MIEPDF

MIEPDF (LET)

Library Function

Compute the Mielke's beta-kappa probability density function with shape parameters k and .

The general form of Mielke's beta-kappa distribution has the following probability density function:

with k and denoting shape parameters and u and denoting the location and scale parameters, respectively.

The standard form of the distribution is

Mielke's beta-kappa distribution is a special case of a reparameterized generalized F distribution of the form a*(F(v₁,v₂)^b). The details of the reparameterization are given in Johnson, Kotz, and Balakrishnan. This reference also discusses several forms of generalized F distributions.

This distribution is also closely related to the Kappa distribution (see Hosking and Wallis for details).

LET <y> = MIEPDF(<x>,<k>,<theta>,<u>,<beta>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, parameter, or variable;
            <k> is a number, parameter, or variable that specifies the first shape parameter;
            <theta> is a number, parameter, or variable that specifies the second shape parameter;
            <u> is a number, parameter, or variable that specifies the location parameter;
            <beta> is a number, parameter, or variable that specifies the scale parameter;
            <y> is a variable or a parameter (depending on what <x> is) where the computed Mielke's beta-kappa pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The <u> and <beta> parameters are optional.

LET A = MIEPDF(3,0.5,2,0,1.5)
LET X2 = MIEPDF(X1,K,THETA)

Mielke's beta-kappa random numbers, probability plots, and goodness of fit tests can be generated with the commands:
LET K = <value>
LET THETA = <value>
LET Y = MIELKE BETA KAPPA RANDOM NUMBERS FOR I = 1 1 N
MIELKE BETA KAPPA PROBABILITY PLOT Y
MIELKE BETA KAPPA PROBABILITY PLOT Y2 X2
MIELKE BETA KAPPA PROBABILITY PLOT Y3 XLOW XHIGH
MIELKE BETA KAPPA KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
MIELKE BETA KAPPA CHI-SQUARE GOODNESS OF FIT Y2 X2
MIELKE BETA KAPPA CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH


The following commands can be used to estimate the k and theta shape parameter for the Mielke's beta-kappa distribution:

LET K1 = <value>
LET K2 = <value>
LET THETA1 = <value>
LET THETA2 = <value>
MIELKE BETA KAPPA PPCC PLOT Y
MIELKE BETA KAPPA PPCC PLOT Y2 X2
MIELKE BETA KAPPA PPCC PLOT Y3 XLOW XHIGH
MIELKE BETA KAPPA KS PLOT Y
MIELKE BETA KAPPA KS PLOT Y2 X2
MIELKE BETA KAPPA KS PLOT Y3 XLOW XHIGH


The default values for K1 and K2 are 0.5 and 10, respectively. The default values for THETA1 and THETA2 are 0.5 and 10, respectively.

The probability plot can then be used to estimate the location and scale (location = PPA0, scale = PPA1).

The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the ppcc plot and the ks plot.

None

None

MIECDF	= Compute Miekle's beta-kappa cumulative distribution function.
MIEPPF	= Compute Miekle's beta-kappa percent point function.
KAPPDF	= Compute the Kappa probability density function.
BETPDF	= Compute the beta probability density function.
FPDF	= Compute the F probability density function.
GAMPDF	= Compute the gamma probability density function.
NCBPDF	= Compute the non-central beta probability density function.
NORPDF	= Compute the normal probability density function.

Hosking and Wallis (1997), "Regional Frequency Analysis", Cambridge University Press, Appendix A10.

Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions: Volume 2", 2nd. Ed., John Wiley and Sons, p. 351.

Distributional Modeling

1996/1: Original implementation as KAPPDF
2008/5: Renamed as MIEPDF (KAPPDF now refers to regular Kappa distribution)
2008/5: Beta parameter now properly treated as a scale parameter (was previously treated as a shape parameter)

 
LET KP = DATA 0.5  1  1.5  2.0
LET T1 = 0.5
LET T2 = 1
LET T3 = 1.5
LET T4 = 2
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 95 95
MULTIPLOT SCALE FACTOR 2
TITLE CASE ASIS
TITLE OFFSET 2
X3LABEL
LINE COLOR BLACK BLUE RED GREEN
.
LOOP FOR LL = 1 1 4
   LET K = KP(LL)
   TITLE K = ^K, Theta = 0.5, 1, 1.5, 2
   PLOT MIEPDF(X,K,T1) FOR X = 0.01  0.01  5  AND
   PLOT MIEPDF(X,K,T2) FOR X = 0.01  0.01  5  AND
   PLOT MIEPDF(X,K,T3) FOR X = 0.01  0.01  5  AND
   PLOT MIEPDF(X,K,T4) FOR X = 0.01  0.01  5
END OF LOOP
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Mielke's Beta-Kappa PDF Functions
    

 
let k = 1.8
let theta = 1.2
let ksav = k
let thetasav = theta
.
let y = mielke beta-kappa rand numb for i = 1 1 200
.
let k1 = 0.5
let k2 = 5
let theta1 = 0.5
let theta2 = 5
.
title automatic
x3label
mielke beta-kappa ppcc plot y
just center
move 50 6
text K = ^shape1, Theta = ^shape2
move 50 2
text KSAV = ^ksav, THETASAV = ^thetasav
.
mielke beta-kappa ks plot y
just center
move 50 6
text K = ^shape1, Theta = ^shape2
move 50 2
text KSAV = ^ksav, THETASAV = ^thetasav
let k = shape1
let theta = shape2
.
char x
line blank
mielke beta-kappa prob plot y
just center
move 50 6
text PPA0 = ^ppa0, PPA1 = ^ppa1, PPCC = ^ppcc
move 50 2
text PPA0BW = ^ppa0bw, PPA1BW = ^ppa1bw
line solid
char blank
.
relative hist y
let amin = minimum y
let loc = max(ppa0bw,amin)
let amax = maximum y
title
limits freeze
pre-erase off
plot miepdf(x,k,theta,loc,ppa1bw) for x = loc  0.01  amax
limits
pre-erase on
.
let ksloc = loc
let ksscale = ppa1bw
mielke beta kappa kolm smir goodness of fit y
    









                   KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            MEILKE BETA-KAPPA
    NUMBER OF OBSERVATIONS              =      200
  
 TEST:
 KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.6514558E-01
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       0.086*              ACCEPT H0
                      0.085**
             5%       0.096*              ACCEPT H0
                      0.095**
             1%       0.115*              ACCEPT H0
                      0.114**
  
     *  - STANDARD LARGE SAMPLE APPROXIMATION  ( C/SQRT(N) )
    ** - MORE ACCURATE LARGE SAMPLE APPROXIMATION  ( C/SQRT(N + SQRT(N/10)) )