
RGTPDFName:
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 reflected generalized Topp and Leone distribution. 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. It is distinguished from the beta distribution in that it can have positive density at the lower limit with a strict positive mode.
<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 reflected 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 = RGTPDF(X,0.5,2) PLOT RGTPDF(X,2,3) FOR X = 0 0.01 1
LET BETA = <value> LET A = <value> LET B = <value> LET Y = REFLECTED GENERALIZED TOPP LEONE ... RANDOM NUMBERS FOR I = 1 1 N REFLECTED GENERALIZED TOPP LEONE PROBABILITY PLOT Y REFLECTED GENERALIZED TOPP LEONE PROBABILITY PLOT Y2 X2 REFLECTED GENERALIZED TOPP LEONE PROBABILITY PLOT ... Y3 XLOW XHIGH REFLECTED GENERALIZED TOPP LEONE ... KOLMOGOROV SMIRNOV GOODNESS OF FIT Y REFLECTED GENERALIZED TOPP LEONE CHISQUARE ... GOODNESS OF FIT Y2 X2 REFLECTED GENERALIZED TOPP LEONE CHISQUARE ... GOODNESS OF FIT Y3 XLOW XHIGH The following commands can be used to estimate the and shape parameters for the reflected generalized Topp and Leone distribution:
LET ALPHA2 = <value>$ LET BETA1 = <value>$ LET BETA2 = <value>$ REFLECTED GENERALIZED TOPP LEONE PPCC PLOT Y$ REFLECTED GENERALIZED TOPP LEONE PPCC PLOT Y2 X2$ REFLECTED GENERALIZED TOPP LEONE PPCC PLOT Y3 XLOW XHIGH$ REFLECTED GENERALIZED TOPP LEONE KS PLOT Y$ REFLECTED GENERALIZED TOPP LEONE KS PLOT Y2 X2$ REFLECTED 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.
Kotz and Van Dorp describe an approximate maximum likelihood method for estimating the and parameters of the standard reflected generalized Topp and Leone distribution. It is assumed that the data are grouped into m intervals [x_{i1},x_{i}] where the ith interval has n_{i} observations. The midpoint of the interval is denoted as . For unbinned data, the intervals consist of the individual observations (and the frequency is equal to 1). The total sample size, N is the sum of the n_{i}. The parameter is the solution to the following equation:
where
The maximum likelihood estimate of is then:
If the data lie outside the (0,1) interval, then we first apply the transformation
To generate the approximate maximum likelihood estimates for ungrouped data in Dataplot, enter the command
For grouped data, enter one of the following commands
REFLECTED GENERALIZED TOPP AND LEONE MLE Y XLOW XHIGH In the first case, X denotes the midpoints of the bins and Y denotes the corresponding frequency. In the second case, XLOW denotes the lower endpoints of the bins and XHIGH denotes the upper endpoints of the bins. If the lower and upper limits are fixed and known, you can enter the following commands:
LET UPPLIMIT = <VALUE> For the unknown case, the minimum and maximum of the data will be used (an epsilon value will be subtracted/added to the minimum/maximum). Alternatively, you can use the estimates of the lower/upper limits generated by either the PPCC plot method or the KS plot methods and specify the LOWLIMIT and UPPLIMIT as above. To specify starting values for and , enter the commands
LET BETASV = <VALUE> For example, the estimates obtained from the PPCC plot or the KS plot can be used as starting values for the maximum likelihood estimates.
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 RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1.5 LET BETA = 6 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1.5 LET BETA = 2 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1.5 LET BETA = 1 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 2 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 1 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 0.75 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 0.5 LET BETA = 0.25 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . LET ALPHA = 1 LET BETA = 1 TITLE Alpha = ^alpha, Beta = ^beta PLOT RGTPDF(X,ALPHA,BETA) FOR X = 0 0.01 1 . END OF MULTIPLOTProgram 2: let alpha = 1.5 let beta = 2 . let y = reflected generalized topp leone rand numb for i = 1 1 200 . let alphsav = alpha let betasav = beta reflected 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 reflected 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 reflected generalized topp leone kolm smir goodness of fit y . bootstrap reflected 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: REFLECTED GENERALIZED TOPP AND LEONE NUMBER OF OBSERVATIONS = 200 TEST: KOLMOGOROVSMIRNOV TEST STATISTIC = 0.3394580E01 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: 9/10/2007 