Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
The deviate, associated with the lower tail probability, , for the Normal distribution is defined as the solution to
where
The method used is an extension of that of Wichura (1988). is first replaced by .
(a)
If , is computed by a rational Chebyshev approximation
where and , are polynomials of degree .
(b)
If , is computed by a rational Chebyshev approximation
where and , are polynomials of degree .
(c)
If , is computed as
where and , are polynomials of degree .
is then calculated from , using the relationsship .
For the upper tail probability is returned, while for the two tail probabilities the value is returned, where is the required tail probability computed from the input value of .
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector parameters by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.
4 References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist.37 477–484
On entry: indicates which tail the supplied probabilities represent. Letting denote a variate from a standard Normal distribution, and , then for
, for :
The lower tail probability, i.e., .
The upper tail probability, i.e., .
The two tail (confidence interval) probability, i.e., .
The two tail (significance level) probability, i.e., .
On entry: IFAIL must be set to , . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6 Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry, at least one value of TAIL, XSTD or P was invalid.
Check IVALID for more information.
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
Dynamic memory allocation failed.
7 Accuracy
The accuracy is mainly limited by the machine precision.
8 Further Comments
None.
9 Example
This example reads vectors of values for , and and prints the corresponding deviates.