pnorMix: 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.

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 re

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

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.

Value

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

encoding

latin1

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<="" code=""> 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) theordered 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).

Examples

Run this code

MW.nm3 # the "strange skew" one
plot(MW.nm3)
## now the cumlative :
x <- seq(-4,4, length=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

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