
GL5PDFName:
The standard form of the Hosking generalized logistic distribution has the probability density function:
The general form of the Hosking generalized logistic probability density function can be obtained by replacing x in the above formula with (xloc)/scale.
<SUBSET/EXCEPT/FOR qualification> where <x> is a variable, number or parameter; <alpha> is a number or parameter that specifies the value of the first shape parameter; <loc> is a number or parameter that specifies the value of the location parameter; <scale> is a number or parameter that specifies the value of the scale parameter; <y> is a variable or a parameter (depending on what <x> is) where the computed generalized logistic type 5 pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. The location and scale parameters are optional.
LET X2 = GL5PDF(X1,ALPHA) PLOT GL5PDF(X,2) FOR X = 5 0.01 0.5
LET Y = HOSKING GENERALIZED LOGISTIC RANDOM NUMBERS ... FOR I = 1 1 N HOSKING GENERALIZED LOGISTIC PROBABILITY PLOT Y HOSKING GENERALIZED LOGISTIC KOLMOGOROV SMIRNOV ... GOODNESS OF FIT Y HOSKING GENERALIZED LOGISTIC CHISQUARE ... GOODNESS OF FIT Y The following commands can be used to estimate the shape parameter for the Hosking generalized logistic distribution:
LET ALPHA2 = <value> HOSKING GENERALIZED LOGISTIC PPCC PLOT Y HOSKING GENERALIZED LOGISTIC KS PLOT Y The default values for ALPHA1 and ALPHA2 are 2 and 2, respectively. Bootstrap samples for these plots can be obtained with the commands
LET ALPHA2 = <value> BOOTSTRAP GENERALIZED LOGISTIC TYPE 2 PLOT Y BOOTSTRAP GENERALIZED LOGISTIC TYPE 2 KS PLOT Y Alternatively, Lmoment based estimates can be obtained with the command
Note that fitting becomes more problematic the further that the absolute value of is from zero. The greater the absolute value of , the greater the occurence of extreme values which distort the fitting procedures. Some informal simulations showed good performance for an absolute value less than 1 (excellent for  ≤ 0.5). However, the performance rapidly declines as  gets larger than 1. For that reason, be sure to apply fitting diagnostics (probability plots, goodness of fit tests) when fitting this distribution.
FOR I = 1 1 100 LET Y = GENERALIZED LOGISTIC HOSKING RANDOM NUMBERS ... FOR I = 1 1 100 LET Y = TYPE 2 GENERALIZED LOGISTIC RANDOM NUMBERS ... FOR I = 1 1 100 LET Y = TYPE II GENERALIZED LOGISTIC RANDOM NUMBERS ... FOR I = 1 1 100 LET Y = GENERALIZED LOGISTIC TYPE 2 RANDOM NUMBERS ... FOR I = 1 1 100 LET Y = GENERALIZED LOGISTIC TYPE II RANDOM NUMBERS ... FOR I = 1 1 100
LET A = DATA 1 0.75 0.5 0.25 0 0.25 0.5 0.75 1 MULTIPLOT 3 3 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 3 LABEL CASE ASIS TITLE CASE ASIS TITLE DISPLACEMENT 2 X1LABEL X Y1LABEL Probability Density X1LABEL DISPLACEMENT 12 Y1LABEL DISPLACEMENT 15 . LOOP FOR K = 1 1 9 LET ALPHA = A(K) LET XSTART = 5 LET XSTOP = 5 IF ALPHA > 0 LET XSTOP = 1/ALPHA END OF IF IF ALPHA < 0 LET XSTART = 1/ALPHA END OF IF TITLE Alpha = ^ALPHA PLOT GL5PDF(X,ALPHA) FOR X = XSTART 0.01 XSTOP END OF LOOP END OF MULTIPLOT CASE ASIS MOVE 50 97 JUSTIFICATION CENTER TEXT Generalized Logistic Type 5 PDF's
Date created: 3/27/2006 