Quality and Reliability Methods - SAS

226 Lifetime Distribution Chapter 13
Statistical Details
Nonparametric Fit
A nonparametric fit describes the basic curve of a distribution. For data with no censoring (failures only)
and for data where the observations consist of both failures and right-censoring, JMP uses Kaplan-Meier
estimates. For mixed, interval, or left censoring, JMP uses Turnbull estimates. When your data set contains
only right-censored data, the Nonparametric Estimate report indicates that the nonparametric estimate
cannot be calculated.
The Life Distribution platform uses the midpoint estimates of the step function to construct probability
plots. The midpoint estimate is halfway between (or the average of) the current and previous Kaplan-Meier
estimates.
Parametric Distributions
Parametric distributions provide a more concise distribution fit than nonparametric distributions. The
estimates of failure-time distributions are also smoother. Parametric models are also useful for extrapolation
(in time) to the lower or upper tails of a distribution.
Note: Many distributions in the Life Distribution platform are parameterized by location and scale. For
lognormal fits, the median is also provided. And a threshold parameter is also included in threshold
distributions. Location corresponds to μ, scale corresponds to σ, and threshold corresponds to γ.
Lognormal
Lognormal distributions are used commonly for failure times when the range of the data is several powers of
ten. This distribution is often considered as the multiplicative product of many small positive identically
independently distributed random variables. It is reasonable when the log of the data values appears
normally distributed. Examples of data appropriately modeled by the lognormal distribution include
hospital cost data, metal fatigue crack growth, and the survival time of bacteria subjected to disinfectants.
The pdf curve is usually characterized by strong right-skewness. The lognormal pdf and cdf are
fxμ ( ; , σ)
=
1
-----φ
xσ nor
log( x)
– μ
----------------------- , x > 0
σ
log( x)
– μ
Fxμσ ( ; , ) = Φ nor
-----------------------
σ
,
where
φ nor
( z)
=
1 z 2 ----------
2π
exp
–----
2
and
z
Φ nor
( z) = φ nor
( w) dw
–∞
are the pdf and cdf, respectively, for the standardized normal, or nor(μ = 0, σ = 1) distribution.