# Methods and formulas for parametric models in Parametric Growth Curve

## Homogeneous Poisson process

The homogeneous Poisson process (HPP) is a Poisson process with a constant intensity function, λ. The intervals between failures are independent, identically distributed, random variables that follow an exponential distribution with mean = 1/λ.

Because the intensity function of the homogeneous Poisson process is constant, this model is appropriate only when intervals between failures do not systematically increase or decrease. The homogeneous Poisson process is not appropriate for systems that are either improving or deteriorating.

## Power-law process

A nonhomogeneous Poisson process with the following intensity function:

The intensity function represents the rate of failures or repairs. The value of the shape (β) depends on whether your system is improving, deteriorating, or remaining stable.

• If 0 < β < 1, the failure/repair rate is decreasing. Thus, your system is improving over time.
• If β = 1, the failure/repair rate is constant. Thus, your system is remaining stable over time.
• If β > 1, the failure/repair rate is increasing. Thus, your system is deteriorating over time.
###### Note

With the default (maximum likelihood) estimation method, the power-law process is also referred to as the AMSAA model or Crow-AMSAA model. (In the original Crow-AMSAA model the scale parameter is lambda= 1/Theta^(beta).) When only a single system is considered and the least squares estimation method is used, the power-law process is called the Duane model.

### Notation

TermDescription
βishape
θiscale
Ni number of failures in the interval (0,t] for the ith system
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