Methods and formulas for the distributions in Tolerance Intervals (Nonnormal Distribution)

Select the method or formula of your choice.

Maximum likelihood estimates

Maximum likelihood estimates of the parameters in the distribution are calculated by maximizing the likelihood function with respect to the parameters. For a given data set, the maximum likelihood estimates are the most likely values for the distribution parameters.

The Newton-Raphson algorithm is used to calculate maximum likelihood estimates of the distribution parameters. The Newton-Raphson algorithm is an iterative numerical method for calculating the maximum of a function. 1

Note

Minitab calculates the parameter estimates using the maximum likelihood method for all the distributions except the lognormal distribution. For the lognormal distribution, Minitab calculates unbiased parameter estimates.

Probability distributions

Lognormal distribution

PDF
CDF
Mean
Stdev
TermDescription
μScale parameter
σShape parameter

Gamma distribution

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CDF
Mean αβ
Stdev αβ2
TermDescription
αShape parameter
βScale parameter

Exponential distribution

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CDF
Mean θ
Stdev θ
TermDescription
θScale parameter

Smallest extreme value distribution

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CDF
Mean
Stdev

TermDescription
μLocation parameter
σScale parameter
γEuler's constant (approximately equals 0.5772)

Weibull distribution

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CDF
Mean
Stdev
TermDescription
αScale parameter
βShape parameter

Largest extreme value distribution

PDF
CDF
Mean
Stdev

TermDescription
μLocation parameter
σScale parameter
γEuler's constant (approximately equals 0.5772)

Logistic distribution

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CDF
Mean μ
Stdev
TermDescription
μLocation parameter
σScale parameter

Loglogistic distribution

PDF
CDF
Mean
Stdev
TermDescription
μLocation parameter
σScale parameter
1 W. Murray, Ed. (1972). Numerical Methods for Unconstrained Optimization. Academic Press.
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