# 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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