Exponential Log-Likelihood Functions and their Partials

The One-Parameter Exponential

This log-likelihood function is composed of three summation portions:

where:

Fe is the number of groups of times-to-failure data points
Ni is the number of times-to-failure in the ith time-to-failure data group
λ is the failure rate parameter (unknown a priori, the only parameter to be found)
Ti is the time of the ith group of time-to-failure data
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 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
and is the ending of the ith interval

The solution will be found by solving for a parameter so that Note that for FI = 0 there exists a closed form solution.

The Two-Parameter Exponential

This log-likelihood function for the two-parameter exponential distribution is very similar to that of the one-parameter distribution and is composed of three summation portions:

where,

Fe is the number of groups of times-to-failure data points
Ni is the number of times-to-failure in the ith time-to-failure data group
λ is the failure rate parameter (unknown a priori, the first of two parameters to be found)
γ is the location parameter (unknown a priori, the second of two parameters to be found)
Ti is the time of the ith group of time-to-failure data
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 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
and is the ending of the ith interval

The two-parameter solution will be found by solving for a pair of parameters (, ), such that For the one-parameter case, solve for

(1)

and:

(2)

Examination of Eqn. (1) will reveal that:

or Eqn. (2) will be equal to zero only if either:

or:

This is an unwelcome fact, alluded to earlier in the chapter, that essentially indicates that there is no realistic solution for the two-parameter MLE for exponential. The above equations indicate that there is no non-trivial MLE solution that satisfies both

It can be shown that the best solution for γ, satisfying the constraint that γ T1 is γ = T1. To then solve for the two-parameter exponential distribution via MLE, one can set γ equal to the first time-to-failure, and then find a λ such that

Using this methodology, a maximum can be achieved along the λ-axis, and a local maximum along the γ-axis at γ = T1, constrained by the fact that γ T1. The 3D Plot utility in Weibull++ illustrates this behavior of the log-likelihood function, as shown next:

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