LAMPPF – M

public subroutine LAMPPF(P, Alamba, Ppf)

NAME

lamppf(3f) - [M_datapac:PERCENT_POINT] compute the Tukey-Lambda percent
point function

SYNOPSIS

   SUBROUTINE LAMPPF(P,Alamba,Ppf)

    REAL(kind=wp),intent(in) :: Alamba
    REAL(kind=wp),intent(in) :: P
    REAL(kind=wp),intent(out) :: Ppf

DESCRIPTION

LAMPPF(3f) computes the percent point function value for the (Tukey)
lambda distribution with tail length parameter value = ALAMBA.

In general, the probability density function for this distribution
is not simple.

The percent point function for this distribution is

    g(P) = ((P**ALAMBA)-((1-P)**ALAMBA))/ALAMBA

Note that the percent point function of a distribution is identically
the same as the inverse cumulative distribution function of the
distribution.

INPUT ARGUMENTS

P        The value (between 0.0 and 1.0) at which the percent point
         function is to be evaluated.

         If ALAMBA is positive, then P should be between 0.0 and 1.0,
         inclusively.

         If ALAMBA is non-positive, then P should be between 0.0 and
         1.0, exclusively.

ALAMBA   The value of lambda (the tail length parameter).

OUTPUT ARGUMENTS

PPF      The percent point function value ppf for the Tukey lambda
         distribution

EXAMPLES

Sample program:

program demo_lamppf
!@(#) line plotter graph of function
use M_datapac, only : lamppf, plott, label
implicit none
integer,parameter :: n=200
real              :: x(n), y(n)
real              :: alamba
integer           :: i
   alamba=3.3333
   call label('lamppf')
   x=[(real(i)/real(n+1),i=1,n)]
   do i=1,n
      call lamppf(x(i),alamba,y(i))
   enddo
   call plott(x,y,n)
end program demo_lamppf

Results:

 The following is a plot of Y(I) (vertically) versus X(I) (horizontally)
                   I-----------I-----------I-----------I-----------I
  0.9950249E+00 -                                                XXX
  0.9537728E+00 I                                              XXX
  0.9125207E+00 I                                           XXX
  0.8712686E+00 I                                        XXXX
  0.8300166E+00 I                                      XXX
  0.7887645E+00 I                                    XXX
  0.7475125E+00 -                                  XXX
  0.7062603E+00 I                                XXX
  0.6650083E+00 I                               XX
  0.6237562E+00 I                              XX
  0.5825042E+00 I                            XX
  0.5412520E+00 I                           XX
  0.5000000E+00 -                         XXX
  0.4587479E+00 I                        XX
  0.4174958E+00 I                       XX
  0.3762438E+00 I                     XX
  0.3349917E+00 I                    XX
  0.2937396E+00 I                  XXX
  0.2524875E+00 -                XXX
  0.2112355E+00 I              XXX
  0.1699834E+00 I            XXX
  0.1287313E+00 I         XXXX
  0.8747923E-01 I       XXX
  0.4622716E-01 I    XXX
  0.4975124E-02 -  XXX
                   I-----------I-----------I-----------I-----------I
           -0.2951E+00 -0.1475E+00  0.0000E+00  0.1475E+00  0.2951E+00

AUTHOR

The original DATAPAC library was written by James Filliben of the
Statistical Engineering Division, National Institute of Standards
and Technology.

MAINTAINER

John Urban, 2022.05.31

LICENSE

CC0-1.0

REFERENCES

Filliben, Simple and Robust Linear Estimation of the Location Parameter of a Symmetric Distribution (Unpublished PH.D. Dissertation, Princeton University), 1969, pages 21-44, 229-231, pages 53-58.
Filliben, ‘The Percent Point Function’, (Unpublished Manuscript), 1970, pages 28-31.
Hastings, Mosteller, Tukey, and Windsor, ‘Low Moments for Small Samples: A Comparative Study of Order Statistics’, Annals of Mathematical Statistics, 18, 1947, pages 413-426.

NAME

lamran(3f) - [M_datapac:RANDOM] generate Tukey-Lambda random numbers

SYNOPSIS

   SUBROUTINE LAMRAN(N,Alamba,Iseed,X)

    INTEGER,intent(in)        :: N
    REAL(kind=wp),intent(in)  :: Alamba
    INTEGER,intent(inout)     :: Iseed
    REAL(kind=wp),intent(out) :: X(:)

DESCRIPTION

LAMRAN(3f) generates a random sample of size N from the (Tukey)
lambda distribution with tail length parameter value = ALAMBA.

In general, the probability density function for this distribution
is not simple. the percent point function for this distribution is

    g(P) = ((P**ALAMBA)-((1-P)**ALAMBA))/ALAMBA

INPUT ARGUMENTS

N       The desired integer number of random numbers to be generated.

ALAMBA  The value of LAMBDA (the tail length parameter).

ISEED   An integer seed value. Should be set to a non-negative value
        to start a new sequence of values. Will be set to -1 on return
        to indicate the next call should continue the current random
        sequence walk.

OUTPUT ARGUMENTS

X       A vector (of dimension at least N) into which the generated
        random sample of size N from the (Tukey) lambda distribution
        will be placed.

EXAMPLES

Sample program:

program demo_lamran
use m_datapac, only : lamran, plott, label, plotxt, sort
implicit none
integer,parameter :: n=400
real :: x(n)
integer :: iseed
real :: gamma
   call label('lamran')
   gamma=3.4
   iseed=12345
   call lamran(n,gamma,iseed,x)
   call plotxt(x,n)
   call sort(x,n,x) ! sort to show distribution
   call plotxt(x,n)
end program demo_lamran

Results:

 THE FOLLOWING IS A PLOT OF X(I) (VERTICALLY) VERSUS I (HORIZONTALLY
                   I-----------I-----------I-----------I-----------I
  0.2940770E+00 -              XX       X X                     X
  0.2701390E+00 I   XXX    X        X              X     X X    X
  0.2462010E+00 I    X          XX X X  X            X
  0.2222630E+00 I          X        XX X  X X  X    X X X X
  0.1983251E+00 I     XX     XX  XX         X                    X
  0.1743871E+00 I      XX X  XX    X  X      X X X X   X  XX XXXX
  0.1504491E+00 -       X    X   XX   X  X        X       XX  X X
  0.1265111E+00 I  X   X XXX    X  X    X   XX     XX  XX   X X
  0.1025732E+00 I    X        XX  XXX     XX XXXX  XX X  X      XXXX
  0.7863519E-01 I   X XX XXXXX       X   X     X  X  XXX     X X XX
  0.5469720E-01 I    X      XXX     X X     XX   X X  XX    X XX X
  0.3075922E-01 I  X  XXX X    XXXX  XXX  XX   XXX    X  XX  XX X
  0.6821245E-02 -   XXXX X XX   X  X X X      XX  XX   XX  XX    X X
 -0.1711673E-01 I     X   X X XX  X   XX XX X     XX X         X  XX
 -0.4105473E-01 I        XX    X X   XXXXXX XXXX XX  X  X   X XXX
 -0.6499270E-01 I          X XX X  X    X XX XXXX   X    XX  X
 -0.8893067E-01 I  X   X  X          XX  X    X XX  X X XX   X   X X
 -0.1128686E+00 I   X     X X X     X       X             X X   XX
 -0.1368066E+00 -   XX  X          X     X X X  XXX    X  X   X
 -0.1607446E+00 I       X    XX         X     X     X  X X  XX X  X
 -0.1846826E+00 I  X  X     XX XXX X X X   X         X   X X   X   X
 -0.2086205E+00 I  X  XX X X        X   X        X  XX   XXX
 -0.2325585E+00 I    X  X         X     X  X    X
 -0.2564965E+00 I                XX          X               X    X
 -0.2804345E+00 -   X      X X           X                 X X    X
                   I-----------I-----------I-----------I-----------I
            0.1000E+01  0.1008E+03  0.2005E+03  0.3002E+03  0.4000E+03

 THE FOLLOWING IS A PLOT OF X(I) (VERTICALLY) VERSUS I (HORIZONTALLY
                   I-----------I-----------I-----------I-----------I
  0.2940770E+00 -                                                  X
  0.2701390E+00 I                                                XX
  0.2462010E+00 I                                               XX
  0.2222630E+00 I                                              XX
  0.1983251E+00 I                                             XX
  0.1743871E+00 I                                          XXX
  0.1504491E+00 -                                        XXX
  0.1265111E+00 I                                      XXX
  0.1025732E+00 I                                   XXXX
  0.7863519E-01 I                                XXXX
  0.5469720E-01 I                              XXX
  0.3075922E-01 I                          XXXX
  0.6821245E-02 -                       XXXX
 -0.1711673E-01 I                    XXXX
 -0.4105473E-01 I                 XXXX
 -0.6499270E-01 I               XXX
 -0.8893067E-01 I             XXX
 -0.1128686E+00 I            XX
 -0.1368066E+00 -          XXX
 -0.1607446E+00 I        XXX
 -0.1846826E+00 I      XXX
 -0.2086205E+00 I    XXX
 -0.2325585E+00 I    X
 -0.2564965E+00 I   X
 -0.2804345E+00 -  XX
                   I-----------I-----------I-----------I-----------I
            0.1000E+01  0.1008E+03  0.2005E+03  0.3002E+03  0.4000E+03

AUTHOR

The original DATAPAC library was written by James Filliben of the
Statistical Engineering Division, National Institute of Standards
and Technology.

MAINTAINER

John Urban, 2022.05.31

LICENSE

CC0-1.0

REFERENCES

Tocher, The Art of Simulation, 1963, pages 14-15.
Hammersley and Handscomb, Monte Carlo Methods, 1964, page 36.
Filliben, Simple and Robust Linear Estimation of the Location Parameter of a Symmetric Distribution (Unpublished PH.D. Dissertation, Princeton University), 1969, pages 21-44, 53-58.
Filliben, ‘The Percent Point Function’, (unpublished manuscript), 1970, pages 28-31.

Arguments

Type	Intent		Name
real(kind=wp),	intent(in)	::	P
real(kind=wp),	intent(in)	::	Alamba
real(kind=wp),	intent(out)	::	Ppf

Source Code

SUBROUTINE LAMPPF(P,Alamba,Ppf)
REAL(kind=wp),intent(in) :: Alamba
REAL(kind=wp),intent(in) :: P
REAL(kind=wp),intent(out) :: Ppf
!---------------------------------------------------------------------
!
!     CHECK THE INPUT ARGUMENTS FOR ERRORS
!
      IF ( Alamba>0.0_wp .OR. P>0.0_wp ) THEN
         IF ( Alamba>0.0_wp .OR. P<1.0_wp ) THEN
            IF ( Alamba<=0.0_wp .OR. P>=0.0_wp ) THEN
               IF ( Alamba<=0.0_wp .OR. P<=1.0_wp ) THEN
                  IF ( -0.001_wp<Alamba .AND. Alamba<0.001_wp ) THEN
                     Ppf = LOG(P/(1.0_wp-P))
                     RETURN
                  ELSE
                     Ppf = (P**Alamba-(1.0_wp-P)**Alamba)/Alamba
                     GOTO 99999
                  ENDIF
               ENDIF
            ENDIF
         ENDIF
      ENDIF
      WRITE (G_IO,99001)
      99001 FORMAT (' ***** FATAL ERROR--The first input argument to LAMPPF(3f) is outside the allowable (0,1) interval *****')
      WRITE (G_IO,99002) P
      99002 FORMAT (' ***** The value of the argument is ',E15.8,' *****')
      RETURN
!
99999 END SUBROUTINE LAMPPF

LAMPPF Subroutine

Contents

Source Code

public subroutine LAMPPF(P, Alamba, Ppf)

NAME

SYNOPSIS

DESCRIPTION

INPUT ARGUMENTS

OUTPUT ARGUMENTS

EXAMPLES

AUTHOR

MAINTAINER

LICENSE

REFERENCES

NAME

SYNOPSIS

DESCRIPTION

INPUT ARGUMENTS

OUTPUT ARGUMENTS

EXAMPLES

AUTHOR

MAINTAINER

LICENSE

REFERENCES

Arguments

Source Code