LMRvuCOP: Conditional L-moments of the Distribution of V given U

Description

Compute the conditional L-moments 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 conditional L-moments are computed for $\lambda_1$ (mean) from EvuCOP (refer to mathematics therein) and for $r \ge 2$ are computed by

$$\lambda_r = \int_0^1 F(k) \times (1-F(k)) \times L_r(F(k))\,\mathrm{d}k\mbox{,}$$

where $F(k)$ becomes replaced in copBasic with the conditional distribution function of a copula by Nelsen (2006, pp. 13, 40--41) with respect to $u$:

$$0 \le \frac{\delta}{\delta u} \mathbf{C}(u,v) \le 1\mbox{,}$$

which is derCOP and $L_r(u)$ is defined (Jones, 2004; Asquith, 2011) as

$$L_r(z) = \frac{1}{r-1}\sum_{j=0}^{r-2} (-1)^j {r-1 \choose j}{r-1 \choose j+1}z^{r-2-j}(1-z)^j\mbox{.}$$

Asquith (2011) mistakenly lists the leading multiplier $1/(1-r)$ in lieu of $1/(1-r)$ (correct) as shown above; the lmomco package contains a suitable errata as discovered in copBasic development for LMRvuCOP and LMRuvCOP.

Usage

LMRvuCOP(u=seq(0.01, 0.99, by=0.01), cop=NULL, para=NULL, nsim=1E5,
      subdivisions=200L, rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)

Value

Value(s) in a list for the median and conditional L-moments given $U=u$ are returned including

median: The conditional median from med.regressCOP;
L1: The conditional mean $\lambda_1|U=u$ from EvuCOP;
L2: The conditional L-scale $\lambda_2|U=u$;
T3: The conditional L-skew $\tau_3|U=u$;
T4: The conditional L-kurtosis $\tau_4|U=u$;

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;
nsim: Number of simulations for Monte Carlo integration if numerical integration were to fail;
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.

Author

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.

Jones, M.C., 2004, On some expressions for variance, covariance, skewness and L-moments: Journal of Statistical Planning and Inference, v.126, pp. 97--106, tools:::Rd_expr_doi("10.1016/j.jspi.2003.09.001").