Positive Definite Quadratic Forms Distribution
Density function, distribution function, quantile function and random generator for positive definite QFs.
dQF(x, obj) pQF(q, obj) qQF(p, obj, eps_quant = 1e-06, maxit_quant = 10000) rQF(n, lambdas, etas = rep(0, length(lambdas)))
x, q |
vector of quantiles. |
obj |
|
p |
vector of probabilities. |
eps_quant |
relative error for quantiles. |
maxit_quant |
maximum number of Newton-Raphson iterations allowed to compute quantiles. |
n |
number of observations. |
lambdas |
vector of positive weights. |
etas |
vector of non-centrality parameters. Default all zeros. |
The quadratic form CDF and PDF are evaluated by numerical inversion of the Mellin transform.
The absolute error specified in compute_MellinQF is guaranteed for values of q and x inside the range_q.
If the quantile is outside range_q, computations are carried out, but a warning is sent.
The function uses the Newton-Raphson algorithm to compute the QF quantiles related to probabilities p.
dQF provides the values of the density function at a quantile x.
pQF provides the cumulative distribution function at a quantile q.
qQF provides the quantile corresponding to a probability level p.
rQF provides a sample of n independent realizations from the QF.
See compute_MellinQF for details on the Mellin computation.
library(QF) # Definition of the QF lambdas_QF <- c(rep(7, 6),rep(3, 2)) etas_QF <- c(rep(6, 6), rep(2, 2)) # Computation Mellin transform eps <- 1e-7 rho <- 0.999 Mellin <- compute_MellinQF(lambdas_QF, etas_QF, eps = eps, rho = rho) xs <- seq(Mellin$range_q[1], Mellin$range_q[2], l = 100) # PDF ds <- dQF(xs, Mellin) plot(xs, ds, type="l") # CDF ps <- pQF(xs, Mellin) plot(xs, ps, type="l") # Quantile qs <- qQF(ps, Mellin) plot(ps, qs, type="l") #Comparison computed quantiles vs real quantiles plot((qs - xs) / xs, type = "l")
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