lcomCOPpv {copBasic}

Simulating the Sample Distribution(s) of L-correlation, L-coskew, and L-cokurtosis for a Copula

Description

EXPERIMENTAL: The function provides two themes of sampling distribution characterization by simulation of the first three L-comoment ratios (L-correlation \tau_{2[\ldots]}, L-coskew \tau_{3[\ldots]} and L-cokurtosis \tau_{4[\ldots]}) of a copula. Subsequently, the sampling distribution can be used for inference.

First, semi-optional Monte Carlo integration estimation of the L-comoments of the parent copula are computed. Second, simulations involving the sample size n presumed the size of the actual sample from which the estimates of the sample L-comoments given as arguments. These simulations result in a report of the L-moments (not L-comoments) of the sampling distribution and these then are used to compute p-values for the L-comoment matrices provided by the user as a function argument.

Usage

lcomCOPpv(n, lcom, cop=NULL, para=NULL, repcoe=5E3, type="gno",
                   mcn=1E4, mcrep=10, usemcmu=FALSE, digits=5, ...)

Arguments

n

The sample size n. This argument is semi-optional because n = 0 can be given to skip corresponding simulations and the ntable on return will only contain NA; this feature permits rapid extraction of the Ntable and thus the lcom contents are simply not used;

lcom

The sample L-comoments (see below);

cop

A copula function;

para

Vector of parameters, if needed, to pass to the copula;

repcoe

The replication coefficient \phi affecting the number of simulations of size n;

type

The distribution type used for modeling the distribution of the sampling values. The generalized normal (see distribution type "gno" in package lmomco) accommodates some skewness compared to the symmetry of the normal ("nor") just in case situations arise in which non-ignorable skewness in the sample distribution exists. The distribution abbreviations of package lmomco are recognized for the type argument, but in reality the "nor" and "gno" should be more than sufficient;

mcn

The sample size N passed to the bilmoms function for the Monte Carlo integration. If N = 0 then the Monte Carlo integration is not used, otherwise the minimum sample size is internally reset to N=4 so that first four L-moments are computable;

mcrep

The number of replications of the Monte Carlo simulation by bilmoms;

usemcmu

A logical toggling whether the mean value computed from the replicated Monte Carlo integrations is used instead of the mean values for the small sample simulation for the p-value computations;

digits

The number of digits to round numerical entries in the returned tables and can be NA for no rounding; and

...

Additional arguments to pass to the bilmoms function or to the copula.

Details

The notation r[\ldots] refers to two specific types of L-comoment definitions and a blend between the two. The notation r[12] means that the rth L-comoment for random variables \{X^{(1)}, X^{(2)}\} where X^{(2)} is the sorted variable and X^{(1)} is shuffled by the sorting index. Conversely, the notation r[21] means that the rth L-comoment for random variables \{X^{(1)}, X^{(2)}\} where X^{(1)} is the sorted variable and X^{(2)} is shuffled by the sorting index. The notation r[12:21] means that the average between the r[21] and r[21] is computed, which might prove useful in circumstances of known or expected symmetry of the L-comoments.

Continuing, \hat\tau_{2[12]} is the sample L-correlation, \hat\tau_{3[12]} is the sample L-coskew, and \hat\tau_{4[12]} is the sample L-cokurtosis all with respect to the sorting of the second variable. The computation of these L-comoment matricies can be made by functions such as function lcomoms2() in the lmomco package. The number of replications for the simulations involving the n sample size is computed by

m = \phi/\sqrt{n}\mbox{,}

where \phi is the repcoe replication factor or coefficient. If usemcmu is TRUE then mcn > 0 else usemcmu is reset to FALSE.

Value

An R list is returned.

text

A string functioning as a label for the remaining tables;

Ntable

Another R list holding tables of the L-moments of the L-comoments derived from Monte Carlo integration for samples of size N = mcn. The simulations are replicated mcrep times; and

ntable

Another R list holding tables of the L-moments of the L-comoments derived from the small sample simulations for samples of size n = n as well as the p-values estimated by a generalized normal distribution (see lmomco package documentation) of the L-moments using either the small sample means or the mean of the replicated Monte Carlo integrations as dictated by usemcmu. In all circumstances, however, the results for the small sample simluations are tabulated in ntable only the p-value will be reflective the setting of usemcmu.

Note

A significance column for the p-values is added to the right side of the returned ntable and is used to guide the eye in interpretation of results. The significant codes having the following definitions for a two-tailed form:

  "_" > 0.1;  ".", 0.1;  "*", 0.05;  "**", 0.01;  "***", 0.001

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

See Also

lcomCOP, COP, kullCOP, vuongCOP

Examples

# See Note section of vuongCOP() for an extended discussion of copula inference
## Not run: 
Tau <- 0.6410811; para <- GHcop(tau=Tau)$para # This Tau is from a situation of
# two river tributaries. These three L-comoments with univariate L-moments on the
T2 <- c(1,  0.79908960, 0.79908960, 1) # diagonals are derived from those river
# tributaries and downstream of the junction data.
T3 <- c(0, -0.04999318, 0.07689082, 0)
T4 <- c(0,  0.01773833, 0.04756257, 0) # Is the Ho:GHcop rejectable?
LCOM <- list(T2=matrix(T2, nrow=2), T3=matrix(T3, nrow=2), T4=matrix(T4, nrow=2))
set.seed(30312)
ZZ1 <- lcomCOPpv(75, LCOM, cop=GHcop, para=para, repcoe=2000, usemcmu=FALSE)
print(ZZ1)
set.seed(30312)
ZZ2 <- lcomCOPpv(75, LCOM, cop=GHcop, para=para, repcoe=2000, usemcmu=TRUE)
print(ZZ2)
# The results here suggest that the GHcop is not rejectable.
## End(Not run)