naginterfaces.library.stat.prob_​chisq_​noncentral_​lincomb

naginterfaces.library.stat.prob_chisq_noncentral_lincomb(a, mult, rlamda, c, tol=0.0, maxit=500)

prob_chisq_noncentral_lincomb returns the lower tail probability of a distribution of a positive linear combination of ${\chi }^2$ random variables.

For full information please refer to the NAG Library document for g01jc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01jcf.html

Parameters

afloat, array-like, shape $\left(n\right)$

The weights, $a_1,a_2,\dots ,a_n$.

multint, array-like, shape $\left(n\right)$

The degrees of freedom, $m_1,m_2,\dots ,m_n$.

rlamdafloat, array-like, shape $\left(n\right)$

The noncentrality parameters, ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$.

cfloat

$c$, the point for which the lower tail probability is to be evaluated.

tolfloat, optional

The relative accuracy required by you in the results. If prob_chisq_noncentral_lincomb is entered with $tol$ greater than or equal to $1.0$ or less than $10\times \text{machine precision}$ (see machine.precision), the value of $10\times \text{machine precision}$ is used instead.

maxitint, optional

The maximum number of terms that should be used during the summation.

Returns

pfloat

The lower tail probability associated with the linear combination of $n$ ${\chi }^2$ random variables with $m_{\text{j}}$ degrees of freedom, and noncentrality parameters ${\lambda }_{\text{j}}$, for $\text{j}=1,2,\dots ,n$.

pdffloat

The value of the probability density function of the linear combination of ${\chi }^2$ variables.

Raises

NagValueError

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit\ge 1$.

(errno $1$)

On entry, $c=⟨value⟩$.

Constraint: $c\ge 0.0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, $rlamda[⟨value⟩]=⟨value⟩$.

Constraint: $rlamda\left[\text{i}\right]\ge 0.0$, for $\text{i}=0,\dots ,n-1$.

(errno $2$)

On entry, $mult[⟨value⟩]=⟨value⟩$.

Constraint: $mult\left[\text{i}\right]\ge 1$, for $\text{i}=0,\dots ,n-1$.

(errno $2$)

On entry, $a[⟨value⟩]=⟨value⟩$.

Constraint: $a\left[\text{i}\right]>0.0$, for $\text{i}=0,\dots ,n-1$.

(errno $3$)

The central ${\chi }^2$ calculation has failed to converge. This is an unlikely exit. A larger value of $tol$ should be tried.

Warns

NagAlgorithmicWarning

(errno $4$)

The solution has failed to converge within $maxit$ iterations. A larger value of $maxit$ or $tol$ should be used. The returned value should be a reasonable approximation to the correct value.

(errno $5$)

The solution appears to be too close to $0$ or $1$ for accurate calculation. The value returned is $0$ or $1$ as appropriate.

Notes

For a linear combination of noncentral ${\chi }^2$ random variables with integer degrees of freedom the lower tail probability is

$$P(\sum _{j=1}^n{a}_j{\chi }^2(m_j,{\lambda }_j)\le c)\text{,}$$

where $a_j$ and $c$ are positive constants and where ${\chi }^2(m_j,{\lambda }_j)$ represents an independent ${\chi }^2$ random variable with $m_j$ degrees of freedom and noncentrality parameter ${\lambda }_j$. The linear combination may arise from considering a quadratic form in Normal variables.

Ruben’s method as described in Farebrother (1984) is used. Ruben has shown that (1) may be expanded as an infinite series of the form

$$\sum _{k=0}^{\infty }{d}_{k}F(m+2k,c/\beta )\text{,}$$

where $F(m+2k,c/\beta )=P({\chi }^2(m+2k)<c/\beta )$, i.e., the probability that a central ${\chi }^2$ is less than $c/\beta$.

The value of $\beta$ is set at

$$\beta ={\beta }_B=\frac{2}{(1/a_{min}+1/a_{max})}$$

unless ${\beta }_B>1.8a_{min}$, in which case

$$\beta ={\beta }_A=a_{min}$$

is used, where $a_{min}=min\left\{a_j\right\}$ and $a_{max}=setmax\left(a_j\right)$, for $\text{j}=1,2,\dots ,n$.

References

Farebrother, R W, 1984, The distribution of a positive linear combination of ${\chi }^2$ random variables, Appl. Statist. (33(3))