INVERSE GAUSSIAN MOMENT ESTIMATES

INVERSE GAUSSIAN MOMENT ESTIMATES (LET)

Let Subcommand

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

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

Note that Dataplot supports two parameterizations of the inverse Gaussian distribution. The original parameterization of Tweedie has shape parameters \(\mu\) and \(\gamma\) (some references use \(\lambda\) for this parameter). The Chan parameterization has shape parameters \(\mu\) and \(\sigma\) where

\( \sigma = \sqrt{\frac{\mu^3}{\gamma}} \)

This command returns estimates for \(\mu\), \(\sigma\), and \(\gamma\) so that estimates based on either parameterization can be obtained.

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 \(\mu\)
Y(3)	=	3-parameter moment estimate for \(\sigma\)
Y(4)	=	3-parameter moment estimate for \(\gamma\)
Y(5)	=	3-parameter modified moment estimate for location
Y(6)	=	3-parameter modified moment estimate for \(\mu\)
Y(7)	=	3-parameter modified moment estimate for \(\sigma\)
Y(8)	=	3-parameter modified moment estimate for \( \mu \)

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 moment and modified moment estimates are computed using the codes provided on pages 360-361 of Cohen and Whitten.

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

LET Y = INVERSE GAUSSIAN MOMENT ESTIMATES X

None

None

Cohen and Whitten (1988), "Parameter Estimation in Reliability and Life Span Models," Marcel Dekker, chapter 5 and pp. 361-362.

WEIBULL MOMENT ESTIMATE	= Generate Weibull moment estimates.
GAMMA MOMENT ESTIMATE	= Generate Gamma moment estimates.
LOGNORMAL MOMENT ESTIMATE	= Generate lognormal 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.

Reliability

2014/4

 
. Purpose:  Test INVERSE GAUSSIAN MOMENT 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 = inverse gaussian moment estimates z
.
let numdec = 5
.
let locmom   = y(1); let locmom   = round(locmom,numdec)
let mumom    = y(2); let mumom    = round(mumom,numdec)
let sigmamom = y(3); let sigmamom = round(sigmamom,numdec)
let gammamom = y(4); let gammamom = round(gammamom,numdec)
let locmmom  = y(5); let locmmom  = round(locmmom,numdec)
let mummom   = y(6); let mummom   = round(mummom,numdec)
let sigmmmom = y(7); let sigmmmom = round(sigmmmom,numdec)
let gammmmom = y(8); 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 "Inverse Gaussian 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 Inverse Gaussian Moment Estimates:"
print "Location:         ^locmom"
print "Mu:               ^mumom"
print "Sigma:            ^sigmamom"
print "Gamma:            ^gammamom"
print " "
print " "
print "3-Parameter Inverse Gaussian Modified Moment Estimates:"
print "Location:         ^locmmom"
print "Mu:               ^mummom"
print "Sigma:            ^sigmmmom"
print "Gamma:            ^gammmmom"
    

The following output is generated.

Inverse Gaussian 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 Inverse Gaussian Moment Estimates:
Location:         0.07099
Mu:               0.35213
Sigma:            0.12528
Gamma:            2.78197
 
 
3-Parameter Inverse Gaussian Modified Moment Estimates:
Location:         0.12162
Mu:               0.30151
Sigma:            0.12528
Gamma:            1.74641