Some Additional Distributions.
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.
Steven E. Pav shabbychef@gmail.com
A collection of distributions which can be approximated via Edgeworth and Cornish-Fisher expansions
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