## 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 ~ 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;

## Lambda Prime

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)

## Upsilon

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.

## K Prime

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.

## 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 ~ 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.

## 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 ~ 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.

## 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 [email protected]

## References

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

sadists documentation built on May 29, 2017, 9:39 p.m.