This function evaluates the inverse of the chi-squared cumulative distribution function.
Function Return Value
CHIIN — Function
value. (Output)
The probability that a chi-squared random
variable takes a value less than or equal to CHIIN is P.
Required Arguments
P — Probability
for which the inverse of the chi-squared distribution function is to be
evaluated. (Input)
P must be in the open
interval (0.0, 1.0).
DF — Number of
degrees of freedom of the chi-squared distribution. (Input)
DF must be
greater than or equal to 0.5.
FORTRAN 90 Interface
Generic: CHIIN (P, DF)
Specific: The specific interface names are S_CHIIN and D_CHIIN.
FORTRAN 77 Interface
Single: CHIIN (P, DF)
Double: The double precision name is DCHIIN.
Description
Function CHIIN evaluates the inverse distribution function of a chi-squared random variable with DF degrees of freedom, that is, with P = P and v = DF, it determines x (equal to CHIIN(P, DF)), such that
where Γ(⋅) is the gamma function. The probability that the random variable takes a value less than or equal to x is P.
For v < 40, CHIIN uses bisection (if v ≤ 2 or P > 0.98) or regula falsi to find the point at which the chi-squared distribution function is equal to P. The distribution function is evaluated using routine CHIDF.
For 40 ≤ v < 100, a modified Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.18) to the normal distribution is used, and routine ANORIN is used to evaluate the inverse of the normal distribution function. For v ≥ 100, the ordinary Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.17) is used.
Comments
Informational errors
Type Code
4 1 Over 100 iterations have occurred without convergence. Convergence is assumed.
Example
In this example, we find the 99-th percentage points of a chi-squared random variable with 2 degrees of freedom and of one with 64 degrees of freedom.
USE UMACH_INT
USE CHIIN_INT
IMPLICIT NONE
INTEGER NOUT
REAL DF, P, X
!
CALL UMACH (2, NOUT)
P = 0.99
DF = 2.0
X = CHIIN(P,DF)
WRITE (NOUT,99998) X
99998 FORMAT (' The 99-th percentage point of chi-squared with 2 df ' &
, 'is ', F7.3)
DF = 64.0
X = CHIIN(P,DF)
WRITE (NOUT,99999) X
99999 FORMAT (' The 99-th percentage point of chi-squared with 64 df ' &
, 'is ', F7.3)
END
Output
The 99-th percentage point of chi-squared with 2 df
is 9.210
The 99-th percentage point of chi-squared with 64 df
is 93.217
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