ALPPDF

ALPPDF (LET)

Library Function

Compute the alpha probability density function with shape parameter .

The alpha distribution has the following probability density function:

with denoting the shape parameter and where

= the standard normal cumulative distribution function

= the standard normal probability density function

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

If Y has a normal distribution with location and scale parameters and truncated to the left of 0, then X = 1/Y has an alpha distribution with shape parameter = / and scale parameter .

This distribution has application in reliability.

LET <y> = ALPPDF(<x>,<alpha>,<loc>,<scale>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, parameter, or variable containing positive values;
            <y> is a variable or a parameter (depending on what <x> is) where the computed alpha pdf value is stored;
            <alpha> 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 = ALPPDF(0.3,2.5)
LET A = ALPPDF(X1,2.5,0,10)
PLOT ALPPDF(X,2.5,0,3) FOR X = 0.1 0.1 10

The 11/2007 version changed the syntax for this function from
LET A = ALPPDF(X,ALPHA,BETA,LOC,SCALE)

to

LET A = ALPPDF(X,ALPHA,LOC,SCALE)

This was done since BETA is in fact a scale parameter.

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


The following commands can be used to estimate the alpha shape parameter for the alpha distribution:

LET ALPHA1 = <value>
LET ALPHA2 = <value>
ALPHA PPCC PLOT Y
ALPHA PPCC PLOT Y2 X2
ALPHA PPCC PLOT Y3 XLOW XHIGH
ALPHA KS PLOT Y
ALPHA KS PLOT Y2 X2
ALPHA KS PLOT Y3 XLOW XHIGH


The default values for ALPHA1 and ALPHA2 are 0.5 and 10.

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

The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the parameter estimates obtained from the ppcc plot and the ks plot methods.

None

None

ALPCDF	= Compute the alpha cumulative distribution function.
ALPPPF	= Compute the alpha percent point function.
ALPHAZ	= Compute the alpha hazard function.
ALPCHAZ	= Compute the alpha cumulative hazard function.
PEXPDF	= Compute the exponential power probability density function.
WEIPDF	= Compute the Weibull probability density function.
LGNPDF	= Compute the log-normal probability density function.
NORPDF	= Compute the normal probability density function.
NORCDF	= Compute the normal cumulative distribution function.

Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions--Volume 1", Second Edition, John Wiley and Sons, p. 173.

Salvia (1985), "Reliability applications of the Alpha Distribution", IEEE Transactions on Reliability, Vol. R-34, No. 3, pp. 251-252.

Reliability, accelerated life testing

1998/4
2007/11: Corrected the second shape parameter to be the scale parameter

 
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  = 0.5
TITLE ALPHA = ^alpha
PLOT ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 1
TITLE ALPHA = ^alpha
PLOT ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 2
TITLE ALPHA = ^alpha
PLOT ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 5
TITLE ALPHA = ^alpha
PLOT ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Alpha Probability Density Functions
    

 
let alpha = 2.4
let y = alpha random numbers for i = 1 1 200
let y = 10*y
let alphasav = alpha
let amax = maximum y
.
alpha ppcc plot y
let alpha1 = alpha - 1
let alpha1 = max(alpha1,0.1)
let alpha2 = alpha + 1
y1label Correlation Coefficient
x1label Alpha
alpha ppcc plot y
justification center
move 50 6
let alpha = shape
text Alphahat = ^alpha (True Value: ^alphasav)
.
char x
line bl
y1label Data
x1label Theoretical
alpha prob plot y
move 50 6
text Location = ^ppa0, Scale = ^ppa1
move 50 2
text PPCC = ^ppcc
char bl
line so
label
.
relative histogram y
limits freeze
pre-erase off
plot alppdf(x,alpha,ppa0,ppa1) for x = 0.01 .01 amax
limits 
pre-erase on
.
let ksloc = ppa0
let ksscale = ppa1
alpha kolm smir 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:            ALPHA
    NUMBER OF OBSERVATIONS              =      200
  
 TEST:
 KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.9138396E-01
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       0.086*              REJECT 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)) )