TDF

This function evaluates the Student’s t cumulative distribution function.

Function Return Value

TDF — Function value, the probability that a Student’s t random variable takes a value less than or equal to the input T. (Output)

Required Arguments

T — Argument for which the Student’s t distribution function is to be evaluated. (Input)

DF — Degrees of freedom. (Input)
DF must be greater than or equal to 1.0.

Optional Arguments

COMPLEMENT — Logical. If .TRUE., the complement of the Student’s t cumulative distribution function is evaluated. If .FALSE., the Student’s t cumulative distribution function is evaluated. (Input)
See the Description section for further details on the use of COMPLEMENT.
Default: COMPLEMENT = .FALSE..

FORTRAN 90 Interface

Generic: TDF (T, DF [, …])

Specific: The specific interface names are S_TDF and D_TDF.

FORTRAN 77 Interface

Single: TDF (T, DF)

Double: The double precision name is DTDF.

Description

Function TDF evaluates the cumulative distribution function of a Student’s t random variable with DF degrees of freedom. If the square of T is greater than or equal to DF, the relationship of a t to an F random variable (and subsequently, to a beta random variable) is exploited, and routine BETDF is used. Otherwise, the method described by Hill (1970) is used. Let ν = DF. If ν is not an integer, if ν is greater than 19, or if ν is greater than 200, a Cornish‑Fisher expansion is used to evaluate the distribution function. If ν is less than 20 and ABS(T) is less than 2.0, a trigonometric series (see Abramowitz and Stegun 1964, equations 26.7.3 and 26.7.4, with some rearrangement) is used. For the remaining cases, a series given by Hill (1970) that converges well for large values of T is used.

If COMPLEMENT = .TRUE., the value of TDF at the point x is 1 ‑ p, where 1 ‑ p is the probability that the random variable takes a value greater than x. In those situations where the desired end result is 1 ‑ p, the user can achieve greater accuracy in the right tail region by using the result returned by TDF with the optional argument COMPLEMENT set to .TRUE. rather than by using 1 ‑ p where p is the result returned by TDF with COMPLEMENT set to .FALSE..

 

Figure 1,  Student’s t Distribution Function

Example

In this example, we find the probability that a t random variable with 6 degrees of freedom is greater in absolute value than 2.447. We use the fact that t is symmetric about 0.

 

USE TDF_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER NOUT

REAL DF, P, T

!

CALL UMACH (2, NOUT)

T = 2.447

DF = 6.0

P = 2.0*TDF(-T,DF)

WRITE (NOUT,99999) P

99999 FORMAT (' The probability that a t(6) variate is greater ', &

'than 2.447 in', /, ' absolute value is ', F6.4)

END

Output

 

The probability that a t(6) variate is greater than 2.447 in absolute value is 0.0500