S\(\alpha\)S Point5 Distribution Functions

Functions to compute the probability density function, cumulative distribution function, and quantile function for the S\(\alpha\)S Point5 distribution.

The S\(\alpha\)S Point5 distribution is a special case of a stable distribution with shape parameter \(\alpha=1/2\), \(\beta=0\).

It has the probability density function (PDF): $$f(x; \mu, \gamma) = \frac{1}{\sqrt{2\pi}} x^{-3/2} e^{-\frac{1}{2x}}$$ (Note: The boost documentation reference shows a standard form, generalised by location \(\mu\) and scale \(\gamma\)).

This distribution has heavier tails than the Cauchy distribution. Note that the S\(\alpha\)S Point5 distribution does not have a defined mean or standard deviation.

Accuracy and Implementation Notes: The error is within 4 epsilon.

Usage

saspoint5_distribution(location = 0, scale = 1)

saspoint5_pdf(x, location = 0, scale = 1)

saspoint5_lpdf(x, location = 0, scale = 1)

saspoint5_cdf(x, location = 0, scale = 1)

saspoint5_lcdf(x, location = 0, scale = 1)

saspoint5_quantile(p, location = 0, scale = 1)

Arguments

location

location parameter (default is 0)

scale

scale parameter (default is 1)

x

quantile

p

probability (0 <= p <= 1)

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See also

Boost Documentation for more details on the mathematical background.

Examples

# SaS Point5 distribution with location 0 and scale 1
dist <- saspoint5_distribution(0, 1)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.6686904
logcdf(dist, 0.5)
#> [1] -0.402434
pdf(dist, 0.5)
#> [1] 0.1707624
logpdf(dist, 0.5)
#> [1] -1.767482
hazard(dist, 0.5)
#> [1] 0.5154165
chf(dist, 0.5)
#> [1] 1.104702
median(dist)
#> [1] 0
mode(dist)
#> [1] 0
range(dist)
#> [1] -Inf  Inf
quantile(dist, 0.2)
#> [1] -2.397479
support(dist)
#> [1] -Inf  Inf

# Convenience functions
saspoint5_pdf(3)
#> [1] 0.02379919
saspoint5_lpdf(3)
#> [1] -3.738104
saspoint5_cdf(3)
#> [1] 0.8164545
saspoint5_lcdf(3)
#> [1] -0.2027841
saspoint5_quantile(0.5)
#> [1] 0