula is about 0.5048.

Author(s)

W.H. Asquith

See Also

EuvCOP, derCOP

Examples

# Highly asymmetric and reflected Clayton copula for which visualization
para <- list(cop=CLcop, para=30, alpha=0.2, beta=0.6, reflect=3)
# UV <- simCOP(5000, cop=breveCOP, para=para, cex=0.5); abline(v=0.25, col="red")
EvuCOP(0.25, cop=breveCOP, para=para)  # 0.5982261
# confirms that at U=0.25 that an intuitive estimate would be about 0.6.

## Not run: 
  # Secondary validation of the EvuCOP() and EuvCOP() implementation
  UV <- simCOP(200, cop=PSP)
  u <- seq(0.005, 0.995, by=0.005)
  v <- sapply(u, function(t) integrate(function(k)
                         1 - derCOP(cop=PSP, t, k),  lower=0, upper=1)$value)
  lines(u,v, col="red", lwd=7, lty=1)   # red line
  v <- seq(0.005, 0.995, by=0.005)
  u <- sapply(v, function(t) integrate(function(k)
                         1 - derCOP2(cop=PSP, k, t), lower=0, upper=1)$value)
  lines(u,v, col="red", lwd=7, lty=2)   # dashed red line

  uv <- seq(0.005, 0.995, by=0.005)     # solid and dashed white lines
  lines(EvuCOP(uv, cop=PSP, asuv=TRUE), col="white",  lwd=3,   lty=1)
  lines(EuvCOP(uv, cop=PSP, asuv=TRUE), col="white",  lwd=3,   lty=3)

  # median regression lines for comparison, green and green dashed lines
  lines(med.regressCOP( uv, cop=PSP), col="seagreen", lwd=1.5, lty=1)
  lines(med.regressCOP2(uv, cop=PSP), col="seagreen", lwd=1.5, lty=4) #
## End(Not run)

## Not run: 
  uv <- seq(0.005, 0.995, by=0.005) # stress testing eample with singularity
  UV <- simCOP(50, cop=M_N5p12b, para=2)
  lines(EvuCOP(uv, cop=M_N5p12b, para=2, asuv=TRUE), col="red",     lwd=5)
  lines(EuvCOP(uv, cop=M_N5p12b, para=2, asuv=TRUE), col="skyblue", lwd=1) #
## End(Not run)

## Not run: 
  uv <- seq(0.005, 0.995, by=0.005) # more asymmetry ---- mean regression in UV and VU
  para <- list(cop=GHcop, para=23, alpha=0.1, beta=0.6, reflect=3)
  para <- list(cop=breveCOP, para=para)
  UV <- simCOP(200, cop=COP, para=para)
  lines(EuvCOP(uv,  cop=COP, para=para, asuv=TRUE), col="red",  lwd=2)
  lines(EvuCOP(uv,  cop=COP, para=para, asuv=TRUE), col="blue", lwd=2) #
## End(Not run)

## Not run: 
  # Open questions? The derCOP() and derCOPinv() functions of the package have long
  # been known to work "properly." But let us think again on the situation of
  # permutation symmetry about the equal value line. Recalling that this symmetry is
  # orthogonal to the equal value line, it remains open whether there could be
  # asymmetry in the vertical (or horizontal). Let us draw some median regression lines
  # and see that the do not plot perfectly on the equal value line, but coudl this be
  # down to numerical issues and by association the simulation of the copula itself
  # that is also using derCOPinv() (conditional simulation method). Then, we can plot
  # the expections and we see that these are not equal to the medians, but again are
  # close. *** Do results here indicate edges of numerical performance? ***
  t <- seq(0.01, 0.99, by=0.01)
  UV <- simCOP(10000,   cop=N4212cop, para=4, pch=21, lwd=0.8, col=8, bg="white")
  lines(med.regressCOP( cop=N4212cop, para=4, asuv=TRUE), col="red")
  lines(med.regressCOP2(cop=N4212cop, para=4, asuv=TRUE), col="red")
  abline(0, 1, col="deepskyblue", lwd=3); abline(v=0.5, col="deepskyblue", lwd=4)
  lines(EvuCOP(t, cop=N4212cop, para=4, asuv=TRUE), pch=16, col="darkgreen")
  lines(EuvCOP(t, cop=N4212cop, para=4, asuv=TRUE), pch=16, col="darkgreen") #
## End(Not run)