Var (MLE) and Cov (μ,σ) are the variances and covariances of the MLEs of μ, σ, α, and β taken from the appropriate element of the inverse of the Fisher information matrix.
Percentile case for the normal, logistic, and smallest extreme value distributions
The sample size needed to estimate the percentile, tp, is calculated as follows:
For a two-sided confidence interval:
For a one-sided confidence interval:
Calculations for the standard error of the percentile
When the specifications for the analysis include the sample size, then the analysis
solves for the standard error of the percentile. In this case, the following formula
gives the asymptotic variance of the percentile:
Avar(tp) = Avar(MLE*)
Notation
tp
percentile
MLE*
maximum likelihood estimate (MLE) of tp
Avar(MLE*)
asymptotic variance of the MLE at design (or use) stress level
Φ-1nor
inverse CDF of the normal distribution
DT
Half the width of the (1–α)100% confidence interval for the percentile
Percentile case for the Weibull, exponential, lognormal, and loglogistic
distributions
The sample size needed to estimate the percentile, tp, is calculated
as follows:
For a two-sided confidence interval:
For a one-sided confidence interval:
where DT depends on whether you specify the distance between the estimate
and the upper limit or the distance between the estimate and the lower
limit.
Calculations for the standard error of the percentile
When the specifications for the analysis include the sample size, then the analysis
solves for the standard error of the percentile. In this case, the following formula
gives the asymptotic variance of the natural log of the percentile:
Avar(tp) = (tp)2Avar(ln(tp))
Notation
Term
Description
tp
percentile
MLE*
maximum likelihood estimate (MLE) of tp
Avar(MLE*)
asymptotic variance of the MLE at design (or use) stress level
Φ-1nor
inverse CDF of the normal distribution
Dupper
distance between the estimate and the upper bound
Dlower
distance between the estimate and the lower bound
Reliability case
The MLE of the standardized time when you estimate reliability is calculated as follows:
For a two-sided confidence interval:
For a one-sided confidence interval:
where
Calculations for the standard error of the reliability
When the specifications for the analysis include the sample size, then the analysis
solves for the standard error of the reliability. In this case, the following
formula gives the asymptotic variance of the reliability:
Avar(Reliability) =
(ϕ(zMLE*))2Avar(zMLE*)
where the definition of ϕ depends on the distribution for the
analysis.
Distribution
ϕ
Normal or lognormal
pdf of the normal distribution
Logistic or loglogistic
pdf of the logistic distribution
Weibull, smallest extreme value, or exponential
pdf of the smallest extreme value distribution
Notation
Term
Description
MLE*
maximum likelihood estimate (MLE) of standardized time
(ZMLE*)
ZMLE* for the normal, logistic, and smallest extreme value distributions
standardized time = (t − μ) / σ
ZMLE* for the Weibull, exponential, lognormal, and loglogistic
distributions
