p_m1  R Documentation 
Computes the probability that (at least) one (out of n)
observation(s) of the latent variable U lies in the nonprincipal
branch region. The 'm1
' in p_m1
stands for 'minus 1', i.e,
the nonprincipal 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 nonprincipal region equals at most
p_{1}(γ, n=1) = P≤ft(U < \frac{1}{γ}\right),
where U is the zeromean, 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 nonprincipal 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  (1p_{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  (1p_{1}(γ, n=1))^N reduces to p_{1}(γ) for N=1.
nonnegative 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 nonprincipal 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 nonprincipal branch.
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