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Arrhenius-Weibull
The $$pdf$$  for 2-parameter Weibull distribution is given by:



The scale parameter (or characteristic life) of the Weibull distribution is $$\eta $$.

The Arrhenius-Weibull model pdf can then be obtained by setting $$\eta =L(V)$$  in Eqn. (arrhenius):


 * $$\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}$$

and substituting for $$\eta $$  in Eqn. (Weibullpdf):


 * $$f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}$$

An illustration of the $$pdf$$  for different stresses is shown in Fig. 6.  As expected, the  $$pdf$$  at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3 in Fig. 6). This behavior is observed when the parameter $$B$$  of the Arrhenius model is positive.



The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in Chapter 5.

Maximum Likelihood Estimation Method
The Arrhenius-Weibull log-likelihood function is as follows:


 * $$\begin{align}

& \Lambda = & \underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}}}}{{\left( \frac{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta }}}} \right] \\ & & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }] \end{align}$$

where:


 * $$R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)}^{\beta }}}}$$


 * $$R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)}^{\beta }}}}$$

and: •	 $${{F}_{e}}$$ is the number of groups of exact times-to-failure data points.

•	 .. is the number of times-to-failure data points in the  $${{i}^{th}}$$  time-to-failure data group.

•	 $$\beta $$ is the Weibull shape parameter (unknown, the first of three parameters to be estimated).

•	 $$B$$ is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

•	 $$C$$ is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).

•	 $${{V}_{i}}$$ is the stress level of the  $${{i}^{th}}$$  group.

•	 $${{T}_{i}}$$ is the exact failure time of the  $${{i}^{th}}$$  group.

•	 $$S$$ 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 running 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 (parameter estimates) will be found by solving for $$\widehat{\beta },$$   $$\widehat{B},$$   $$\widehat{C}$$  so that  $$\tfrac{\partial \Lambda }{\partial \beta }=0,$$   $$\tfrac{\partial \Lambda }{\partial B}=0$$  and  $$\tfrac{\partial \Lambda }{\partial C}=0$$, where:


 * $$\begin{align}

& \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{C\cdot {{e}^{\tfrac{B}}}} \right) \\ & & -\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}}}} \right) \\ & & -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}}}} \right) \\ & & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \end{align}$$


 * $$\begin{align}

& \frac{\partial \Lambda }{\partial B}= & -\beta \underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}\frac{1}+\beta \underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}}}} \right)}^{\beta }} \\ & & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} \end{align}$$


 * $$\begin{align}

& \frac{\partial \Lambda }{\partial C}= & -\frac{\beta }{C}\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} \\ & & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} \end{align}$$