Description Details Sum of (non-central) chi-square to powers Lambda Prime Upsilon K Prime Doubly Noncentral t Doubly Noncentral F Parameter recycling Legal Mumbo Jumbo Note Author(s) References

Some Additional Distributions.

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

Let *X_i ~ chi^2(delta_i, v_i)*
be independently distributed non-central chi-squares, where *v_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 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 ~ x^2(v)*, and
*Z* is a standard normal.

*T = Z + t sqrt(y/v)*

takes a lambda prime distribution with parameters
*v, t*.
A lambda prime random variable can be viewed as a confidence
variable on a non-central t because

*t = (Z' + T)/sqrt(y/v)*

The upsilon distribution generalizes the lambda prime to the
case of the sum of multiple chi variables. That is,
suppose *y_i ~ x^2(v_i)*
independently and independently of *Z*, a standard normal.
Then

*T = Z + sum_i t_i sqrt(y_i/v_i)*

takes an upsilon distribution with parameter vectors
*<v_1, v_2, ..., v_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 ~ x^2(v1)*, and
*x ~ t(v2,(a/b) sqrt(y/v1))*.
Then the random variable

*T = b x*

takes a K prime distribution with parameters
*v1, v2, a, b*. In Lecoutre's terminology,
*T ~ K'_v1,v2(a,b)*

Equivalently, we can think of

*T = (bZ + a sqrt(chi2_v1/v1)) / sqrt(chi2_v2/v2)*

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 *v1=inf* we recover a rescaled non-central t distribution;
when *b=0*, we get a rescaled square root of a central F
distribution; when *v2=inf*, 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 ~ N(u,1)* independently
of *Y ~ x^2(k,theta)*, then
the random variable

*T = 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 ~ x^2(n1,theta1)* independently
of *Y ~ x^2(n2,theta2)*, then
the random variable

*T = (X/n1) / (Y/n2)*

takes a doubly non-central F distribution with parameters
*n1, n2, theta1, theta2*.

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 [email protected]

Paolella, Marc. Intermediate Probability: A Computational Approach. Wiley, 2007. http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470026375.html

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. http://arxiv.org/abs/1003.4890v1

Walck, C. "HAndbook on Statistical Distributions for experimentalists." 1996. http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

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