EuvCOP {copBasic}

Expected value of U given V

Description

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

E[Y] = \int_0^\infty yf(y)\mathrm{d}y = \int_0^\infty \bigl(1-G_y(Y)\bigr)\mathrm{d}y\mbox{,}

which for the setting here becomes

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

This function solves the integral using the derCOP2 function. This avoids a call of the derCOPinv2 through its uniroot() inversion of the derivative. The example shown for EuvCOP() below does a validation check using conditional simulation, which is dependence (of course) of the design of the copBasic package, as part of simple isolation of a horizontal slice of the simulation and computing the mean of the V within the slice.

Usage

EuvCOP(v=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

v

Nonexceedance probability v in the Y 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);

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 derCOP2.

Value

Value(s) for the expectation are returned.

Note

The author is well aware that the name of this function does not contain the number 2 as the family of functions also sharing this with respect to v nature. It was a design mistake in 2008 to have used the 2. The uv in the function name is the moniker for this with respect to v.

There can be the rare examples of the numerical integration failing. In such circumstances, Monte Carlo integration is performed and the returned vector becomes a named vector with the sim identifying values stemming from the simulation.

  para <- list(cop=RFcop, para=0.9)
  para <- list(cop=COP, para=para, reflect=1, alpha=0, beta=0.3)
  EuvCOP(c(0.0001, 0.0002, 0.001, 0.01, 0.1), cop=composite1COP, para=para)
  #            sim
  #[1] 0.001319395 0.002238238 0.006905300 0.034608078 0.173451788

Author(s)

W.H. Asquith

See Also

EvuCOP, derCOP2

Examples

# We can show algorithmic validation using a highly asymmetric case of a
# copula having its parameter also nearly generating a singular component.
v <- c(0.2, 0.8); n <- 5E2; set.seed(1)
para <- list(cop=HRcop, para=120, alpha=0.4, beta=0.05)
UV   <- simCOP(n, cop=composite1COP, para=para, graphics=FALSE) # set TRUE to view

sapply(v, function(vv) EuvCOP(vv, cop=composite1COP, para=para))
# [1] 0.3051985 0.7059999

sapply(v, function(vv) mean( UV$U[UV$V > vv - 50/n & UV$V < vv + 50/n] ) )
# [1] 0.2796518 0.7092755 # general validation is thus shown as n-->large

# If visualized, we see in the lower corner than heuristics suggest a mean further
# to the right of the "singularity" for v=0.2 than v=0.80. For v=0.80, the
# "singularity" appears tighter given the upper tail dependency contrast of the
# coupla in the symmetrical case (alpha=0, beta=0) and changing the parameter to
# a Spearman Rho (say) similar to the para settting in this example. So, 0.70 for
# the mean given v=0.80 makes sense. Further notice that the two estimates of the
# mean are further apparent for v=0.2 relative to v=0.80. Again, this makes sense
# when the copula is visualized even at small n let alone large n.

# See additional Examples under EvuCOP().

## Not run: 
  set.seed(1)
  n <- 5000; Vlo <- rep(0.001, n); Vhi <- rep(0.95, n); Theta <- 3
  Ulo <- simCOPmicro(Vlo, cop=JOcopB5, para=Theta); dlo <- density(Ulo)
  Uhi <- simCOPmicro(Vhi, cop=JOcopB5, para=Theta); dhi <- density(Uhi)
  dlo$x[dlo$x < 0] <- 0; dhi$x[dhi$x < 0] <- 0
  dlo$x[dlo$x > 1] <- 1; dhi$x[dhi$x > 1] <- 1

  summary(Ulo)
  Ulomu <- EuvCOP(Vlo[1], cop=JOcopB5, para=Theta); print(Ulomu)
  #      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
  # 0.0000669 0.0887330 0.2006123 0.2504796 0.3802847 0.9589315
  #                  Ulomu -----> 0.2502145
  summary(Uhi)
  Uhimu <- EuvCOP(Vhi[1], cop=JOcopB5, para=Theta); print(Uhimu)
  #      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
  #   0.01399  0.90603    0.93919 0.9157600   0.95946   0.99411
  #                  Uhimu -----> 0.9154093

  UV <- simCOP(n, cop=JOcopB5, para=Theta,
                  cex=0.6, pch=21, bg="palegreen", col="darkgreen")
  abline(h=Vlo[1], col="salmon", lty=3) # near the bottom to form datum for density
  abline(h=Vhi[1], col="purple", lty=3) # near the   top  to form datum for density
  lines(dlo$x,   dlo$y/max(dlo$y)/2 +    Vlo[1],  col="salmon", lwd=2)
  # re-scaled density along the line already drawn near the bottom (Vlo)
  # think rug plotting to bottom the values plotting very close to the line
  lines(dhi$x, 1-dhi$y/max(dhi$y)/2 - (1-Vhi[1]), col="purple", lwd=2)
  # re-scaled  density along the line already drawn near the  top  (Vhi)
  # think rug plotting to bottom the values plotting very close to the line
  uv <- seq(0.001, 0.999, by=0.001) # for trajectory of E[U | V=v]
  lines(EuvCOP(uv, cop=JOcopB5, para=Theta), uv, col="blue", lwd=3.5, lty=2)
  points(Ulomu, Vlo[1], pch=16, col="salmon", cex=2)
  points(Uhimu, Vhi[1], pch=16, col="purple", cex=2) #
## End(Not run)