TNPR

This function evaluates the noncentral Student's t probability density function.

Function Return Value

TNPR — Function value, the value of the probability density function. (Output)

Required Arguments

T — Argument for which the noncentral Student's t probability density function is to be evaluated. (Input)

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

DELTA — Noncentrality parameter. (Input)

FORTRAN 90 Interface

Generic: TNPR (T, DF, DELTA)

Specific: The specific interface names are S_TNPR and D_TNPR.

Description

The noncentral Student's t distribution is a generalization of the Student's t distribution.

If w is a normally distributed random variable with unit variance and mean δ and u is a chi‑square random variable with ν degrees of freedom that is statistically independent of w, then

 

is a noncentral t‑distributed random variable with ν degrees of freedom and noncentrality parameter δ, that is, with ν = DF, and δ = DELTA. The probability density function for the noncentral t‑distribution is:

 

where

 

and t = T.

For δ = 0, the PDF reduces to the (central) Student’s t PDF:

 

and, for t = 0, the PDF becomes:

 

Example

This example calculates the noncentral Student’s t PDF for a distribution with 2 degrees of freedom and noncentrality parameter δ = 10.

 

      USE TNPR_INT

      USE UMACH_INT

      IMPLICIT NONE

     

      INTEGER  :: NOUT, I

      REAL     :: X(6)=(/ -.5, 1.5, 3.5, 7.5, 51.5, 99.5 /)

      REAL     :: DF, DELTA, PDFV

 

      CALL UMACH (2, NOUT)

      DF = 2.0

      DELTA = 10.0

 

      WRITE (NOUT,'("DF: ", F4.0, " DELTA: ", F4.0 // &

             " X PDF(X)")') DF, DELTA

 

      DO I = 1, 6

         PDFV = TNPR(X(I), DF, DELTA)

         WRITE (NOUT,'(1X, F4.1, 2X, E12.5)') X(I), PDFV

      END DO

      END

Output

 

DF:   2.  DELTA:  10.

 

   X        PDF(X)

 -0.5   0.16399E-23

  1.5   0.74417E-09

  3.5   0.28972E-02

  7.5   0.78853E-01

 51.5   0.14215E-02

 99.5   0.20290E-03