naginterfaces.library.stat.prob_chisq_lincomb¶
- naginterfaces.library.stat.prob_chisq_lincomb(rlam, d, c, method='D')[source]¶
prob_chisq_lincomb
calculates the lower tail probability for a linear combination of (central) variables.For full information please refer to the NAG Library document for g01jd
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01jdf.html
- Parameters
- rlamfloat, array-like, shape
The weights, , for , of the central variables.
- dfloat
, the multiplier of the central variables.
- cfloat
, the value of the constant.
- methodstr, length 1, optional
Indicates whether Pan’s, Imhof’s or an appropriately selected procedure is to be used.
Pan’s method is used.
Imhof’s method is used.
Pan’s method is used if , for are at least distinct and ; otherwise Imhof’s method is used.
- Returns
- probfloat
The lower tail probability for the linear combination of central variables.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: , or .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, for all values of , for .
- Notes
Let be independent Normal variables with mean zero and unit variance, so that have independent -distributions with unit degrees of freedom.
prob_chisq_lincomb
evaluates the probability thatIf this is equivalent to the probability that
Alternatively let
then
prob_chisq_lincomb
returns the probability thatTwo 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 then a non-adaptive method described in
quad.dim1_fin_smooth
is used to compute the value of the integral otherwisequad.dim1_fin_bad
is used.Pan’s procedure can only be used if the are sufficiently distinct;
prob_chisq_lincomb
requires the to be at least distinct; see Further Comments. If the are at least distinct and , 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