Methods and formulas for parameter estimates in Parametric Distribution Analysis (Right Censoring)

Parameter estimates

Formula

Distribution Parameters

Smallest extreme value

Normal

Logistic

μ = location,

σ = scale, σ > 0

Lognormal

Loglogistic

μ = location, μ > 0

σ = scale, σ > 0

3-parameter lognormal

3-parameter loglogistic

μ = location, μ > 0

σ = scale, σ > 0

λ = threshold.

Weibull

α = scale, α = exp(μ)

β = shape, β = 1/σ

3-parameter Weibull

α = scale, α = exp(μ)

β = shape, β = 1/σ

λ = threshold,

Exponential

θ = scale, θ > 0

2-parameter exponential

θ = scale, θ > 0

λ = threshold,

Standard error of parameter estimates

The standard error is the standard deviation of the estimate of the parameter. The standard error provides a measure of the variability in each estimate.

, , , , , and denote the standard error of the MLE of μ, σ, α, β, θ, and λ. Each standard error is calculated as the square root of the appropriate diagonal element of the inverse of the Fisher information matrix.

Confidence limits for parameter estimates

Formula

Distribution Parameter Lower confidence limit Upper Confidence limit
Smallest extreme value, normal, logistic, lognormal, loglogistic Location, μ
Scale, σ
3-parameter lognormal, 3-parameter loglogistic Location, μ
Scale, σ
Threshold, λ
Weibull Shape, β
Scale, α

3-parameter Weibull

Shape, β

Scale, α

Threshold, λ

Exponential Scale
2-parameter exponential Scale, θ
Threshold, λ
Note

For some data, the likelihood function is unbounded and, therefore, yields inconsistent estimates for distributions with a threshold parameter (such as the 2-parameter exponential). When this happens, the variance-covariance matrix of the estimated parameters cannot be determined numerically. In that case, Minitab assumes that is fixed, resulting in SE () = 0. The upper and lower bound for is .

Notation

TermDescription
zx the upper critical value for the standard normal distribution where 100x % is the confidence level and 0 < x < 1.
By using this site you agree to the use of cookies for analytics and personalized content.  Read our policy