Density function, distribution function, quantiles and random number generation for the skewnormal (SN) and the extended skewnormal (ESN) distribution.
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x 
vector of quantiles. Missing values ( 
p 
vector of probabilities. Missing values ( 
xi 
vector of location parameters. 
omega 
vector of scale parameters; must be positive. 
alpha 
vector of slant parameter(s); 
tau 
a single value representing the ‘hidden mean’ parameter
of the ESN distribution; 
dp 
a vector of length 3 (in the SN case) or
4 (in the ESN case), whose components represent
the individual parameters described above. If 
n 
a positive integer representing the sample size. 
tol 
a scalar value which regulates the accuracy of the result of

log 
logical flag used in 
engine 
a character string which selects the computing engine;
this is either 
solver 
a character string which selects the numerical method used for
solving the quantile equation; possible options are 
... 
additional parameters passed to 
density (dsn
), probability (psn
), quantile (qsn
)
or random sample (rsn
) from the skewnormal distribution with given
xi
, omega
and alpha
parameters or from the extended
skewnormal if tau!=0
Typical usages are
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psn
and qsn
make use of function T.Owen
or biv.nt.prob
In qsn
, the choice solver="NR"
selects the NewtonRaphson method
for solving the quantile equation, while option solver="RFB"
alternates a step of regula falsi with one of bisection.
The "NR"
method is generally more efficient, but "RFB"
is
occasionally required in some problematic cases.
The family of skewnormal distributions is an extension of the normal
family, via the introdution of a alpha
parameter which regulates
asymmetry; when alpha=0
, the skewnormal distribution reduces to
the normal one. The density function of the SN distribution
in the ‘normalized’ case having xi=0
and omega=1
is
2φ(x)Φ(α x), if φ and Φ denote the
standard normal density and distribution function.
An early discussion of the skewnormal distribution is given by
Azzalini (1985); see Section 3.3 for the ESN variant,
up to a slight difference in the parameterization.
An updated exposition is provided in Chapter 2 of Azzalini and Capitanio (2014); the ESN variant is presented Section 2.2. See Section 2.3 for an historical account. A multivariate version of the distribution is examined in Chapter 5.
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171178.
Azzalini, A. with the collaboration of Capitanio, A. (2014). The SkewNormal and Related Families. Cambridge University Press, IMS Monographs series.
Functions used by psn
:
T.Owen
, biv.nt.prob
Related distributions: dmsn
, dst
,
dmst
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