Appendix: Log-Likelihood Equations

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:

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\frac{\beta }{\eta }{\left(\frac{T_i}{\eta }\right)}^{\beta -1}{e}^{-{\left(\frac{T_i}{\eta }\right)}^{\beta }}\right]-\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(\frac{T_i^{\prime }}{\eta }\right)}^{\beta }\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\ln [e^{-{\left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)}^{\beta }}]\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\beta$ is the Weibull shape parameter (unknown a priori, the first of two parameters to be found)
• $\eta$ is the Weibull scale parameter (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval failure data groups
• $N_i^{\prime \prime }$ is the number of intervals in $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

For the purposes of MLE, left censored data will be considered to be intervals with $T_{Li}^{\prime \prime }=0.$

The solution will be found by solving for a pair of parameters $(\hat{\beta },\hat{\eta })$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \beta }=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial \eta }=0.$ 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.

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \beta }=& \frac{1}{\beta }\underset{i=1}{\sum ^{F_e}}{N}_i+\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left(\frac{T_i}{\eta }\right)\\ & -\underset{i=1}{\sum ^{F_e}}{N}_i{\left(\frac{T_i}{\eta }\right)}^{\beta }\ln \left(\frac{T_i}{\eta }\right)-\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(\frac{T_i^{\prime }}{\eta }\right)}^{\beta }\ln \left(\frac{T_i^{\prime }}{\eta }\right)\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{-{\left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)}^{\beta }\ln \left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)e^{-{\left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)}^{\beta }}+{\left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)}^{\beta }\ln \left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)e^{-{\left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)}^{\beta }}}{e^{-{\left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)}^{\beta }}}\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \eta }=& \frac{-\beta }{\eta }\underset{i=1}{\sum ^{F_e}}{N}_i+\frac{\beta }{\eta }\underset{i=1}{\sum ^{F_e}}{N}_i{\left(\frac{T_i}{\eta }\right)}^{\beta }\\ & +\frac{\beta }{\eta }\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(\frac{T_i^{\prime }}{\eta }\right)}^{\beta }\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{\left(\frac{\beta }{\eta }\right){\left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)}^{\beta }{e}^{-{\left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)}^{\beta }}-\left(\frac{\beta }{\eta }\right){\left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)}^{\beta }{e}^{-{\left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)}^{\beta }}}{e^{-{\left(\frac{T_{Li}^{\prime \prime }}{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }}{\eta }\right)}^{\beta }}}\end{array}$

The Three-Parameter Weibull

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

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\frac{\beta }{\eta }{\left(\frac{T_i-\gamma }{\eta }\right)}^{\beta -1}{e}^{-{\left(\frac{T_i-\gamma }{\eta }\right)}^{\beta }}\right]-\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(\frac{T_i^{\prime }-\gamma }{\eta }\right)}^{\beta }\\ & \\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\ln [e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}]\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\beta$ is the Weibull shape parameter (unknown a priori, the first of three parameters to be found)
• $\eta$ is the Weibull scale parameter (unknown a priori, the second of three parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $\gamma$ is the Weibull location parameter (unknown a priori, the third of three parameters to be found)
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval data groups
• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• and $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

The solution is found by solving for $(\hat{\beta },\hat{\eta },\hat{\gamma })$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \beta }=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial \eta }=0,$ and $\frac{\partial \mathrm{\Lambda }}{\partial \gamma }=0.$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \beta }=& \frac{1}{\beta }\underset{i=1}{\sum ^{F_e}}{N}_i+\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left(\frac{T_i-\gamma }{\eta }\right)-\underset{i=1}{\sum ^{F_e}}{N}_i{\left(\frac{T_i-\gamma }{\eta }\right)}^{\beta }\ln \left(\frac{T_i-\gamma }{\eta }\right)\\ & -\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(\frac{T_i^{\prime }-\gamma }{\eta }\right)}^{\beta }\ln \left(\frac{T_i^{\prime }-\gamma }{\eta }\right)\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }\ln \left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}{e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }\ln \left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}{e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \eta }=& \frac{-\beta }{\eta }\underset{i=1}{\sum ^{F_e}}{N}_i+\frac{\beta }{\eta }\underset{i=1}{\sum ^{F_e}}{N}_i{\left(\frac{T_i-\gamma }{\eta }\right)}^{\beta }+\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(\frac{T_i^{\prime }-\gamma }{\eta }\right)}^{\beta }\left(\frac{\beta }{\eta }\right)\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{\frac{\beta }{\eta }{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }\ln \left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}{e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}\\ & -\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{\frac{\beta }{\eta }{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }\ln \left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}{e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \gamma }=& (1-\beta )\underset{i=1}{\sum ^{F_e}}\left(\frac{N_i}{T_i-\gamma }\right)+\underset{i=1}{\sum ^{F_e}}{N}_i{\left(\frac{T_i-\gamma }{\eta }\right)}^{\beta }\left(\frac{\beta }{T_i-\gamma }\right)\\ & +\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(\frac{T_i^{\prime }-\gamma }{\eta }\right)}^{\beta }\left(\frac{\beta }{T_i^{\prime }-\gamma }\right)\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{\frac{\beta }{T_{Li}^{\prime \prime }-\gamma }{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }{e}^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}-\frac{\beta }{T_{Ri}^{\prime \prime }-\gamma }{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }{e}^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}{e^{-{\left(\frac{T_{Li}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}-e^{-{\left(\frac{T_{Ri}^{\prime \prime }-\gamma }{\eta }\right)}^{\beta }}}\end{array}$

It should be pointed out that the solution to the three-parameter Weibull via MLE is not always stable and can collapse if $\beta \sim 1.$ 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 $\beta \le 2.$ In general, there are no MLE solutions in the region of $0<\beta <1.$ When $1<\beta <2,$ 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 $\gamma$ is estimated using non-linear regression. Once $\gamma$ is obtained, the MLE estimates of $\hat{\beta }$ and $\hat{\eta }$ are computed using the transformation $T_i^{\prime }=(T_i-\gamma ).$

Exponential Log-Likelihood Functions and their Partials

The One-Parameter Exponential

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

$\ln (L)=\mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\lambda e^{-\lambda T_i}\right]-\underset{i=1}{\sum ^S}{N}_i^{\prime }\lambda T_i^{\prime }+\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\ln [e^{-\lambda T_{Li}^{\prime \prime }}-e^{-\lambda T_{Ri}^{\prime \prime }}]$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\lambda$ is the failure rate parameter (unknown a priori, the only parameter to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval data groups
• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

The solution will be found by solving for a parameter $\hat{\lambda }$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \lambda }=0.$ Note that for $FI=0$ there exists a closed form solution.

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \lambda }=& \underset{i=1}{\sum ^{F_e}}{N}_i(\frac{1}{\lambda }-T_i)-\underset{i=1}{\sum ^S}{N}_i^{\prime }{T}_i^{\prime }\\ & -\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\left[\frac{T_{Li}^{\prime \prime }{e}^{-\lambda T_{Li}^{\prime \prime }}-T_{Ri}^{\prime \prime }{e}^{-\lambda T_{Ri}^{\prime \prime }}}{e^{-\lambda T_{Li}^{\prime \prime }}-e^{-\lambda T_{Ri}^{\prime \prime }}}\right]\end{array}$

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:

$\begin{array}{rlr}& \ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\lambda e^{-\lambda (T_i-\gamma )}\right]-\underset{i=1}{\sum ^S}{N}_i^{\prime }\lambda (T_i^{\prime }-\gamma )\\ & & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\ln [e^{-\lambda (T_{Li}^{\prime \prime }-\gamma )}-e^{-\lambda (T_{Ri}^{\prime \prime }-\gamma )}],\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\lambda$ is the failure rate parameter (unknown a priori, the first of two parameters to be found)
• $\gamma$ is the location parameter (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval data groups
• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

To find the two-parameter solution, look at the partial derivatives $\frac{\partial \mathrm{\Lambda }}{\partial \lambda }$ and $\frac{\partial \mathrm{\Lambda }}{\partial \gamma }$:

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \lambda }=& \underset{i=1}{\sum ^{F_e}}{N}_i[\frac{1}{\lambda }-(T_i-\gamma )]\\ & -\underset{i=1}{\sum ^S}{N}_i^{\prime }(T_i^{\prime }-\gamma )\\ & -\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\left[\frac{(T_{Li}^{\prime \prime }-\gamma )e^{-\lambda (T_{Li}^{\prime \prime }-{\gamma }_0)}-(T_{Ri}^{\prime \prime }-\gamma )e^{-\lambda (T_{Ri}^{\prime \prime }-\gamma )}}{e^{-\lambda (T_{Li}^{\prime \prime }-\gamma )}-e^{-\lambda (T_{Ri}^{\prime \prime }-\gamma )}}\right]\end{array}$

and

$\begin{array}{r}\frac{\partial \mathrm{\Lambda }}{\partial \gamma }=\underset{i=1}{\sum ^{F_e}}{N}_i\lambda +\underset{i=1}{\sum ^S}{N}_i^{\prime }\lambda +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\lambda \end{array}$.

From here we see that $\frac{\partial \mathrm{\Lambda }}{\partial \gamma }$ is a positive, constant function of $\gamma$. As alluded to in the chapter on the exponential distribution, this implies that the log-likelihood function $\mathrm{\Lambda }$ is, for fixed $\lambda$, an increasing function of $\gamma$. Thus the MLE for $\gamma$ is its largest possible value $T_1$. Therefore, to find the full MLE solution $(\hat{\lambda },\hat{\gamma })$ for the two-parameter exponential distribution, one should set $\gamma$ equal to the first failure time and then find (numerically) a $\lambda$ such that $\frac{\partial \mathrm{\Lambda }}{\partial \lambda }=0$.

The 3D Plot utility in Weibull++ further 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:

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\frac{1}{\sigma }\varphi \left(\frac{T_i-\mu }{\sigma }\right)\right]\\ & +\underset{i=1}{\sum ^S}{N}_i^{{}^{\prime }}\ln [1-\mathrm{\Phi }\left(\frac{T_i^{{}^{\prime }}-\mu }{\sigma }\right)]\\ & +\underset{i=1}{\sum ^{F_i}}{N}_i^{{}^{\prime \prime }}\ln [\mathrm{\Phi }\left(\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }\right)-\mathrm{\Phi }\left(\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }\right)]\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\mu$ is the mean parameter (unknown a priori, the first of two parameters to be found)
• $\sigma$ is the standard deviation parameter (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $F_i$ is the number of interval data groups
• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

The solution will be found by solving for a pair of parameters $({\mu }_0,{\sigma }_0)$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \mu }=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=0.$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \mu }=& \frac{1}{{\sigma }^2}\underset{i=1}{\sum ^{F_e}}{N}_i(T_i-\mu )\\ & +\frac{1}{\sigma }\underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{\varphi \left(\frac{T_i^{\prime }-\mu }{\sigma }\right)}{1-\mathrm{\Phi }\left(\frac{T_i^{\prime }-\mu }{\sigma }\right)}\\ & -\frac{1}{\sigma }\underset{i=1}{\sum ^{F_i}}{N}_i^{\prime \prime }\frac{\varphi \left(\frac{T_{Ri}^{\prime \prime }-\mu }{\sigma }\right)-\varphi \left(\frac{T_{Li}^{\prime \prime }-\mu }{\sigma }\right)}{\mathrm{\Phi }\left(\frac{T_{Ri}^{\prime \prime }-\mu }{\sigma }\right)-\mathrm{\Phi }\left(\frac{T_{Li}^{\prime \prime }-\mu }{\sigma }\right)}\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=& \underset{i=1}{\sum ^{F_e}}{N}_i(\frac{{(T_i-\mu )}^2}{{\sigma }^3}-\frac{1}{\sigma })\\ & +\frac{1}{\sigma }\underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{\left(\frac{T_i^{\prime }-\mu }{\sigma }\right)\varphi \left(\frac{T_i^{\prime }-\mu }{\sigma }\right)}{1-\mathrm{\Phi }\left(\frac{T_i^{\prime }-\mu }{\sigma }\right)}\\ & -\frac{1}{\sigma }\underset{i=1}{\sum ^{F_i}}{N}_i^{\prime \prime }\frac{\left(\frac{T_{Ri}^{\prime \prime }-\mu }{\sigma }\right)\varphi \left(\frac{T_{Ri}^{\prime \prime }-\mu }{\sigma }\right)-\left(\frac{T_{Li}^{\prime \prime }-\mu }{\sigma }\right)\varphi \left(\frac{T_{Li}^{\prime \prime }-\mu }{\sigma }\right)}{\mathrm{\Phi }\left(\frac{T_{Ri}^{\prime \prime }-\mu }{\sigma }\right)-\mathrm{\Phi }\left(\frac{T_{Li}^{\prime \prime }-\mu }{\sigma }\right)}\end{array}$

where:

$\varphi \left(x\right)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}$

and:

$\mathrm{\Phi }(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{e}^{-\frac{t^2}{2}}dt$

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:

$\hat{\mu }=\hat{\overline{T}}=\frac{1}{N}\underset{i=1}{\sum ^N}{T}_i$

and:

$\begin{array}{rl}{\hat{\sigma }}_T^2=& \frac{1}{N}\underset{i=1}{\sum ^N}{(T_i-\overline{T})}^2\\ {\hat{\sigma }}_T=& \sqrt{\frac{1}{N}\underset{i=1}{\sum ^N}{(T_i-\overline{T})}^2}\end{array}$

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:

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\frac{1}{T_i{\sigma }_{T^{\prime }}}\varphi \left(\frac{\ln \left(T_i\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)\right]\\ & +\underset{i=1}{\sum ^S}{N}_i^{\prime }\ln [1-\mathrm{\Phi }\left(\frac{\ln \left(T_i^{\prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)]\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\ln [\mathrm{\Phi }\left(\frac{\ln \left(T_{Ri}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)-\mathrm{\Phi }\left(\frac{\ln \left(T_{Li}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)]\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• ${\mu }^{\prime }$ is the mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found)
• ${\sigma }_{T^{\prime }}$ is the standard deviation of the natural logarithms of the times-to-failure (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval data groups
• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

The solution will be found by solving for a pair of parameters $({\mu }^{\prime },{\sigma }_{T^{\prime }})$ so that $\frac{\partial \mathrm{\Lambda }}{\partial {\mu }^{\prime }}=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial {\sigma }_{T^{\prime }}}=0$:

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial {\mu }^{\prime }}=& \frac{1}{{\sigma }_{T^{\prime }}^2}\underset{i=1}{\sum ^{F_e}}{N}_i(\ln (T_i)-{\mu }^{\prime })\\ & +\frac{1}{{\sigma }_{T^{\prime }}}\underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{\varphi \left(\frac{\ln \left(T_i^{\prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}{1-\mathrm{\Phi }\left(\frac{\ln \left(T_i^{\prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}\\ & -\underset{i=1}{\sum ^{FI}}\frac{N_i^{\prime \prime }}{\sigma }\frac{\varphi \left(\frac{\ln \left(T_{Ri}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)-\varphi \left(\frac{\ln \left(T_{Li}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}{\mathrm{\Phi }\left(\frac{\ln \left(T_{Ri}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)-\mathrm{\Phi }\left(\frac{\ln \left(T_{Li}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial {\sigma }_{T^{\prime }}}=& \underset{i=1}{\sum ^{F_e}}{N}_i(\frac{{(\ln (T_i)-{\mu }^{\prime })}^2}{{\sigma }_{T^{\prime }}^3}-\frac{1}{{\sigma }_{T^{\prime }}})\\ & +\frac{1}{{\sigma }_{T^{\prime }}}\underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{\left(\frac{\ln \left(T_i^{\prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)\varphi \left(\frac{\ln \left(T_i^{\prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}{1-\mathrm{\Phi }\left(\frac{\ln \left(T_i^{\prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}\\ & -\frac{1}{{\sigma }_{T^{\prime }}}\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\frac{\left(\frac{\ln \left(T_{Ri}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)\varphi \left(\frac{\ln \left(T_{Ri}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)-\left(\frac{\ln \left(T_{Li}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)\varphi \left(\frac{\ln \left(T_{Li}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}{\mathrm{\Phi }\left(\frac{\ln \left(T_{Ri}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)-\mathrm{\Phi }\left(\frac{\ln \left(T_{Li}^{\prime \prime }\right)-{\mu }^{\prime }}{{\sigma }_{T^{\prime }}}\right)}\end{array}$

where:

$\varphi \left(x\right)=\frac{1}{\sqrt{2\pi }}\cdot e^{-\frac{x^2}{2}}$

and:

$\mathrm{\Phi }(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{e}^{-\frac{t^2}{2}}dt$

Mixed Weibull Log-Likelihood Functions and their Partials

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

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\underset{k=1}{\sum ^Q}{\rho }_k\frac{{\beta }_k}{{\eta }_k}{\left(\frac{T_i}{{\eta }_k}\right)}^{{\beta }_k-1}{e}^{-{\left(\frac{T_i}{{\eta }_k}\right)}^{{\beta }_k}}\right]\\ & +\underset{i=1}{\sum ^S}{N}_i^{\prime }\ln \left[\underset{k=1}{\sum ^Q}{\rho }_k{e}^{-{\left(\frac{T_i^{\prime }}{{\eta }_k}\right)}^{{\beta }_k}}\right]\\ & +\underset{i=1}{\sum ^{FI}}{N}_i^{\prime \prime }\ln \left[\underset{k=1}{\sum ^Q}{\rho }_k\frac{{\beta }_k}{{\eta }_k}{\left(\frac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{\eta }_k}\right)}^{{\beta }_k-1}{e}^{-{\left(\frac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{\eta }_k}\right)}^{{\beta }_k}}\right]\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $Q$ is the number of subpopulations
• ${\rho }_k$ is the proportionality of the $k^{th}$ subpopulation (unknown a priori, the first set of three sets of parameters to be found)
• ${\beta }_k$ is the Weibull shape parameter of the $k^{th}$ subpopulation (unknown a priori, the second set of three sets of parameters to be found)
• ${\eta }_k$ is the Weibull scale parameter (unknown a priori, the third set of three sets of parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of groups of interval data points
• $N_i^{\prime \prime }$ is the number of intervals in $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

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

$(\widehat{{\rho }_{1,}}\widehat{{\beta }_1},\widehat{{\eta }_1},\widehat{{\rho }_{2,}}\widehat{{\beta }_2},\widehat{{\eta }_2},...,\widehat{{\rho }_{Q,}}\widehat{{\beta }_Q},\widehat{{\eta }_Q})$

so that:

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial {\rho }_1}=& 0,\frac{\partial \mathrm{\Lambda }}{\partial {\beta }_1}=0,\frac{\partial \mathrm{\Lambda }}{\partial {\eta }_1}=0\\ \frac{\partial \mathrm{\Lambda }}{\partial {\rho }_2}=& 0,\frac{\partial \mathrm{\Lambda }}{\partial {\beta }_2}=0,\frac{\partial \mathrm{\Lambda }}{\partial {\eta }_2}=0\\ ⋮\\ \frac{\partial \mathrm{\Lambda }}{\partial {\rho }_{Q-1}}=& 0,\frac{\partial \mathrm{\Lambda }}{\partial {\beta }_{Q-1}}=0,\frac{\partial \mathrm{\Lambda }}{\partial {\eta }_{Q-1}}=0\\ \frac{\partial \mathrm{\Lambda }}{\partial {\beta }_Q}=& 0,\text{ and }\frac{\partial \mathrm{\Lambda }}{\partial {\eta }_Q}=0\end{array}$

Logistic Log-Likelihood Functions and their Partials

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

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\ln \left(\frac{e^{\frac{T_i-\mu }{\sigma }}}{\sigma {(1+e^{\frac{T_i-\mu }{\sigma }})}^2}\right)-\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\ln (1+e^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }})\\ & +\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\ln (\frac{1}{1+e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}-\frac{1}{1+e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}})\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\mu$ is the logistic shape parameter (unknown a priori, the first of two parameters to be found)
• $\eta$ is the logistic scale parameter (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval failure data group
• $N_i^{\prime \prime }$ is the number of intervals in $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

For the purposes of MLE, left censored data will be considered to be intervals with $T_{Li}^{\prime \prime }=0.$

The solution of the maximum log-likelihood function is found by solving for ($\hat{\mu },\hat{\sigma })$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \mu }=0,\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=0.$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \mu }=& -\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i+\frac{2}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{e^{\frac{T_i-\mu }{\sigma }}}{1+e^{\frac{T_i-\mu }{\sigma }}}+\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\frac{e^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}}{1+e^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}}\\ & -\frac{\underset{i=1}{\stackrel{F_I}{{}_{}^{}}}{N}_i^{{}^{\prime \prime }}}{\sigma }+\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}(\frac{e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}{1+e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}+\frac{e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}{1+e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}})\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=& -\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{T_i-\mu }{{\sigma }^2}-\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i+\frac{2}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{\frac{T_i-\mu }{\sigma }{e}^{\frac{T_i-\mu }{\sigma }}}{1+e^{\frac{T_i-\mu }{\sigma }}}\\ & +\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\frac{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }{e}^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}}{1+e^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}}\\ & \frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}(\frac{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }{e}^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}{1+e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}+\frac{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }{e}^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}{1+e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}\\ & -\frac{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }{e}^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}-\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }{e}^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}{e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}-e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}})\end{array}$

The Loglogistic Log-Likelihood Functions and their Partials

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

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\ln \left(\frac{e^{\frac{\ln (T_i)-\mu }{\sigma }}}{\sigma t{(1+e^{\frac{\ln (T_i)-\mu }{\sigma }})}^2}\right)\\ & -\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\ln (1+e^{\frac{\ln (T_i^{{}^{\prime }})-\mu }{\sigma }})\\ & +\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\ln (\frac{1}{1+e^{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}-\frac{1}{1+e^{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }}})\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\mu$ is the loglogistic shape parameter (unknown a priori, the first of two parameters to be found)
• $\sigma$ is the loglogistic scale parameter (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval failure data groups,
• $N_i^{\prime \prime }$ is the number of intervals in $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

For the purposes of MLE, left censored data will be considered to be intervals with $T_{Li}^{\prime \prime }=0.$

The solution of the maximum log-likelihood function is found by solving for ($\hat{\mu },\hat{\sigma })$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \mu }=0,\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=0.$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \mu }=& -\frac{\underset{i=1}{\stackrel{F_e}{{}_{}^{}}}{N}_i}{\sigma }+\frac{2}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{e^{\frac{\ln (T_i)-\mu }{\sigma }}}{1+e^{\frac{\ln (T_i)-\mu }{\sigma }}}\\ & +\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\frac{e^{\frac{\ln (T_i^{{}^{\prime }})-\mu }{\sigma }}}{1+e^{\frac{\ln (T_i^{{}^{\prime }})-\mu }{\sigma }}}-\frac{F_I}{\sigma }\\ & +\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}(\frac{e^{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}{1+e^{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}+\frac{e^{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}{1+e^{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }}})\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=& -\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{\ln (T_i)-\mu }{{\sigma }^2}-\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i+\frac{2}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{\frac{\ln (T_i)-\mu }{\sigma }{e}^{\frac{\ln (T_i)-\mu }{\sigma }}}{1+e^{\frac{\ln (T_i)-\mu }{\sigma }}}\\ & +\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\frac{\frac{\ln (T_i^{{}^{\prime }})-\mu }{\sigma }{e}^{\frac{\ln (T_i^{{}^{\prime }})-\mu }{\sigma }}}{1+e^{\frac{\ln (T_i^{{}^{\prime }})-\mu }{\sigma }}}\\ & \frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}(\frac{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }{e}^{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}{1+e^{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}+\frac{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }{e}^{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}{1+e^{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}\\ & -\frac{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }{e}^{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }}-\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }{e}^{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }}}{e^{\frac{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }{\sigma }}-e^{\frac{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }{\sigma }}})\end{array}$

The Gumbel Log-Likelihood Functions and their Partials

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

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\ln \left(\frac{e^{\frac{T_i-\mu }{\sigma }-e^{\frac{T_i-\mu }{\sigma }}}}{\sigma }\right)\\ & -\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\ln \left(e^{-e^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}}\right)\\ & +\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\ln (e^{-e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}-e^{-e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}})\end{array}$

or:

$\begin{array}{rl}\mathrm{\Lambda }=& \underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i(\frac{T_i-\mu }{\sigma }-e^{\frac{T_i-\mu }{\sigma }})-\ln (\sigma )\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\\ & +\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}{e}^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}\\ & +\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\ln (e^{-e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}-e^{-e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}})\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\mu$ is the Gumbel shape parameter (unknown a priori, the first of two parameters to be found)
• $\sigma$ is the Gumbel scale parameter (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval failure data groups
• $N_i^{\prime \prime }$ is the number of intervals in $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

For the purposes of MLE, left censored data will be considered to be intervals with $T_{Li}^{\prime \prime }=0.$

The solution of the maximum log-likelihood function is found by solving for ($\hat{\mu },\hat{\sigma })$ so that:

$\frac{\partial \mathrm{\Lambda }}{\partial \mu }=0,\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=0.$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \mu }=& -\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i+\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i{e}^{\frac{T_i-\mu }{\sigma }}-\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}{e}^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}\\ & +\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\left(\frac{e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }-e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}-e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }-e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}}{e^{-e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}-e^{-e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}}\right)\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \sigma }=& -\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{T_i-\mu }{{\sigma }^2}-\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}+\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\frac{T_i-\mu }{\sigma }{e}^{\frac{T_i-\mu }{\sigma }}\\ & -\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\frac{T_i^{{}^{\prime }}-\mu }{\sigma }{e}^{\frac{T_i^{{}^{\prime }}-\mu }{\sigma }}+\frac{1}{\sigma }\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\\ & \left(\frac{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }{e}^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }-e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}-\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }{e}^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }-e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}}{e^{-e^{\frac{T_{L_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}-e^{-e^{\frac{T_{R_i}^{{}^{\prime \prime }}-\mu }{\sigma }}}}\right)\end{array}$

The Gamma Log-Likelihood Functions and their Partials

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

$\begin{array}{rl}\ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i\ln \left(\frac{e^{k(\ln (T_i)-\mu )-e^{e^{\ln (T_i)-\mu }}}}{T_i\mathrm{\Gamma }(k)}\right)\\ & +\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\ln (1-\mathrm{\Gamma }({}_{1}k;e^{\ln (T_i^{{}^{\prime }})-\mu )}))\\ & +\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\ln ({\mathrm{\Gamma }}_1(k;e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu })-{\mathrm{\Gamma }}_1(k;e^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }))\end{array}$

or:

$\begin{array}{rl}\mathrm{\Lambda }=& \underset{i=1}{\stackrel{F_e}{-\sum ^{}_{}}}{N}_i\ln (T_i)\underset{i=1}{\stackrel{F_e}{-\sum ^{}_{}}}{N}_i\ln (\mathrm{\Gamma }(k))+k\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i(\ln (T_i)-\mu )\\ & \underset{i=1}{\stackrel{F_e}{-\sum ^{}_{}}}{N}_i{e}^{\ln (T_i)-\mu }\\ & +\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\ln (1-{\mathrm{\Gamma }}_1(k;e^{\ln (T_i^{{}^{\prime }})-\mu }))\\ & +\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\ln ({\mathrm{\Gamma }}_1(k;e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu )})-{\mathrm{\Gamma }}_1(k;e^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu )}))\end{array}$

where:

• $F_e$ is the number of groups of times-to-failure data points
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group
• $\mu$ is the gamma shape parameter (unknown a priori, the first of two parameters to be found)
• $k$ is the gamma scale parameter (unknown a priori, the second of two parameters to be found)
• $T_i$ is the time of the $i^{th}$ group of time-to-failure data
• $S$ is the number of groups of suspension data points
• $N_i^{\prime }$ is the number of suspensions in $i^{th}$ group of suspension data points
• $T_i^{\prime }$ is the time of the $i^{th}$ suspension data group
• $FI$ is the number of interval failure data groups
• $N_i^{\prime \prime }$ is the number of intervals in $i^{th}$ group of data intervals
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval
• and $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval

For the purposes of MLE, left censored data will be considered to be intervals with $T_{Li}^{\prime \prime }=0.$

The solution of the maximum log-likelihood function is found by solving for ($\hat{\mu },\hat{\sigma })$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \mu }=0,\frac{\partial \mathrm{\Lambda }}{\partial k}=0.$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial \mu }=& -k\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i+\underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i{e}^{\ln (T_i)-\mu }\\ & +\frac{1}{\mathrm{\Gamma }(k)}\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\frac{e^{k(\ln (T_i^{{}^{\prime }})-\mu )-e^{\ln (T_i^{{}^{\prime }})-\mu )})}}{1-{\mathrm{\Gamma }}_1(k;e^{\ln (T_i^{{}^{\prime }})-\mu })}\\ & +\frac{1}{\mathrm{\Gamma }(k)}\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\{\frac{e^{k{e}^{e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }}-e^{e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu }}}}{{\mathrm{\Gamma }}_1(k;e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu })-{\mathrm{\Gamma }}_1(k;e^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu })}\\ & -\frac{e^{k{e}^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }-e^{e^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }}}}{{\mathrm{\Gamma }}_1(k;e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu })-{\mathrm{\Gamma }}_1(k;e^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu })}\}\end{array}$

$\begin{array}{rl}\frac{\partial \mathrm{\Lambda }}{\partial k}=& \underset{i=1}{\sum ^{}_{}^{F_e}}{N}_i(\ln (T_i)-\mu )-\frac{{\mathrm{\Gamma }}^{{}^{\prime }}(k)\underset{i=1}{\stackrel{F_e}{{}_{}^{}}}{N}_i}{\mathrm{\Gamma }(k)}\\ & -\underset{i=1}{\sum ^{}_{}^S}{N}_i^{{}^{\prime }}\frac{\frac{\partial {\mathrm{\Gamma }}_1(k;e^{\ln (T_i^{{}^{\prime }})-\mu })}{\partial k}}{1-{\mathrm{\Gamma }}_1(k;e^{\ln (T_i^{{}^{\prime }})-\mu })}\\ & +\underset{i=1}{\sum ^{}_{}^{F_I}}{N}_i^{{}^{\prime \prime }}\left(\frac{\frac{\partial {\mathrm{\Gamma }}_1(k;e^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu })}{\partial k}-\frac{\partial {\mathrm{\Gamma }}_1(k;e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu })}{\partial k}}{{\mathrm{\Gamma }}_1(k;e^{\ln (T_{R_i}^{{}^{\prime \prime }})-\mu })-{\mathrm{\Gamma }}_1(k;e^{\ln (T_{L_i}^{{}^{\prime \prime }})-\mu }))}\right)\end{array}$