Methods and formulas for parameter estimates in Parametric Growth Curve

Maximum likelihood (exact data)

For exact data, every solution, , of the following system of equations maximizes the likelihood.
where:

The standard errors are the standard deviations of the estimate of the parameter. The standard errors are calculated as the square root of the appropriate diagonal element of the inverse of the Fisher information matrix.

Notation

TermDescription
Yiretirement time for the ith system
Tij jth failure time for the ith system
ninumber of events for the ith system
Nnumber of systems

Maximum likelihood (interval data)

For interval data, the maximum likelihood estimates, , satisfy the following equations:

The standard errors are the standard deviations of the estimate of the parameter. The standard errors are calculated as the square root of the appropriate diagonal element of the inverse of the Fisher information matrix.

Notation

TermDescription
Yiretirement time for the ith system
tijinterval endpoints for the ith system
ki number of failures for the ith system
Nijnumber of failures in an interval
Ntotal number of systems (in each growth curve)

Conditional maximum likelihood

where:
The standard error for is:
where
with mi = ni - 1 if Yi = Tini or mi = ni otherwise

Notation

TermDescription
Yiretirement time for the ith system
Tij jth failure time for the ith system
ninumber of events for the ith system
Ntotal number of systems (in each growth curve)

Least squares

where

Xij = logTij

Yij = log[Tij -1Ni(Tij)]

Notation

TermDescription
Ni(Tij)number of failures in the interval (0, Tij]
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