p_1 {LambertW} | R Documentation |
Computes the probability that one (or more) U lie in the non-principal branch region. See Details.
p_1(delta, distname = "normal", nu = NULL, n=1)
delta |
skewness parameter |
distname |
character string stating the input distribution: "t" or "normal" . |
nu |
degrees of freedom (if distname="t" ) |
n |
number of RVs/observations; default n=1 |
For one observation this probability equals
p_{-1}(delta, n=1) = P(U < -frac{1}{|delta|}),
where U sim F_U(u) is a zero-mean, unit-variance RV.
For N independent RVs U_1, ..., U_N, it is interesting to know the probability that all of them lie in the principal region, or equivalently one minus the probability that at least one of the observations lies in the non-principal branch. Hence, for N > 1 define,
p_{-1}(delta, n=N) = P(U_i >=q -frac{1}{|delta|}, forall i ) = 1 - P(U_i < -frac{1}{|delta|}, forall i ) = 1- (1-p_{-1}(delta, n=1))^N.
Hence, the cdf of a geometric RV can be used (pgeom
).
For N=1 this coincides with the original definition.
p_{-1} for n
observations (in practice very small given empirical evidence).
Numerical problems can occur for delta < 0.03 (always gives 0
; rounding errors).
Georg M. Goerg
Goerg, G.M. (2009). “Lambert W Random Variables - A new class of skewed distribution functions”. Unpublished
# for n=1 observation p_1(0) # this probability is identical to 0 p_1(0.01) # in theory not identical to 0; but machine precision too low p_1(0.05) # extremely small p_1(0.1) # not 0, but very small; delta = 0.1 typical value for sample data p_1(1.5) # 1 out of 4 samples is a non-principal input; delta=1.5 is not common in practice # for n=100 observations p_1(0, n=100) # this probability is identical to 0 p_1(0.1, n=100) # still small p_1(1.5, n=100) # we can be almost 100