Dataplot Vol 2 Vol 1

BU3PDF

Name:
BU3PDF (LET)
Type:
Library Function
Purpose:
Compute the Burr type 3 probability density function with shape parameters r and k.
Description:
The standard Burr type 3 distribution has the following probability density function:

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.

Syntax:
LET <y> = BU3PDF(<x>,<r>,<k>,<loc>,<scale>)
<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.

Examples:
LET A = BU3PDF(0.3,0.2,1.7)
LET Y = BU3PDF(X,0.5,2.2,0,5)
PLOT BU3PDF(X,2.3,1.4) FOR X = 0 0.01 5
Note:
Burr type 3 random numbers, probability plots, and goodness of fit tests can be generated with the commands:

LET R = <value>
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 CHI-SQUARE GOODNESS OF FIT Y2 X2
BURR TYPE 3 CHI-SQUARE 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 R1 = <value>
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 heavy-tailed, the following command can be useful before using the BURR TYPE 3 KS PLOT:

SET PPCC PLOT LOCATION SCALE BIWEIGHT

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.

Default:
None
Synonyms:
BURR TYPE III is a synonym for BURR TYPE 3.
Related Commands:
 BU3CDF = Compute the Burr type 3 cumulative distribution function. BU3PPF = Compute the Burr type 3 percent point function. BU2PDF = Compute the Burr type 2 probability density function. BU3PDF = Compute the Burr type 3 probability density function. BU4PDF = Compute the Burr type 4 probability density function. BU5PDF = Compute the Burr type 5 probability density function. BU6PDF = Compute the Burr type 6 probability density function. BU7PDF = Compute the Burr type 7 probability density function. BU8PDF = Compute the Burr type 8 probability density function. BU9PDF = Compute the Burr type 9 probability density function. B10PDF = Compute the Burr type 10 probability density function. B11PDF = Compute the Burr type 11 probability density function. B12PDF = Compute the Burr type 12 probability density function. RAYPDF = Compute the Rayleigh probability density function. WEIPDF = Compute the Weibull probability density function. EWEPDF = Compute the exponentiated Weibull probability density function.
Reference:
Burr (1942), "Cumulative Frequency Functions", Annals of Mathematical Statistics, 13, pp. 215-232.

Johnson, Kotz, and Balakrishnan (1994), "Contiunuous Univariate Distributions--Volume 1", Second Edition, Wiley, pp. 53-54.

Devroye (1986), "Non-Uniform Random Variate Generation", Springer-Verlang, pp. 476-477.

Applications:
Distributional Modeling
Implementation Date:
2007/10
Program 1:
```
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
pre-erase off
plot bu3pdf(x,r,k,ppa0,ppa1) for x = 0.01 .01 amax
limits
pre-erase on
.
let ksloc = ppa0
let ksscale = ppa1
burr type 3 kolmogorov smirnov goodness of fit y

KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT 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:
KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3263575E-01

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
Last updated: 12/17/2007