Compute the theoretrical L-moments for a vector. A theoretrical L-moment in integral form is
$$ \lambda_r = \frac{1}{r}
\sum^{r-1}_{k=0}{(-1)^k {r-1 \choose k}
\frac{r!\:I_r}{(r-k-1)!\,k!}
} \mbox{,}$$
in which
$$ I_r = \int^1_0 x(F) \times F^{r-k-1}(1-F)^{k}\,\mathrm{d}F \mbox{,}$$
where \(x(F)\) is the quantile function of the random variable \(X\) for nonexceedance probability \(F\), and \(r\) represents the order of the L-moments. This function actually dispatches to theoTLmoms with trim=0 argument.
theoLmoms(para, nmom=5, verbose=FALSE, minF=0, maxF=1)An R
list is returned.
Vector of the TL-moments. First element is \(\lambda_1\), second element is \(\lambda_2\), and so on.
Vector of the L-moment ratios. Second element is \(\tau_2\), third element is \(\tau_3\) and so on.
Level of symmetrical trimming used in the computation, which will equal zero (the ordinary L-moments).
An attribute identifying the computational source of the L-moments: “theoTLmoms”.
A distribution parameter object such as from vec2par.
The number of moments to compute. Default is 5.
Toggle verbose output. Because the R function integrate is used to perform the numerical integration, it might be useful to see selected messages regarding the numerical integration.
The end point of nonexceedance probability in which to perform the integration. Try setting to non-zero (but very small) if the integral is divergent.
The end point of nonexceedance probability in which to perform the integration. Try setting to non-unity (but still very close [perhaps 1 - minF]) if the integral is divergent.
W.H. Asquith
Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105--124.
theoTLmoms
para <- vec2par(c(0,1),type='nor') # standard normal
TL00 <- theoLmoms(para) # compute ordinary L-moments
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