R: Non-principal branch probability

p_m1 {LambertW} | R Documentation

## 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)

### Arguments

`gamma` | scalar; skewness parameter.
`beta` | numeric vector (deprecated); parameter \boldsymbol β 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 β 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}(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.

### Value

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

### 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.

[Package LambertW version 0.6.7 Index]