sadists | R Documentation |

Some Additional Distributions.

A collection of distributions which can be approximated via Edgeworth and Cornish-Fisher expansions

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;

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}}`

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.

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.

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`

.

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`

.

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.

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.

This package is maintained as a hobby.

Steven E. Pav shabbychef@gmail.com

**Maintainer**: Steven E. Pav shabbychef@gmail.com (ORCID)

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

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