unimed(3f) - [M_datapac:STATISTICS] generates the N order statistic
medians from the uniform (rectangular) distribution on the unit
interval (0,1).
SUBROUTINE UNIMED(N,X)
INTEGER,intent(in) :: N
REAL(kind=wp),intent(out) :: X(:)
UNIMED(3f) generates the N order statistic medians from the uniform
(rectangular) distribution on the unit interval (0,1).
This distribution has mean = 0.5 and standard deviation = sqrt(1/12)
= 0.28867513. This distribution has the probability density function
f(X) = 1.
UNIMED(3f) is a support subroutine for all of the probability plot
subroutines in datapac; it is rarely used by the data analyst directly.
A probability plot for a general distribution is a plot of the ordered
observations versus the order statistic medians for that distribution.
The i-th order statistic median for a general distribution is obtained
by transforming the i-th uniform order statistic median by the percent
point function of the desired distribution--hence the importance of
being able to generate uniform order statistic medians.
It is of theoretical interest to note that the i-th uniform order
statistic median in a sample of size N is identically the median of
the beta distribution with parameters i and N-i+1.
N The desired integer number of uniform order statistic medians
to be generated.
X A vector (of dimension at least N) into which the generated
uniform order statistic medians will be placed.
Sample program:
program demo_unimed
use M_datapac, only : unimed, label, plotxt
implicit none
integer,parameter :: N=100
real :: X(N)
call label('unimed')
call unimed(N,X)
call plotxt(x,n)
end program demo_unimed
Results:
THE FOLLOWING IS A PLOT OF X(I) (VERTICALLY) VERSUS I (HORIZONTALLY
I-----------I-----------I-----------I-----------I
0.9930925E+00 - XX
0.9520015E+00 I XXX
0.9109104E+00 I XXX
0.8698193E+00 I XXX
0.8287283E+00 I XXX
0.7876373E+00 I XXX
0.7465463E+00 - XXX
0.7054552E+00 I XXX
0.6643642E+00 I XXX
0.6232731E+00 I XXX
0.5821820E+00 I XXX
0.5410910E+00 I XXX
0.5000000E+00 - XXX
0.4589090E+00 I XXX
0.4178179E+00 I XXX
0.3767269E+00 I XXX
0.3356358E+00 I XXX
0.2945448E+00 I XXX
0.2534538E+00 - XXX
0.2123627E+00 I XXX
0.1712717E+00 I XXX
0.1301807E+00 I XXX
0.8908957E-01 I XXX
0.4799855E-01 I XXX
0.6907523E-02 - XX
I-----------I-----------I-----------I-----------I
0.1000E+01 0.2575E+02 0.5050E+02 0.7525E+02 0.1000E+03
The original DATAPAC library was written by James Filliben of the
Statistical Engineering Division, National Institute of Standards
and Technology.
John Urban, 2022.05.31
CC0-1.0
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| integer, | intent(in) | :: | N | |||
| real(kind=wp), | intent(out) | :: | X(:) |
SUBROUTINE UNIMED(N,X) INTEGER,intent(in) :: N REAL(kind=wp),intent(out) :: X(:) REAL(kind=wp) :: ai, an, gam INTEGER :: i, imax, irev, nevodd, nhalf !--------------------------------------------------------------------- ! ! CHECK THE INPUT ARGUMENTS FOR ERRORS ! IF ( N<1 ) THEN WRITE (G_IO,99001) 99001 FORMAT (' ***** FATAL ERROR--The first input argument to UNIMED(3f) is non-positive *****') WRITE (G_IO,99002) N 99002 FORMAT (' ***** The value of the argument is ',I0,' *****') RETURN ELSE IF ( N==1 ) THEN WRITE (G_IO,99003) 99003 FORMAT (' ***** NON-FATAL DIAGNOSTIC--The first input argument to UNIMED(3f) has the value 1 *****') ENDIF an = N ! ! COMPUTE THE MEDIANS FOR THE FIRST AND LAST ORDER STATISTICS ! X(N) = 0.5_wp**(1.0_wp/an) X(1) = 1.0_wp - X(N) ! ! DETERMINE IF AN ODD OR EVEN SAMPLE SIZE ! nhalf = (N/2) + 1 nevodd = 2*(N/2) IF ( N/=nevodd ) X(nhalf) = 0.5_wp IF ( N<=3 ) RETURN ! ! COMPUTE THE MEDIANS FOR THE OTHER ORDER STATISTICS ! gam = 0.3175_wp imax = N/2 DO i = 2 , imax ai = i irev = N - i + 1 X(i) = (ai-gam)/(an-2.0_wp*gam+1.0_wp) X(irev) = 1.0_wp - X(i) ENDDO ENDIF END SUBROUTINE UNIMED