ALPCHAZ

ALPCHAZ (LET)

Library Function

Compute the alpha cumulative hazard function with shape parameter .

The alpha distribution has the following cumulative hazard function:

with denoting the shape parameter and where is the cumulative distribution function of the standard normal distribution.

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> = ALPCHAZ(<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 cumulative 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.

LET A = ALPCHAZ(0.3,2.5)
LET A = ALPCHAZ(X1,2.5,0,10)
PLOT ALPCHAZ(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 = ALPCHAZ(X,ALPHA,BETA,LOC,SCALE)

to

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

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

None

None

ALPPDF	= Compute the alpha probability density function.
ALPCDF	= Compute the alpha cumulative distribution function.
ALPPPF	= Compute the alpha percent point function.
ALPHAZ	= Compute the alpha 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.

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.

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