| p_m1 {LambertW} | R Documentation |
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.
p_m1(gamma, beta, distname, n = 1, use.mean.variance = TRUE)
gamma |
scalar; skewness parameter. |
beta |
numeric vector (deprecated); parameter \boldsymbol β of
the input distribution. See |
distname |
character; name of input distribution; see
|
n |
number of RVs/observations. |
use.mean.variance |
logical; if |
The probability that one observation of the latent RV U lies in the non-principal region equals at most
p_{-1}(γ, n=1) = P≤ft(U < -\frac{1}{|γ|}\right),
where U is the zero-mean, unit variance version of the input X \sim F_X(x \mid \boldsymbol β) – see References.
For N independent RVs U_1, …, U_N, the probability that at least one data point came from the non-principal region equals
p_{-1}(γ, n=N) = P≤ft(U_i < -\frac{1}{|γ|} \; for \; at \; least \; one \; i \right)
This equals (assuming independence)
P≤ft(U_i < -\frac{1}{|γ|} \; for \; at \; least \; one \; i \right) = 1 - P≤ft(U_i ≥q -\frac{1}{|γ|}, \forall i \right) = 1 - ∏_{i=1}^{N} P≤ft(U_i ≥q -\frac{1}{|γ|} \right)
= 1 - ∏_{i=1}^{N} ≤ft(1 - p_{-1}(γ, n=1) \right) = 1 - (1-p_{-1}(γ, 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 |γ| <
0.03 (returns 0 due to rounding errors).
Note that 1 - (1-p_{-1}(γ, n=1))^N reduces to p_{-1}(γ) for N=1.
non-negative float; the probability p_{-1} for n observations.
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.