AVar (MLE) is the asymptotic variance and ACov (,) is the asymptotic covariance of the MLEs of μ, σ, θ, and β taken from the appropriate element of the inverse of the Fisher information matrix. For more information, see Meeker and Escobar1.
Percentile case
The sample size needed to estimate the percentile, tp, is calculated as follows:
Normal, logistic, and smallest extreme value distributions
For a two-sided confidence interval
For a one-sided confidence interval
where
Term
Description
N
sample size
tp,mle
ML estimate of tp
DT
distance between the estimate and the upper (or lower) bound of the (1 – α)100% confidence interval
Φ-1
inverse CDF of the chosen model
Φ-1 nor
inverse CDF of the normal distribution
Weibull, lognormal, and loglogistic models
For a two-sided confidence interval
For a one-sided confidence interval
where
Term
Description
N
sample size
tp,mle
ML estimate of tp
RT
precision when the upper (or lower) bound of the (1 – α)100% confidence interval falls X percent away from the MLE. For an upper bound, RT =1 + X/100. For a lower bound, RT = 1/(1-X/100).
Φ-1
inverse CDF for the chosen model
Φ-1 nor
inverse CDF of the normal distribution
Reliability case
For a two-sided confidence interval
For a one-sided confidence interval
where
for the lower bound
for the upper bound
for normal, logistic, and smallest extreme value distributions
for Weibull, lognormal, and loglogistic distributions
Term
Description
N
sample size
μmle
MLE estimate of mean (normal, logistic), location (smallest extreme value), or log-location (lognormal, loglogistic)
σmle
MLE estimate of scale parameter
DT
precision
Φ-1
inverse CDF of the chosen model
Φ-1 nor
inverse CDF of the normal distribution
1 W.Q. Meeker and L.A. Escobar (1998). Statistical Methods for Reliability Data. John Wiley & Sons, Inc.
,
) is the asymptotic covariance of the MLEs of μ, σ, θ, and β taken from the appropriate element of the inverse of the Fisher information matrix. For more information, see Meeker and Escobar1. 
for the lower bound 
for the upper bound 
for normal, logistic, and smallest extreme value distributions 
for Weibull, lognormal, and loglogistic distributions 