Methods and formulas for parameters to estimate in Accelerated Life Test Plan

Variance-covariance matrix

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

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:

Notation

TermDescription
tppercentile
MLE*maximum likelihood estimate (MLE) of tp
Avar(MLE*) asymptotic variance of the MLE at design (or use) stress level
Φ-1norinverse CDF of the normal distribution
DTdistance between the estimate and the upper or lower bound of the (1 – α)100% confidence interval, based on the bound that you specified for the analysis

Weibull, exponential, lognormal and loglogistic models

  • For a two-sided confidence interval:
  • For a one-sided confidence interval:
    where

Notation

TermDescription
tppercentile
MLE*maximum likelihood estimate (MLE) of ln (tp)
Avar(MLE*) asymptotic variance of the MLE at design (or use) stress level
Φ-1norinverse CDF of the normal distribution
DTdistance between the estimate and the upper or lower bound of the (1 – α)100% confidence interval, based on the bound that you specified for the analysis

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

Notation

TermDescription
MLE*standardized time
Avar(MLE*) asymptotic variance of the MLE
Φ-1norinverse CDF of the normal distribution
DTdistance between the estimate and the upper (or lower) bound of the (1 – α)100% confidence interval
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