GAMMA MOMENT ESTIMATES

GAMMA MOMENT ESTIMATES (LET)

Let Subcommand

Estimate the parameters of the 3-parameter gamma distribution based on summary statistics.

In most cases, we prefer to estimate the parameters of the 3-parameter gamma distribution using the 3-PARAMETER GAMMA MLE Y command. However, this assumes that we have the full data set. In some cases, we may only have summary statistics available.

The input array, say X, should contain the following values:

X(1)	=	the sample mean
X(2)	=	the sample standard deviation
X(3)	=	the sample skewness
X(4)	=	the sample minimum
X(5)	=	the sample size

If one of the values is not available, then you can enter either CPUMIN or the statistic missing value. For example, if the skewness is not available, you can do one of the following:

PROBE CPUMIN
LET CPUMIN = PROBVAL
LET X(3) = CPUMIN


or

SET STATISTIC MISSING VALUE -9999
LET X(3) = -9999


The following output vector, say Y, is returned:

Y(1)	=	3-parameter moment estimate for location
Y(2)	=	3-parameter moment estimate for scale
Y(3)	=	3-parameter moment estimate for shape
Y(4)	=	3-parameter modified moment estimate for location
Y(5)	=	3-parameter modified moment estimate for scale
Y(6)	=	3-parameter modified moment estimate for shape

Any of these moment estimates that cannot be computed will be set to CPUMIN. This can happen if certain summary statistics are not provided or if the equation solvers are not able to find a solution.

The 3-parameter gamma moment and modified moment estimates are computed using the codes provided on pages 363-364 of Cohen and Whitten.

LET <y> = GAMMA MOMENT ESTIMATES <x>
                  <SUBSET/EXCEPT/FOR qualification>
where <x> is the variable containing the summary statistics;
            <y> is a variable containing the gamma moment estimates;
and where the <SUBSET/EXCEPT/FOR qualification> is optional and rarely used for this command.

LET Y = GAMMA MOMENT ESTIMATES X

None

None

Cohen and Whitten (1988), "Parameter Estimation in Reliability and Life Span Models," Marcel Dekker, p. 61 and pp. 363-364.

WEIBULL MOMENT ESTIMATE	= Generate Weibull moment estimates.
LOGNORMAL MOMENT ESTIMATE	= Generate lognormal moment estimates.
INVERSE GAUSSIAN MOMENT ESTIMATE	= Generate Gamma moment estimates.
MAXIMUM LIKELIHOOD	= Perform maximum likelihood estimation for various distributions.
BEST DISTRIBUTIONAL FIT	= Perform a best distributional fit analysis.
PPCC PLOT	= Generate a probability plot correlation coefficient plot.
PROBABILITY PLOT	= Generate a probability plot.

Reliability

2014/4

 
. Purpose:  Test GAMMA MOMEMNT ESTIMATES command
.
. Step 1:   Read data
.
.           Data from
.
.           Cohen and Whitten (1988), "Parameter Estimation in
.           Reliability  and Life Span Models", Dekker, p. 54.
.
serial read x
0.654  0.613  0.315  0.449  0.297
0.402  0.379  0.423  0.379  0.3235
0.269  0.740  0.418  0.412  0.494
0.416  0.338  0.392  0.484  0.265
end of data
.
let xmean = mean x
let xsd   = sd   x
let xmin  = mini x
let xskew = skew x
let n = size x
let z = data xmean xsd xskew xmin n
.
let y = gamma moment estimates z
.
let numdec = 5
.
let locmom   = y(1); let locmom   = round(locmom,numdec)
let scalemom = y(2); let scalemom = round(scalemom,numdec)
let gammamom = y(3); let gammamom = round(gammamom,numdec)
let locmmom  = y(4); let locmmom  = round(locmmom,numdec)
let scalmmom = y(5); let scalmmom = round(scalmmom,numdec)
let gammmmom = y(6); let gammmmom = round(gammmmom,numdec)
.
let xmean = round(xmean,numdec)
let xsd   = round(xsd,numdec)
let xskew = round(xskew,numdec)
let xmin  = round(xmin,numdec)
.
print "Gamma Parameter Estimates From Summary Data"
print " "
print " "
print "Sample Mean:      ^xmean"
print "Sample SD:        ^xsd"
print "Sample Skewness:  ^xskew"
print "Sample Minimum:   ^xmin"
print "Sample Size:      ^n"
print " "
print " "
print "3-Parameter Gamma Moment Estimates:"
print "Location:         ^locmom"
print "Scale:            ^scalemom"
print "Shape:            ^gammamom"
print " "
print " "
print "3-Parameter Gamma Modified Moment Estimates:"
print "Location:         ^locmmom"
print "Scale:            ^scalmmom"
print "Shape:            ^gammmmom"
    

The following output is generated.

 
Gamma Parameter Estimates From Summary Data
 
 
Sample Mean:      0.42313
Sample SD:        0.12528
Sample Skewness:  1.06732
Sample Minimum:   0.265
Sample Size:      20
 
 
3-Parameter Gamma Moment Estimates:
Location:         0.18837
Scale:            0.06686
Shape:            3.51129
 
 
3-Parameter Gamma Modified Moment Estimates:
Location:         0.20962
Scale:            0.07351
Shape:            2.90435