Dataplot Vol 2 Vol 1

# ALPHAZ

Name:
ALPHAZ (LET)
Type:
Library Function
Purpose:
Compute the alpha hazard function with shape parameter .
Description:
The alpha distribution has the following hazard function:

with denoting the shape parameter and where

t =

= 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.

Syntax:
LET <y> = ALPHAZ(<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 hazard 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.

Examples:
LET A = ALPHAZ(0.3,2.5)
LET A = ALPHAZ(X1,2.5,0,10)
PLOT ALPHAZ(X,2.5,0,3) FOR X = 0.1 0.1 10
Note:
The 11/2007 version changed the syntax for this function from

LET A = ALPHAZ(X,ALPHA,BETA,LOC,SCALE)

to

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

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

Default:
None
Synonyms:
None
Related Commands:
 ALPPDF = Compute the alpha probability density function. ALPCDF = Compute the alpha cumulative distribution function. ALPPPF = Compute the alpha percent point 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.
Reference:
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.

Applications:
Reliability, accelerated life testing
Implementation Date:
1998/4
2007/11: Corrected the second shape parameter to be the scale parameter
Program:
```
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 ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 1
TITLE ALPHA = ^alpha
PLOT ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 2
TITLE ALPHA = ^alpha
PLOT ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 5
TITLE ALPHA = ^alpha
PLOT ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Alpha Hazard Functions
```

Date created: 11/27/2007
Last updated: 11/27/2007
Please email comments on this WWW page to alan.heckert@nist.gov.