p_m1 function

Non-principal branch probability

Non-principal branch probability

Computes the probability that (at least) one (out of n) observation(s) of the latent variable UU 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)

Arguments

  • gamma: scalar; skewness parameter.
  • beta: numeric vector (deprecated); parameter β\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 β\boldsymbol \beta to do the transformation (Goerg 2011). If FALSE, it uses the alternative definition from Goerg (2016) with location and scale parameter.

Returns

non-negative float; the probability p−1p_{-1} for n observations.

Details

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(U<−1∣γ∣), p_{-1}(\gamma, n=1)= P\left(U < -\frac{1}{|\gamma|}\right),

where UU is the zero-mean, unit variance version of the input c("XsimFX(xmidboldsymbol\nX \\sim F_X(x \\mid \\boldsymbol\n", "beta)\\beta)") -- see References.

For NN independent RVs U1,…,UNU_1, \ldots, U_N, the probability that at least one data point came from the non-principal region equals

p−1(γ,n=N)=P(Ui<−1∣γ∣  for  at  least  one  i) p_{-1}(\gamma, n=N) = P\left(U_i < -\frac{1}{|\gamma|} \; for \; at \;least \; one \; i \right)

This equals (assuming independence)

P(Ui<−1∣γ∣  for  at  least  one  i)=1−P(Ui≥−1∣γ∣,∀i)=1−∏i=1NP(Ui≥−1∣γ∣) 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−∏i=1N(1−p−1(γ,n=1))=1−(1−p−1(γ,n=1))N. = 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 c("∣gamma∣<\n|\\gamma| <\n", "0.030.03") (returns 0 due to rounding errors).

Note that 1−(1−p−1(γ,n=1))N1 - (1-p_{-1}(\gamma, n=1))^N reduces to p−1(γ)p_{-1}(\gamma)

for N=1N=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.
LambertW packageRead PDF manual
  • Maintainer: Georg M. Goerg
  • License: GPL (>= 2)
  • Last published: 2023-11-30

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