LambertW (version 0.6.9-1)

p_m1: Non-principal branch probability

Description

Computes the probability that (at least) one (out of n) observation(s) of the latent variable \(U\) lies in the non-principal branch region. The 'm1' in p_m1 stands for 'minus 1', i.e, the non-principal branch.

See Goerg (2011) and Details for mathematical derivations.

Usage

p_m1(gamma, beta, distname, n = 1, use.mean.variance = TRUE)

Value

non-negative float; the probability \(p_{-1}\) for n observations.

Arguments

gamma

scalar; skewness parameter.

beta

numeric vector (deprecated); parameter \(\boldsymbol \beta\) of the input distribution. See check_beta on how to specify beta for each distribution.

distname

character; name of input distribution; see get_distnames.

n

number of RVs/observations.

use.mean.variance

logical; if TRUE it uses mean and variance implied by \(\boldsymbol \beta\) to do the transformation (Goerg 2011). If FALSE, it uses the alternative definition from Goerg (2016) with location and scale parameter.

Details

The probability that one observation of the latent RV U lies in the non-principal region equals at most $$ p_{-1}(\gamma, n=1) = P\left(U < -\frac{1}{|\gamma|}\right), $$ where \(U\) is the zero-mean, unit variance version of the input \(X \sim F_X(x \mid \boldsymbol \beta)\) -- see References.

For \(N\) independent RVs \(U_1, \ldots, U_N\), the probability that at least one data point came from the non-principal region equals $$ p_{-1}(\gamma, n=N) = P\left(U_i < -\frac{1}{|\gamma|} \; for \; at \; least \; one \; i \right) $$

This equals (assuming independence) $$ P\left(U_i < -\frac{1}{|\gamma|} \; for \; at \; least \; one \; i \right) = 1 - P\left(U_i \geq -\frac{1}{|\gamma|}, \forall i \right) = 1 - \prod_{i=1}^{N} P\left(U_i \geq -\frac{1}{|\gamma|} \right) $$ $$ = 1 - \prod_{i=1}^{N} \left(1 - p_{-1}(\gamma, n=1) \right) = 1 - (1-p_{-1}(\gamma, n=1))^N. $$

For improved numerical stability the cdf of a geometric RV (pgeom) is used to evaluate the last expression. Nevertheless, numerical problems can occur for \(|\gamma| < 0.03\) (returns 0 due to rounding errors).

Note that \(1 - (1-p_{-1}(\gamma, n=1))^N\) reduces to \(p_{-1}(\gamma)\) for \(N=1\).

Examples

beta.01 <- c(mu = 0, sigma = 1)
# for n=1 observation
p_m1(0, beta = beta.01, distname = "normal") # identical to 0
# in theory != 0; but machine precision too low
p_m1(0.01, beta = beta.01, distname = "normal") 
p_m1(0.05, beta = beta.01, distname = "normal") # extremely small
p_m1(0.1, beta = beta.01, distname = "normal") # != 0, but very small
# 1 out of 4 samples is a non-principal input;
p_m1(1.5, beta = beta.01, distname = "normal") 
# however, gamma=1.5 is not common in practice

# for n=100 observations
p_m1(0, n=100, beta = beta.01, distname = "normal") # == 0
p_m1(0.1, n=100, beta = beta.01, distname = "normal") # still small
p_m1(0.3, n=100, beta = beta.01, distname = "normal") # a bit more likely
p_m1(1.5, n=100, beta = beta.01, distname = "normal") 
# Here we can be almost 100% sure (rounding errors) that at least one
# y_i was caused by an input in the non-principal branch.