Quality and Reliability Methods - SAS

Chapter 13 Lifetime Distribution 235
Statistical Details
The pdf and cdf for zero-inflated distributions are
ft () = ( 1 – p) 1 -- 1 t σ -- φ
Ft () = p + ( 1 – p)
Φ
( log()
t – μ)
---------------------------
σ
( ---------------------------
log()
t – μ)
σ
where
p is the proportion of zero data values,
t is the time of measurement for the lifetime event,
μ and σ are estimated by calculating the usual maximum likelihood estimations after removing zero
values from the original data,
φ(z) and Φ(z) are the density and cumulative distribution function, respectively, for a standard
distribution. For example, for a Weibull distribution,
φ(z) = exp(z-exp(z)) and Φ(z) = 1 - exp(-exp(z)).
See Lawless (2003, p 34) for a more detailed explanation of using zero-inflated distributions. Substitute
p = 1 - p and S 1 (t) = 1 - Φ(t) to obtain the form shown above.
See Tobias and Trindade (1995, p 232) for additional information about reliability distributions. This
reference gives the general form for mixture distributions. Using the parameterization in Tobias and
Trindade, the form above can be found by substituting α = p, F d (t) = 1, and F N (t) = Φ(t).