naginterfaces.library.stat.prob_​chisq_​lincomb

naginterfaces.library.stat.prob_chisq_lincomb(rlam, d, c, method='D')

prob_chisq_lincomb calculates the lower tail probability for a linear combination of (central) ${\chi }^2$ variables.

For full information please refer to the NAG Library document for g01jd

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g01/g01jdf.html

Parameters

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

The weights, ${\lambda }_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, of the central ${\chi }^2$ variables.

dfloat

$d$, the multiplier of the central ${\chi }^2$ variables.

cfloat

$c$, the value of the constant.

methodstr, length 1, optional

Indicates whether Pan’s, Imhof’s or an appropriately selected procedure is to be used.

$method=\text{'P'}$

Pan’s method is used.

$method=\text{'I'}$

Imhof’s method is used.

$method=\text{'D'}$

Pan’s method is used if ${\lambda }_{\text{i}}^{\ast }$, for $\text{i}=1,2,\dots ,n$ are at least $1\%$ distinct and $n\le 60$; otherwise Imhof’s method is used.

Returns

probfloat

The lower tail probability for the linear combination of central ${\chi }^2$ variables.

Raises

NagValueError

(errno $1$)

On entry, $method=⟨value⟩$.

Constraint: $method=\text{'P'}$, $\text{'I'}$ or $\text{'D'}$.

(errno $1$)

On entry, $d=⟨value⟩$.

Constraint: $d\ge 0.0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, $rlam[\text{i}-1]=d$ for all values of $\text{i}$, for $\text{i}=1,2,\dots ,n$.

Notes

Let $u_1,u_2,\dots ,u_n$ be independent Normal variables with mean zero and unit variance, so that $u_1^2,u_2^2,\dots ,u_n^2$ have independent ${\chi }^2$-distributions with unit degrees of freedom. prob_chisq_lincomb evaluates the probability that

$${\lambda }_1{u}_1^2+{\lambda }_2{u}_2^2+\cdots +{\lambda }_n{u}_n^2<d(u_1^2+u_2^2+\cdots +u_n^2)+c\text{.}$$

If $c=0.0$ this is equivalent to the probability that

$$\frac{{\lambda }_1{u}_1^2+{\lambda }_2{u}_2^2+\cdots +{\lambda }_n{u}_n^2}{u_1^2+u_2^2+\cdots +u_n^2}<d\text{.}$$

Alternatively let

$${\lambda }_i^{\ast }={\lambda }_i-d\text{,}i=1,2,\dots ,n\text{,}$$

then prob_chisq_lincomb returns the probability that

$${\lambda }_1^{\ast }{u}_1^2+{\lambda }_2^{\ast }{u}_2^2+\cdots +{\lambda }_n^{\ast }{u}_n^2<c\text{.}$$

Two methods are available. One due to Pan (1964) (see Farebrother (1980)) makes use of series approximations. The other method due to Imhof (1961) reduces the problem to a one-dimensional integral. If $n\ge 6$ then a non-adaptive method described in quad.dim1_fin_smooth is used to compute the value of the integral otherwise quad.dim1_fin_bad is used.

Pan’s procedure can only be used if the ${\lambda }_i^{\ast }$ are sufficiently distinct; prob_chisq_lincomb requires the ${\lambda }_i^{\ast }$ to be at least $1\%$ distinct; see Further Comments. If the ${\lambda }_i^{\ast }$ are at least $1\%$ distinct and $n\le 60$, then Pan’s procedure is recommended; otherwise Imhof’s procedure is recommended.

References

Farebrother, R W, 1980, Algorithm AS 153. Pan’s procedure for the tail probabilities of the Durbin–Watson statistic, Appl. Statist. (29), 224–227

Imhof, J P, 1961, Computing the distribution of quadratic forms in Normal variables, Biometrika (48), 419–426

Pan, Jie–Jian, 1964, Distributions of the noncircular serial correlation coefficients, Shuxue Jinzhan (7), 328–337