This function evaluates the inverse of the noncentral Student's t cumulative distribution function.

Function Return Value

TNIN — Function value.   (Output)
The probability that a noncentral Student's t random variable takes a value less than or equal to TNIN is P.

Required Arguments

P — Probability for which the inverse of the noncentral Student's t cumulative distribution function is to be evaluated.   (Input)
P must be in the open interval (0.0, 1.0).

IDF — Number of degrees of freedom of the noncentral Student's t cumulative distribution.   (Input) IDF must be positive.

DELTA — The noncentrality parameter.   (Input)

FORTRAN 90 Interface

Generic:                              TNIN (P, IDF, DELTA)

Specific:                             The specific interface names are S_TNIN and D_TNIN.

FORTRAN 77 Interface

Single:                                TNIN (P, IDF, DELTA)

Double:                              The double precision name is DTNIN.

Description

Function TNIN evaluates the inverse distribution function of a noncentral t random variable with IDF degrees of freedom and noncentrality parameter DELTA; that is, with P = P, v = IDF, and
δ = DELTA, it determines t0 (= TNIN(P, IDF, DELTA )), such that

 

 

where Γ(⋅) is the gamma function. The probability that the random variable takes a value less than or equal to t0 is P. See TNDF for an alternative definition in terms of normal and chi-squared random variables. The function TNIN uses bisection and modified regula falsi to invert the distribution function, which is evaluated using routine TNDF.

Comments

Informational error

Type Code

4         1                  Over 100 iterations have occurred without convergence. Convergence is assumed.

Example

In this example, we find the 95-th percentage point for a noncentral t random variable with 6 degrees of freedom and noncentrality parameter 6.

 

      USE TNIN_INT
      USE UMACH_INT
      IMPLICIT   NONE

      INTEGER    IDF, NOUT

      REAL       DELTA, P, T

!

      CALL UMACH (2, NOUT)

      IDF   = 6

      DELTA = 6.0

      P     = 0.95

      T     = TNIN(P,IDF,DELTA)

      WRITE (NOUT,99999) T

!

99999 FORMAT (' The 0.05 noncentral t critical value is ', F6.3, &

            '.')

!

      END

Output

 

The 0.05 noncentral t critical value is 11.995.

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