Appendix: Log-Likelihood Equations

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This appendix covers the log-likelihood functions and their associated partial derivatives for most of the distributions available in Weibull++. These distributions are discussed in more detail in the chapter for each distribution.

Weibull Log-Likelihood Functions and their Partials

The Two-Parameter Weibull

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

where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the Weibull shape parameter (unknown a priori, the first of two parameters to be found)
  • is the Weibull scale parameter (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in group of suspension data points
  • is the time of the suspension data group
  • is the number of interval failure data groups
  • is the number of intervals in group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

For the purposes of MLE, left censored data will be considered to be intervals with

The solution will be found by solving for a pair of parameters so that and It should be noted that other methods can also be used, such as direct maximization of the likelihood function, without having to compute the derivatives.



The Three-Parameter Weibull

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

where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the Weibull shape parameter (unknown a priori, the first of three parameters to be found)
  • is the Weibull scale parameter (unknown a priori, the second of three parameters to be found)
  • is the time of the group of time-to-failure data
  • is the Weibull location parameter (unknown a priori, the third of three parameters to be found)
  • is the number of groups of suspension data points
  • is the number of suspensions in group of suspension data points
  • is the time of the suspension data group
  • is the number of interval data groups
  • is the number of intervals in the group of data intervals
  • is the beginning of the interval
  • and is the ending of the interval

The solution is found by solving for so that and






It should be pointed out that the solution to the three-parameter Weibull via MLE is not always stable and can collapse if In estimating the true MLE of the three-parameter Weibull distribution, two difficulties arise. The first is a problem of non-regularity and the second is the parameter divergence problem, as discussed in Hirose [14].

Non-regularity occurs when In general, there are no MLE solutions in the region of When MLE solutions exist but are not asymptotically normal, as discussed in Hirose [14]. In the case of non-regularity, the solution is treated anomalously.

Weibull++ attempts to find a solution in all of the regions using a variety of methods, but the user should be forewarned that not all possible data can be addressed. Thus, some solutions using MLE for the three-parameter Weibull will fail when the algorithm has reached predefined limits or fails to converge. In these cases, the user can change to the non-true MLE approach (in Weibull++ Application Setup), where is estimated using non-linear regression. Once is obtained, the MLE estimates of and are computed using the transformation

Exponential Log-Likelihood Functions and their Partials

The One-Parameter Exponential

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

where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the failure rate parameter (unknown a priori, the only parameter to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in the group of suspension data points
  • is the time of the suspension data group
  • is the number of interval data groups
  • is the number of intervals in the group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

The solution will be found by solving for a parameter so that Note that for 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:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the failure rate parameter (unknown a priori, the first of two parameters to be found)
  • is the location parameter (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in the group of suspension data points
  • is the time of the suspension data group
  • is the number of interval data groups
  • is the number of intervals in the group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

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

and:

Examination the derivative for will reveal that:

The above equation 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 satisfying the constraint that is To then solve for the two-parameter exponential distribution via MLE, one can set equal to the first time-to-failure, and then find a such that

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

Normal Log-Likelihood Functions and their Partials

The complete normal likelihood function (without the constant) is composed of three summation portions:



where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the mean parameter (unknown a priori, the first of two parameters to be found)
  • is the standard deviation parameter (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in the group of suspension data points
  • is the time of the suspension data group
  • is the number of interval data groups
  • is the number of intervals in the group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

The solution will be found by solving for a pair of parameters so that and



where:

and:


Complete Data

Note that for the normal distribution, and in the case of complete data only (as was shown in Basic Statistical Background), there exists a closed-form solution for both of the parameters or:


and:

Lognormal Log-Likelihood Functions and their Partials

The general log-likelihood function (without the constant) for the lognormal distribution is composed of three summation portions:


where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found)
  • is the standard deviation of the natural logarithms of the times-to-failure (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in the group of suspension data points
  • is the time of the suspension data group
  • is the number of interval data groups
  • is the number of intervals in the group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

The solution will be found by solving for a pair of parameters so that and :



where:

and:

Mixed Weibull Log-Likelihood Functions and their Partials

The log-likelihood function (without the constant) is composed of three summation portions:

where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the number of subpopulations
  • is the proportionality of the subpopulation (unknown a priori, the first set of three sets of parameters to be found)
  • is the Weibull shape parameter of the subpopulation (unknown a priori, the second set of three sets of parameters to be found)
  • is the Weibull scale parameter (unknown a priori, the third set of three sets of parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in group of suspension data points
  • is the time of the suspension data group
  • is the number of groups of interval data points
  • is the number of intervals in group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

The solution will be found by solving for a group of parameters:

so that:

Logistic Log-Likelihood Functions and their Partials

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

where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the logistic shape parameter (unknown a priori, the first of two parameters to be found)
  • is the logistic scale parameter (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in group of suspension data points
  • is the time of the suspension data group
  • is the number of interval failure data group
  • is the number of intervals in group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

For the purposes of MLE, left censored data will be considered to be intervals with

The solution of the maximum log-likelihood function is found by solving for ( so that




The Loglogistic Log-Likelihood Functions and their Partials

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

where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the loglogistic shape parameter (unknown a priori, the first of two parameters to be found)
  • is the loglogistic scale parameter (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in group of suspension data points
  • is the time of the suspension data group
  • is the number of interval failure data groups,
  • is the number of intervals in group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

For the purposes of MLE, left censored data will be considered to be intervals with

The solution of the maximum log-likelihood function is found by solving for ( so that



The Gumbel Log-Likelihood Functions and their Partials

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

or:

where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the Gumbel shape parameter (unknown a priori, the first of two parameters to be found)
  • is the Gumbel scale parameter (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in group of suspension data points
  • is the time of the suspension data group
  • is the number of interval failure data groups
  • is the number of intervals in group of data intervals
  • is the beginning of the interval
  • is the ending of the interval

For the purposes of MLE, left censored data will be considered to be intervals with

The solution of the maximum log-likelihood function is found by solving for ( so that:



The Gamma Log-Likelihood Functions and their Partials

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

or:


where:

  • is the number of groups of times-to-failure data points
  • is the number of times-to-failure in the time-to-failure data group
  • is the gamma shape parameter (unknown a priori, the first of two parameters to be found)
  • is the gamma scale parameter (unknown a priori, the second of two parameters to be found)
  • is the time of the group of time-to-failure data
  • is the number of groups of suspension data points
  • is the number of suspensions in group of suspension data points
  • is the time of the suspension data group
  • is the number of interval failure data groups
  • is the number of intervals in group of data intervals
  • is the beginning of the interval
  • and is the ending of the interval

For the purposes of MLE, left censored data will be considered to be intervals with

The solution of the maximum log-likelihood function is found by solving for ( so that




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