T-NT Weibull

The pdf for the temperature non-thermal relationship and the Weibull distribution is given next.

By setting η = L(U, V) from Eqn. (1), the T-NT Weibull model is given by:

T-NT Weibull Statistical Properties Summary

Mean or MTTF

The mean, , for the T-NT Weibull model is given by:

where is the gamma function evaluated at the value of

Median

The median, for the T-NT Weibull model is given by:

(5)

Mode

The mode, for the T-NT Weibull model is given by:

(6)

Standard Deviation

The standard deviation, for the T-NT Weibull model is given by:

T-NT Weibull Reliability Function

The T-NT Weibull reliability function is given by:

Conditional Reliability Function

The T-NT Weibull conditional reliability function at a specified stress level is given by:

Reliable Life

For the T-NT Weibull model, the reliable life, TR, of a unit for a specified reliability and starting the mission at age zero is given by:

(7)

T-NT Weibull Failure Rate Function

The T-NT Weibull failure rate function, λ(T), is given by:

Parameter Estimation

Maximum Likelihood Estimation Method

Substituting the T-NT relationship into the Weibull log-likelihood function yields:

where:

and:

Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure data points in the ith time-to-failure data group.
β is the Weibull shape parameter (unknown, the first of four parameters to be estimated).
B is the first T-NT parameter (unknown, the second of four parameters to be estimated).
C is the second T-NT parameter (unknown, the third of four parameters to be estimated).
n is the third T-NT parameter (unknown, the fourth of four parameters to be estimated).
Vi is the temperature level of the ith group.
Ui is the non-thermal stress level of the ith group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is the number of suspensions in the ith group of suspension data points.
is the running time of the ith suspension data group.
FI is the number of interval data groups.
is the number of intervals in the ith group of data intervals.
is the beginning of the ith interval.
is the ending of the ith interval.

The solution (parameter estimates) will be found by solving for the parameters B, C, n and β so that = 0, = 0, = 0 and = 0.