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MIEPDFName:
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).
<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 X2 = MIEPDF(X1,K,THETA)
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 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.
Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions: Volume 2", 2nd. Ed., John Wiley and Sons, p. 351.
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 Program 2: 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
Date created: 1/26/2009 |