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Some Additional Distributions
Description
Some Additional Distributions.
Details
A collection of distributions which can be approximated via Edgeworth and Cornish-Fisher expansions
Sum of (non-central) chi-square to powers
Let X_i \sim \chi^2\left(\delta_i, \nu_i\right)
be independently distributed non-central chi-squares, where \nu_i
are the degrees of freedom, and \delta_i are the
non-centrality parameters.
Let w_i and p_i be given constants. Suppose
Y = \sum_i w_i X_i^{p_i}.
Then Y follows a weighted sum of chi-squares to power distribution.
The special case where all the p_i are one is a 'sum of
chi-squares' distribution;
The special case where all the p_i are one half is a 'sum of
chis' distribution;
Lambda Prime
Introduced by Lecoutre, the lambda prime distribution
finds use in inference on the Sharpe ratio under normal
returns.
Suppose y \sim \chi^2\left(\nu\right), and
Z is a standard normal.
T = Z + t \sqrt{y/\nu}
takes a lambda prime distribution with parameters
\nu, t.
A lambda prime random variable can be viewed as a confidence
variable on a non-central t because
t = \frac{Z' + T}{\sqrt{y/\nu}}
Upsilon
The upsilon distribution generalizes the lambda prime to the
case of the sum of multiple chi variables. That is,
suppose y_i \sim \chi^2\left(\nu_i\right)
independently and independently of Z, a standard normal.
Then
T = Z + \sum_i t_i \sqrt{y_i/\nu_i}
takes an upsilon distribution with parameter vectors
[\nu_1, \nu_2, \ldots, \nu_k]', [t_1, t_2, ..., t_k]'.
The upsilon distribution is used in certain tests of the Sharpe ratio for independent observations.
K Prime
Introduced by Lecoutre, the K prime family of distributions generalize
the (singly) non-central t, and lambda prime distributions.
Suppose y \sim \chi^2\left(\nu_1\right), and
x \sim t \left(\nu_2, a\sqrt{y/\nu_1}/b\right).
Then the random variable
T = b x
takes a K prime distribution with parameters
\nu_1, \nu_2, a, b. In Lecoutre's terminology,
T \sim K'_{\nu_1, \nu_2}\left(a, b\right)
Equivalently, we can think of
T = \frac{b Z + a \sqrt{\chi^2_{\nu_1} / \nu_1}}{\sqrt{\chi^2_{\nu_2} / \nu_2}}
where Z is a standard normal, and the normal and the (central) chi-squares are
independent of each other. When a=0 we recover
a central t distribution;
when \nu_1=\infty we recover a rescaled non-central t distribution;
when b=0, we get a rescaled square root of a central F
distribution; when \nu_2=\infty, we recover a
Lambda prime distribution.
Doubly Noncentral t
The doubly noncentral t distribution generalizes the (singly)
noncentral t distribution to the case where the numerator is
the square root of a scaled noncentral chi-square distribution.
That is, if
X \sim \mathcal{N}\left(\mu,1\right) independently
of Y \sim \chi^2\left(k,\theta\right), then
the random variable
T = \frac{X}{\sqrt{Y/k}}
takes a doubly non-central t distribution with parameters
k, \mu, \theta.
Doubly Noncentral F
The doubly noncentral F distribution generalizes the (singly)
noncentral F distribution to the case where the numerator is
a scaled noncentral chi-square distribution.
That is, if
X \sim \chi^2\left(n_1,\theta_1\right) independently
of Y \sim \chi^2\left(n_2,\theta_2\right), then
the random variable
T = \frac{X / n_1}{Y / n_2}
takes a doubly non-central F distribution with parameters
n_1, n_2, \theta_1, \theta_2.
Parameter recycling
It should be noted that the functions provided by sadists do not
recycle their distribution parameters against the
x, p, q or n parameters. This is in contrast to the
common R idiom, and may cause some confusion. This is mostly for reasons
of performance, but also because some of the distributions have vector-valued
parameters; recycling over these would require the user to provide lists
of parameters, which would be unpleasant.
Legal Mumbo Jumbo
sadists is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.
Note
This package is maintained as a hobby.
Author(s)
Steven E. Pav shabbychef@gmail.com
Maintainer: Steven E. Pav shabbychef@gmail.com (ORCID)
References
Lecoutre, Bruno. "Two Useful distributions for Bayesian predictive procedures under normal models." Journal of Statistical Planning and Inference 79, no. 1 (1999): 93-105.
Poitevineau, Jacques, and Lecoutre, Bruno. "Implementing Bayesian predictive procedures: The K-prime and K-square distributions." Computational Statistics and Data Analysis 54, no. 3 (2010): 724-731. https://arxiv.org/abs/1003.4890v1
Walck, C. "Handbook on Statistical Distributions for experimentalists." 1996. https://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
See Also
Useful links: