
DPUPDFName:
where
with m and n denoting the shape parameters, denoting the location parameter, and (  ) denoting the scale parameter. This distribution is uniform between and . It has Paretian tails for both the lower and upper tails. The m parameter controls the shape of the lower tail and the n parameter controls the shape of the upper tail. The case where = 0 and = 1 is referred to as the standard doubly Pareto uniform 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 double Pareto uniform pdf value is stored; <m> is a number, parameter, or variable that specifies the first shape parameter; <n> is a number, parameter, or variable that specifies the second shape paremeter; <alpha> is a number, parameter, or variable that specifies the location parameter; <beta> is a number, parameter, or variable (<beta>  <alpha> is the scale parameter); and where the <SUBSET/EXCEPT/FOR qualification> is optional. If <alpha> and <beta> are omitted, they default to 0 and 1, respectively.
LET Y = DPUPDF(X,1.4,3.2,0,1) PLOT DPUPDF(X,1.4,3.2,5,5) FOR X = 10 0.1 10
LET N = <value> LET Y = DOUBLY PARETO UNIFORM RANDOM NUMBERS ... FOR I = 1 1 N DOUBLY PARETO UNIFORM PROBABILITY PLOT Y DOUBLY PARETO UNIFORM PROBABILITY PLOT Y2 X2 DOUBLY PARETO UNIFORM PROBABILITY PLOT Y3 XLOW XHIGH DOUBLY PARETO UNIFORM KOLMOGOROV SMIRNOV ... GOODNESS OF FIT Y DOUBLY PARETO UNIFORM CHISQUARE ... GOODNESS OF FIT Y2 X2 DOUBLY PARETO UNIFORM CHISQUARE ... GOODNESS OF FIT Y3 XLOW XHIGH For the random numbers, you can optionally specify the alpha and beta parameters with the commands
LET BETA = <value> The following commands can be used to estimate the m and n shape parameters for the doubly Pareto uniform distribution:
LET M2 = <value> LET N1 = <value> LET N2 = <value> DOUBLY PARETO UNIFORM PPCC PLOT Y DOUBLY PARETO UNIFORM PPCC PLOT Y2 X2 DOUBLY PARETO UNIFORM PPCC PLOT Y3 XLOW XHIGH DOUBLY PARETO UNIFORM KS PLOT Y DOUBLY PARETO UNIFORM KS PLOT Y2 X2 DOUBLY PARETO UNIFORM KS PLOT Y3 XLOW XHIGH The default values for M1 and M2 are 0.5 and 10. The default values for N1 and N2 are 0.5 and 10. The probability plot can then be used to estimate the and limits (alpha = PPA0, beta = PPA0 + PPA1). The following options may be useful for these commands.
The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the ppcc plot and ks plot.
Van Dorp, Singh, and Mazzuchi "The DoublyPareto Uniform Distribution with Applications in Uncertainty Analysis and Econometrics", Mediterranean Journal of Mathematics, Vol. 3 (2), pp. 205225.
LABEL CASE ASIS TITLE CASE ASIS TITLE OFFSET 2 . MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 2 . LET ALPHA = 0 LET BETA = 1 . LET M = 0.5 LET N = 0.5 TITLE M = ^m, N = ^n PLOT DPUPDF(X,M,N,ALPHA,BETA) FOR X = 5 0.01 5 . LET M = 2 LET N = 0.5 TITLE M = ^m, N = ^n PLOT DPUPDF(X,M,N,ALPHA,BETA) FOR X = 5 0.01 5 . LET M = 0.5 LET N = 2 TITLE M = ^m, N = ^n PLOT DPUPDF(X,M,N,ALPHA,BETA) FOR X = 5 0.01 5 . LET M = 2 LET N = 2 TITLE M = ^m, N = ^n PLOT DPUPDF(X,M,N,ALPHA,BETA) FOR X = 5 0.01 5 . END OF MULTIPLOT . CASE ASIS JUSTIFICATION CENTER MOVE 50 97 TEXT Doubly Pareto Uniform Probability Density Functions Program 2: let alpha = 0 let beta = 1 let m = 2 let n = 1 let msav = m let nsav = n . let y = doubly pareto uniform rand numb for i = 1 1 200 . set ppcc plot location scale biweight doubly pareto uniform ks plot y let m = shape1 let n = shape2 justification center move 50 6 text Mhat = ^m (M = ^msav) move 50 2 text Nhat = ^n (N = ^nsav) . char x line bl doubly pareto uniform prob plot y let alphahat = ppa0bw let betahat = ppa0bw + ppa1bw move 50 6 text Alphahat = ^alphahat move 50 2 text Betahat = ^betahat char blank line solid . relative hist y limits freeze preerase off line color blue plot dpupdf(x,m,n,alphahat,betahat) for x = 5 0.01 5 line color black limits preerase on . let ksloc = alphahat let ksscale = betahat  alphahat doubly pareto uniform kolm smir goodness of fit y KOLMOGOROVSMIRNOV GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: DOUBLYPARETO UNIFORM NUMBER OF OBSERVATIONS = 200 TEST: KOLMOGOROVSMIRNOV TEST STATISTIC = 0.4028350E01 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: 1/8/2008 