Quality and Reliability Methods - SAS

234 Lifetime Distribution Chapter 13
Statistical Details
TH Loglogistic
are the pdf and cdf, respectively, for the standardized largest extreme value LEV(μ = 0, σ = 1) distribution.
The pdf and cdf of the three-parameter loglogistic distribution are
fxμ ( ; , σ,
γ)
=
1
σ ------------------- ( x – γ)
φ log
--------------------------------
( x – γ )–
μ
, x > γ
logis σ
Fxμσγ ( ; , , ) = Φ logis
log( x – γ)
– μ
--------------------------------
σ
where
φ logis
( z)
=
exp( z)
--------------------------------
[ 1 + exp( z)
] 2
and
Φ logis
( z)
are the pdf and cdf, respectively, for the standardized logistic or logis distribution (μ = 0, σ = 1).
Distributions for Defective Subpopulations
In reliability experiments, there are times when only a fraction of the population has a particular defect
leading to failure. Since all units are not susceptible to failure, using the regular failure distributions is
inappropriate and might produce misleading results. Use the DS distribution options to model failures that
occur on only a subpopulation. The following DS distributions are available:
• DS Lognormal
• DS Weibull
• DS Loglogistic
• DS Fréchet
Zero-Inflated Distributions
exp( z)
1
= ----------------------------- = ----------------------------
[ 1 + exp( z)
] 1 + exp( – z)
Zero-inflated distributions are used when some proportion (p) of the data fail at t = 0. When the data
contain more zeros than expected by a standard model, the number of zeros is inflated. When the
time-to-event data contain zero as the minimum value in the Life Distribution platform, four zero-inflated
distributions are available. These distributions include:
• Zero Inflated Lognormal (ZI Lognormal)
• Zero Inflated Weibull (ZI Weibull)
• Zero Inflated Loglogistic (ZI Loglogistic)
• Zero Inflated Fréchet (ZI Fréchet)