Methods and formulas for parameters to estimate in Estimation Test Plan

Asymptotic variance

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
TermDescription
Nsample size
tp,mleML estimate of tp
DTdistance 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 norinverse CDF of the normal distribution

Weibull, lognormal, and loglogistic models

  • For a two-sided confidence interval
  • For a one-sided confidence interval
where
TermDescription
Nsample size
tp,mleML estimate of tp
RTprecision 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 norinverse 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

TermDescription
Nsample size
μmleMLE estimate of mean (normal, logistic), location (smallest extreme value), or log-location (lognormal, loglogistic)
σmleMLE estimate of scale parameter
DTprecision
Φ-1 inverse CDF of the chosen model
Φ-1 norinverse CDF of the normal distribution
1 W.Q. Meeker and L.A. Escobar (1998). Statistical Methods for Reliability Data. John Wiley & Sons, Inc.
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