LOGNORMAL MOMENT ESTIMATES

LOGNORMAL MOMENT ESTIMATES (LET)

Let Subcommand

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

In most cases, we prefer to estimate the parameters of the 3-parameter lognormal distribution using the 3-PARAMETER LOGNORMAL 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 moment estimate for \( \mu \) (= log(scale))
Y(5)	=	3-parameter modified moment estimate for location
Y(6)	=	3-parameter modified moment estimate for scale
Y(7)	=	3-parameter modified moment estimate for shape
Y(8)	=	3-parameter modified moment estimate for \( \mu \) (= log(scale))

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 352-354 of Cohen and Whitten.

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

LET Y = LOGNORMAL MOMENT ESTIMATES X

None

None

Cohen and Whitten (1988), "Parameter Estimation in Reliability and Life Span Models," Marcel Dekker, p. 61 and pp. 352 -354.

WEIBULL MOMENT ESTIMATE	= Generate Weibull moment estimates.
GAMMA MOMENT ESTIMATE	= Generate Gamma 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 LOGNORMAL 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  0.3225
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 = lognormal 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 sigmamom = y(3); let sigmamom = round(sigmamom,numdec)
let uhatmom  = y(4); let uhatmom  = round(uhatmom,numdec)
let locmmom  = y(5); let locmmom  = round(locmmom,numdec)
let scalmmom = y(6); let scalmmom = round(scalmmom,numdec)
let sigmmmom = y(7); let sigmmmom = round(sigmmom,numdec)
let uhatmmom = y(8); let uhatmmom = round(uhatmmom,numdec)
.
let xmean = round(xmean,numdec)
let xsd   = round(xsd,numdec)
let xskew = round(xskew,numdec)
let xmin  = round(xmin,numdec)
.
print "Lognormal 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 Lognormal Moment Estimates:"
print "Location:         ^locmom"
print "Scale:            ^scalemom"
print "Shape:            ^sigmamom"
print "Uhat:             ^uhatmom"
print " "
print " "
print "3-Parameter Lognormal Modified Moment Estimates:"
print "Location:         ^locmmom"
print "Scale:            ^scalmmom"
print "Shape:            ^sigmmmom"
print "Uhat:             ^uhatmmom"
    

The following output is generated.

Lognormal Parameter Estimates From Summary Data
 
 
Sample Mean:      0.42308
Sample SD:        0.12532
Sample Skewness:  1.06665
Sample Minimum:   0.265
Sample Size:      20
 
 
3-Parameter Lognormal Moment Estimates:
Location:         0.05685
Scale:            0.3465
Shape:            0.33277
Uhat:             -1.05986
 
 
3-Parameter Lognormal Modified Moment Estimates:
Location:         0.1716
Scale:            0.22508
Shape:            0.4709776775
Uhat:             -1.49131