B10PDF

B10PDF (LET)

Library Function

Compute the Burr type 10 probability density function with shape parameter r.

The standard Burr type 10 distribution has the following probability density function:

with r denoting the shape parameter.

This distribution can be generalized with location and scale parameters in the usual way using the relation

If r = 1, the Burr type 10 distribution is equivalent to the Rayleigh distribution. For this reason, the Burr type 10 distribution is also referred to as the generalized Rayleigh distribution. It has found use in reliability applications.

LET <y> = B10PDF(<x>,<r>,<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 10 pdf value is stored;
            <r> is a positive number, parameter, or variable that specifies the 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.

LET A = B10PDF(0.3,0.2)
LET Y = B10PDF(X,0.5,0,5)
PLOT B10PDF(X,2,0,3) FOR X = 0 0.01 5

Burr type 10 random numbers, probability plots, and goodness of fit tests can be generated with the commands:
LET R = <value>
LET Y = BURR TYPE 10 RANDOM NUMBERS FOR I = 1 1 N
BURR TYPE 10 PROBABILITY PLOT Y
BURR TYPE 10 PROBABILITY PLOT Y2 X2
BURR TYPE 10 PROBABILITY PLOT Y3 XLOW XHIGH
BURR TYPE 10 KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
BURR TYPE 10 CHI-SQUARE GOODNESS OF FIT Y2 X2
BURR TYPE 10 CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH


The following commands can be used to estimate the r shape parameter for the Burr type 10 distribution:

LET R1 = <value>
LET R2 = <value>
BURR TYPE 10 PPCC PLOT Y
BURR TYPE 10 PPCC PLOT Y2 X2
BURR TYPE 10 PPCC PLOT Y3 XLOW XHIGH
BURR TYPE 10 KS PLOT Y
BURR TYPE 10 KS PLOT Y2 X2
BURR TYPE 10 KS PLOT Y3 XLOW XHIGH


The default values for R1 and R2 are 0.5 and 10.

The probability plot can then be used to estimate the location and scale (location = PPA0, scale = PPA1).

The 2-parameter Burr type 10 maximum likelihood estimates can be obtained using the command

BURR TYPE 10 MAXIMUM LIKELIHOOD Y

The maximum likelihood estimates are obtained as the solution of the following simultaneous equations (from Raqab and Kundu):





The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the parameter estimates based on the ppcc plot, the ks plot, and the maximum likelihood estimates.

None

BURR TYPE X is a synonym for BURR TYPE 10.

B10CDF	= Compute the Burr type 10 cumulative distribution function.
B10PPF	= Compute the Burr type 10 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.
BU5PDF	= 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.
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.

Burr (1942), "Cumulative Frequency Functions", Annals of Mathematical Statistics, 13, pp. 215-232.

Raqab and Kundu (2006), "Burr Type X Distributions: Revisited", Journal of Probability and Statistical Sciences, Vol. 4, No. 2, pp. 179-193.

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.

Distributional Modeling

2007/10

 
LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
.
LET R  = 0.5
TITLE R = ^r
PLOT B10PDF(X,R) FOR X = 0.01  0.01  5
.
LET R  = 1
TITLE R = ^r
PLOT B10PDF(X,R) FOR X = 0.01  0.01  5
.
LET R  = 2
TITLE R = ^r
PLOT B10PDF(X,R) FOR X = 0.01  0.01  5
.
LET R  = 5
TITLE R = ^r
PLOT B10PDF(X,R) FOR X = 0.01  0.01  5
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Burr Type 10 Probability Density Functions
    

 
let r = 2.1
let rsav = r
.
let y = burr type 10 random numbers for i = 1 1 200
let y = 10*y
let amax = maximum y
.
burr type 10 ppcc plot y
let rtemp = shape - 2
let r1 = max(rtemp,0.05)
let r2 = shape + 2
y1label Correlation Coefficient
x1label R
burr type 10 ppcc plot y
let r = shape
justification center
move 50 6
text Rhat = ^r (R = ^rsav)
move 50 2
text Maximum PPCC = ^maxppcc
.
char x
line bl
burr type 10 prob plot y1
move 50 6
text Location = ^ppa0, Scale = ^ppa1
char bl
line so
.
relative hist y
limits freeze
pre-erase off
plot b10pdf(x,r,ppa0,ppa1) for x = 0.01 .01 amax
limits 
pre-erase on
.
let ksloc = ppa0
let ksscale = ppa1
burr type 10 kolmogorov smirnov goodness of fit y
.
burr type 10 mle y
let ksloc = 0
let ksscale = scaleml
let r = rml
burr type 10 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 10
    NUMBER OF OBSERVATIONS              =      200
  
 TEST:
 KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3035629E-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)) )
  
  
             BURR TYPE 10 PARAMETER ESTIMATION
  
 SUMMARY STATISTICS:
 NUMBER OF OBSERVATIONS                   =      200
 SAMPLE MEAN                              =    12.03479
 SAMPLE STANDARD DEVIATION                =    4.367350
 SAMPLE MINIMUM                           =    2.148263
 SAMPLE MAXIMUM                           =    24.75311
  
 MAXIMUM LIKELIHOOD ESTIMATES:
 ESTIMATE OF R                            =    2.141123
 ESTIMATE OF SCALE                        =   -10.27274
  
  
                   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 10
    NUMBER OF OBSERVATIONS              =      200
  
 TEST:
 KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.9947237
  
    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)) )