
BU3PDFName:
with r and k denoting the shape parameters. This distribution can be generalized with location and scale parameters in the usual way using the relation
If X has a Burr type 12 distribution, then 1/X has a Burr type 3 distribution. For this reason, the Burr type 3 distribution is often referred to as the inverse Burr distribution. The Burr type 3 distribution is also sometimes referred to as the Dagum type I distribution.
<SUBSET/EXCEPT/FOR qualification> where <x> is a number, parameter, or variable; <y> is a variable or a parameter (depending on what <x> is) where the computed Burr type 3 pdf value is stored; <r> is a positive number, parameter, or variable that specifies the first shape parameter; <k> is a positive number, parameter, or variable that specifies the second shape parameter; <loc> is a number, parameter, or variable that specifies the location parameter; <scale> is a positive number, parameter, or variable that specifies the scale parameter; and where the <SUBSET/EXCEPT/FOR qualification> is optional. If <loc> and <scale> are omitted, they default to 0 and 1, respectively.
LET Y = BU3PDF(X,0.5,2.2,0,5) PLOT BU3PDF(X,2.3,1.4) FOR X = 0 0.01 5
LET K = <value> LET Y = BURR TYPE 3 RANDOM NUMBERS FOR I = 1 1 N BURR TYPE 3 PROBABILITY PLOT Y BURR TYPE 3 PROBABILITY PLOT Y2 X2 BURR TYPE 3 PROBABILITY PLOT Y3 XLOW XHIGH BURR TYPE 3 KOLMOGOROV SMIRNOV GOODNESS OF FIT Y BURR TYPE 3 CHISQUARE GOODNESS OF FIT Y2 X2 BURR TYPE 3 CHISQUARE GOODNESS OF FIT Y3 XLOW XHIGH The following commands can be used to estimate the r and k shape parameters for the Burr type 3 distribution:
LET R2 = <value> LET K1 = <value> LET K2 = <value> BURR TYPE 3 PPCC PLOT Y BURR TYPE 3 PPCC PLOT Y2 X2 BURR TYPE 3 PPCC PLOT Y3 XLOW XHIGH BURR TYPE 3 KS PLOT Y BURR TYPE 3 KS PLOT Y2 X2 BURR TYPE 3 KS PLOT Y3 XLOW XHIGH The default values for R1, R2, K1, and K2 are 0.5, 10, 0.5, and 10, respectively. The probability plot can then be used to estimate the location and scale (location = PPA0, scale = PPA1). Since the Burr type 3 distribution can sometimes be heavytailed, the following command can be useful before using the BURR TYPE 3 KS PLOT:
This bases the location and scale estimates on the biweight estimates of the fitted line for the underlying probability plot. In the final probability plot, use PPA0BW and PPA1BW as the estimates of location and scale rather than PPA0 and PPA1. Since the ppcc plot is invariant to location and scale, this option does not apply. The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the parameter estimates based on the ppcc plot and ks plot.
Johnson, Kotz, and Balakrishnan (1994), "Contiunuous Univariate DistributionsVolume 1", Second Edition, Wiley, pp. 5354. Devroye (1986), "NonUniform Random Variate Generation", SpringerVerlang, pp. 476477.
LABEL CASE ASIS TITLE CASE ASIS TITLE OFFSET 2 . MULTIPLOT 4 4 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 4 . LET RVAL = DATA 0.5 1 2 5 LET KVAL = DATA 0.5 1 2 5 . LOOP FOR IROW = 1 1 4 LOOP FOR ICOL = 1 1 4 LET R = RVAL(IROW) LET K = KVAL(ICOL) TITLE R = ^r, K = ^k PLOT BU3PDF(X,R,K) FOR X = 0.01 0.01 5 END OF LOOP END OF LOOP . END OF MULTIPLOT . JUSTIFICATION CENTER MOVE 50 97 TEXT Burr Type 3 Probability Density Functions Program 2: let r = 2.1 let k = 1.3 let rsav = r let ksav = k . let y = burr type 3 random numbers for i = 1 1 200 let y = 10*y let amax = maximum y . y1label Correlation Coefficient x1label R burr type 3 ppcc plot y let r = shape1 let k = shape2 justification center move 50 6 text Rhat = ^r (R = ^rsav), Khat = ^k (K = ^ksav) move 50 2 text Maximum PPCC = ^maxppcc . y1label Data x1label Theoretical char x line bl burr type 3 prob plot y move 50 6 text Location = ^ppa0, Scale = ^ppa1 char bl line so . y1label Relative Frequency x1label relative hist y limits freeze preerase off plot bu3pdf(x,r,k,ppa0,ppa1) for x = 0.01 .01 amax limits preerase on . let ksloc = ppa0 let ksscale = ppa1 burr type 3 kolmogorov smirnov goodness of fit yKOLMOGOROVSMIRNOV GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: BURR TYPE 3 NUMBER OF OBSERVATIONS = 200 TEST: KOLMOGOROVSMIRNOV TEST STATISTIC = 0.3263575E01 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)) )
Date created: 12/17/2007 