
GTLPDFName:
with and denoting the shape parameters and a and b the lower and upper limits, respectively. The case where a = 0 and b = 1 is referred to as the standard generalized Topp and Leone distribution. It has the following probability density function:
The lower and upper limits are related to the location and scale parameters as follows:
scale = b  a Kotz and van Dorp have proposed this distribution as an alternative to the beta distribution.
<SUBSET/EXCEPT/FOR qualification> where <x> is a number, parameter, or variable containing values in the interval (a,b); <y> is a variable or a parameter (depending on what <x> is) where the computed generalized Topp and Leone pdf value is stored; <alpha> is a number, parameter, or variable in the interval (0, 2) that specifies the first shape parameter; <beta> is a positive number, parameter, or variable that specifies the second shape parameter; <a> is a number, parameter, or variable that specifies the lower limit; <b> is a number, parameter, or variable that specifies the upper limit; and where the <SUBSET/EXCEPT/FOR qualification> is optional. If <a> and <b> are omitted, they default to 0 and 1, respectively.
LET Y = GTLPDF(X,0.5,2) PLOT GTLPDF(X,2,3) FOR X = 0 0.01 1
LET BETA = <value> LET A = <value> LET B = <value> LET Y = GENERALIZED TOPP LEONE ... RANDOM NUMBERS FOR I = 1 1 N GENERALIZED TOPP LEONE PROBABILITY PLOT Y GENERALIZED TOPP LEONE PROBABILITY PLOT Y2 X2 GENERALIZED TOPP LEONE PROBABILITY PLOT Y3 XLOW XHIGH GENERALIZED TOPP LEONE ... KOLMOGOROV SMIRNOV GOODNESS OF FIT Y GENERALIZED TOPP LEONE CHISQUARE GOODNESS OF FIT Y2 X2 GENERALIZED TOPP LEONE CHISQUARE ... GOODNESS OF FIT Y3 XLOW XHIGH The following commands can be used to estimate the and shape parameters for the generalized Topp and Leone distribution:
LET ALPHA2 = <value>$ LET BETA1 = <value>$ LET BETA2 = <value>$ GENERALIZED TOPP LEONE PPCC PLOT Y$ GENERALIZED TOPP LEONE PPCC PLOT Y2 X2$ GENERALIZED TOPP LEONE PPCC PLOT Y3 XLOW XHIGH$ GENERALIZED TOPP LEONE KS PLOT Y$ GENERALIZED TOPP LEONE KS PLOT Y2 X2$ GENERALIZED TOPP LEONE KS PLOT Y3 XLOW XHIGH$ The default values for ALPHA1 and ALPHA2 are 0.1 and 2. The default values for BETA1 and BETA2 are 0.5 and 10. The probability plot can then be used to estimate the lower and upper limits (lower limit = PPA0, upper limit = PPA0 + PPA1). The following options may be useful for these commands.
LABEL CASE ASIS TITLE CASE ASIS TITLE OFFSET 2 . MULTIPLOT 3 3 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 3 . LET ALPHA = 2 LET BETA = 3 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1.5 LET BETA = 6 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1.5 LET BETA = 2 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1.5 LET BETA = 1 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 2 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 1 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 0.75 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 0.25 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1 LET BETA = 1 TITLE Alpha = ^alpha, Beta = ^beta PLOT GTLPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . END OF MULTIPLOTProgram 2: let alpha = 1.5 let beta = 2 . let y = generalized topp leone rand numb for i = 1 1 200 . let alphsav = alpha let betasav = beta generalized topp leone ppcc plot y just center move 50 5 let alpha = shape1 let beta = shape2 text maxppcc = ^maxppcc, Alpha = ^alpha, Beta = ^beta move 50 2 text Alphasav = ^alphsav, Betasav = ^betasav . char x line blank generalized topp leone prob plot y move 50 5 text PPA0 = ^ppa0, PPA1 = ^ppa1 move 50 2 let upplim = ppa0 + ppa1 text Lower Limit = ^ppa0, Upper Limit = ^upplim char blank line solid . let ksloc = ppa0 let ksscale = upplim generalized topp leone kolm smir goodness of fit y . bootstrap generalized topp leone plot yThe following output is generated: KOLMOGOROVSMIRNOV GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: GENERALIZED TOPP AND LEONE NUMBER OF OBSERVATIONS = 200 TEST: KOLMOGOROVSMIRNOV TEST STATISTIC = 1.000000 ALPHA LEVEL CUTOFF CONCLUSION 10% 0.086* REJECT H0 0.085** 5% 0.096* REJECT H0 0.095** 1% 0.115* REJECT H0 0.114** *  STANDARD LARGE SAMPLE APPROXIMATION ( C/SQRT(N) ) **  MORE ACCURATE LARGE SAMPLE APPROXIMATION ( C/SQRT(N + SQRT(N/10)) )
Date created: 9/10/2007 