pnorMix: Normal Mixture Cumulative Distribution and Quantiles

In nor1mix: Normal aka Gaussian 1-d Mixture Models

Normal Mixture Cumulative Distribution and Quantiles

Description

Compute cumulative probabilities or quantiles (the inverse) for a normal mixture specified as norMix object.

Usage

pnorMix(q, obj, lower.tail = TRUE, log.p = FALSE)

qnorMix(p, obj, lower.tail = TRUE, log.p = FALSE,
        tol = .Machine$double.eps^0.25, maxiter = 1000, traceRootsearch = 0,
        method = c("interpQspline", "interpspline", "eachRoot", "root2"),
        l.interp = pmax(1, pmin(20, 1000 / m)), n.mu.interp = 100)

Arguments

obj	an object of class norMix.
p	numeric vector of probabilities. Note that for all methods but "eachRoot", qnorMix(p, *) works with the full vector p, typically using (inverse) interpolation approaches; consequently the result is very slightly dependent on p as a whole.
q	numeric vector of quantiles

.

lower.tail	logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].
log.p	logical; if TRUE, probabilities p are given as log(p).
tol, maxiter	tolerance and maximal number of iterations for the root search algorithm, see method below and uniroot.
traceRootsearch	logical or integer in \{0,1,2,3\}, determining the amount of information printed during root search.
method	a string specifying which algorithm is used for the “root search”. Originally, the only method was a variation of "eachRoot", which is the default now when only very few quantiles are sought. For large m.norMix(), the default is set to "root2", currently.
l.interp	positive integer for method = "interQpspline" or "interpspline", determining the number of values in each “mu-interval”.
n.mu.interp	positive integer for method = "interQpspline" or "interpspline", determining the (maximal) number of mu-values to be used as knots for inverse interpolation.

Details

Whereas the distribution function pnorMix is the trivial sum of weighted normal probabilities (pnorm), its inverse, qnorMix is computed numerically: For each p we search for q such that pnorMix(obj, q) == p, i.e., f(q) = 0 for f(q) := pnorMix(obj, q) - p. This is a root finding problem which can be solved by uniroot(f, lower,upper,*). If length(p) <= 2 or method = "eachRoot", this happens one for one for the sorted p's. Otherwise, we start by doing this for the outermost non-trivial (0 < p < 1) values of p.

For method = "interQpspline" or "interpspline", we now compute p. <- pnorMix(q., obj) for values q. which are a grid of length l.interp in each interval [q_j,q_{j+1}], where q_j are the “X-extremes” plus (a sub sequence of length n.mu.interp of) the ordered mu[j]'s. Then, we use montone inverse interpolation (splinefun(q., p., method="monoH.FC")) plus a few (maximally maxiter, typically one!) Newton steps. The default, "interQpspline", additionally logit-transforms the p. values to make the interpolation more linear. This method is faster, particularly for large length(p).

Value

a numeric vector of the same length as p or q, respectively.

Author(s)

Very first version (for length-1 p,q) by Erik Jørgensen [e-mail].

See Also

dnorMix for the density function.

Examples

MW.nm3 # the "strange skew" one
plot(MW.nm3)
## now the cumlative :
x <- seq(-4,4, length.out = 1001)
plot(x, pnorMix(x, MW.nm3), type="l", col=2)
## and some of its inverse :
pp <- seq(.1, .9, by=.1)
plot(qnorMix(pp, MW.nm3), pp)

## The "true" median of a normal mixture:
median.norMix <- function(x) qnorMix(1/2, x)
median.norMix(MW.nm3) ## -2.32

nor1mix documentation built on May 29, 2024, 8:28 a.m.