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
$$Y = \sum_i w_i X_i^{p_i}.$$
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 $y ~ x^2(v)$, and
$Z$ is a standard normal.
$$T = Z + t \sqrt{y/\nu}$$
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
$$t = \frac{Z' + T}{\sqrt{y/\nu}}$$Upsilon
The upsilon distribution generalizes the lambda prime to the
case of the sum of multiple chi variables. That is,
suppose $y_i ~ x^2(v_i)$
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
$, $. 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
$$T = \frac{b Z + a \sqrt{\chi^2_{\nu_1} / \nu_1}}{\sqrt{\chi^2_{\nu_2} / \nu_2}}$$
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 $v1=inf$ 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
$$T = \frac{X}{\sqrt{Y/k}}$$
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
$X ~ x^2(n1,theta1)$ independently
of $Y ~ x^2(n2,theta2)$, then
the random variable
$$T = \frac{X / n_1}{Y / n_2}$$
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.