Quality and Reliability Methods - SAS

230 Lifetime Distribution Chapter 13
Statistical Details
Logistic
where
and
φ sev
( z) = exp[ z – exp( z)
]
Φ sev
( z) = 1 – exp[ –exp( z)
]
are the pdf and cdf, respectively, for the standardized smallest extreme value, SEV(μ = 0, σ = 1) distribution.
The logistic distribution has a shape similar to the normal distribution, but with longer tails. Logistic
regression models for a binary or ordinal response are often used to model life data when negative failure
times are not an issue. The pdf and cdf are
fxμσ ( ; , )
=
1 x – μ
-- φ
σ logis
-----------
σ
, -∞ < μ < ∞ and σ > 0.
Fxμσ ( ; , ) = Φ x – μ
logis
-----------
σ
where
φ logis
( z)
=
exp( z)
--------------------------------
[ 1 + exp( z)
] 2
and
Φ logis
( z)
exp( z)
= ----------------------------- =
[ 1 + exp( z)
]
are the pdf and cdf, respectively, for the standardized logistic or logis distribution (μ = 0, σ = 1).
Largest Extreme Value (LEV)
1
----------------------------
1 + exp( – z)
This right-skewed distribution can be used to model failure times if σ is small relative to μ > 0. This
distribution is not commonly used in reliability but is useful for estimating natural extreme phenomena,
such as a catastrophic flood heights or extreme wind velocities. The pdf and cdf are
fxμσ ( ; , )
=
1 x – μ
-- φ
σ lev
-----------
σ
, -∞ < μ < ∞ and σ > 0.
Fxμσ ( ; , ) = Φ x – μ
lev
-----------
σ
where
φ lev
( z) = exp[ – z – exp( – z)
]