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Columns 1..L are phi_0, ..., phi_{L-1} with phi_k(u) = sqrt(2k+1) P_k(2u - 1), orthonormal on [0, 1]. The generalized moment integral Q(u) phi_{r-1}(u) du is the projection of the quantile function onto phi_{r-1}; it equals Hosking's (1990) classical L-moment up to the sqrt(2r-1) orthonormalisation factor (phi_0 = 1, so the first equals the mean). These orthonormal moments – not Hosking's classical scaling – are what qfmix matches and reports.

Usage

shifted_legendre(u, L)

Arguments

u

Numeric vector in [0, 1].

L

Number of polynomials.

Value

An length(u) x L matrix.