Template:Bounds on the Parameters FMB ED - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

2012-08-10T06:52:01Z

Redirected page to The Exponential Distribution

← Older revision		Revision as of 06:52, 10 August 2012
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	~~====Bounds on the Parameters====~~		#REDIRECT [[The Exponential Distribution]]

	~~For the failure rate <math>\hat{\lambda }</math> the upper (<math>{{\lambda }_{U}}</math>) and lower (<math>{{\lambda }_{L}}</math>) bounds are estimated by~~ [[~~Appendix: Weibull References\|[30]]]:~~


	~~::<math>\begin{align}~~
	~~& {{\lambda }_{U}}= & \hat{\lambda }\cdot {{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}} \\~~
	~~& & \\~~
	~~& {{\lambda }_{L}}= & \frac{\hat{\lambda }}{{{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}}}~~
	~~\end{align}</math>~~

	~~where <math>{{K}_{\alpha }}</math> is defined by:~~


	~~::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>~~


	~~If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds.~~
	~~The variance of <math>\hat{\lambda },</math> <math>Var(\hat{\lambda }),</math> is estimated from the Fisher matrix, as follows:~~


	~~::<math>Var(\hat{\lambda })={{\left( -\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \right)}^{-1}}</math>~~


	~~where <math>\Lambda </math> is the log-likelihood function of the exponential distribution.~~

	Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{t}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e., <math>Var(\hat{\gamma })=0.</math> (See the discussion in [[Appendix:_Log-Likelihood_Equations\|the appendix]] ~~for more information.)~~

Lisa Hacker: /* Bounds on the Parameters */

2012-08-03T08:53:01Z

Bounds on the Parameters

← Older revision		Revision as of 08:53, 3 August 2012
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	where <math>\Lambda </math> is the log-likelihood function of the exponential distribution~~, described in [[Appendix:_Log-Likelihood_Equations\|an appendix]]~~.		where <math>\Lambda </math> is the log-likelihood function of the exponential distribution.

	Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{t}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e. <math>Var(\hat{\gamma })=0.</math> (See the discussion in [[Appendix:_Log-Likelihood_Equations\|the appendix]] for more information.)		Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{t}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e., <math>Var(\hat{\gamma })=0.</math> (See the discussion in [[Appendix:_Log-Likelihood_Equations\|the appendix]] for more information.)

Lisa Hacker: /* Bounds on the Parameters */

2012-08-03T08:52:35Z

Bounds on the Parameters

← Older revision		Revision as of 08:52, 3 August 2012
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	where <math>\Lambda </math> is the log-likelihood function of the exponential distribution, described in [[Appendix: ~~Distribution Log~~-~~Likelihood Equations~~\|~~Appendix for log-likelihood function~~]].		where <math>\Lambda </math> is the log-likelihood function of the exponential distribution, described in [[Appendix:_Log-Likelihood_Equations\|an appendix]].

	Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{t}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e. <math>Var(\hat{\gamma })=0.</math> (See the discussion in [[Appendix: ~~Distribution Log~~-~~Likelihood Equations~~\|~~Appendix for log-likelihood function~~]] for more information.)		Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{t}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e. <math>Var(\hat{\gamma })=0.</math> (See the discussion in [[Appendix:_Log-Likelihood_Equations\|the appendix]] for more information.)

Harry Guo: /* Bounds on the Parameters */

2012-02-07T23:49:23Z

Bounds on the Parameters

← Older revision		Revision as of 23:49, 7 February 2012
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	====Bounds on the Parameters====		====Bounds on the Parameters====

	For the failure rate <math>\hat{\lambda }</math> the upper (<math>{{\lambda }_{U}}</math>) and lower (<math>{{\lambda }_{L}}</math>) bounds are estimated by [30]:		For the failure rate <math>\hat{\lambda }</math> the upper (<math>{{\lambda }_{U}}</math>) and lower (<math>{{\lambda }_{L}}</math>) bounds are estimated by [[Appendix: Weibull References\|[30]]]:


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	where <math>\Lambda </math> is the log-likelihood function of the exponential distribution, described in Appendix C.		where <math>\Lambda </math> is the log-likelihood function of the exponential distribution, described in [[Appendix: Distribution Log-Likelihood Equations\|Appendix for log-likelihood function]].

	Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{T}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e. <math>Var(\hat{\gamma })=0.</math> (See the discussion in Appendix C for more information.)		Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{t}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e. <math>Var(\hat{\gamma })=0.</math> (See the discussion in [[Appendix: Distribution Log-Likelihood Equations\|Appendix for log-likelihood function]] for more information.)

Nicolette Young: Created page with '====Bounds on the Parameters==== For the failure rate $\hat{\lambda }$ the upper ( ${{\lambda }_{U}}$ ) and lower ( ${{\lambda }_{L}}$ ) bounds are …'

2012-01-04T16:05:55Z

Created page with '====Bounds on the Parameters==== For the failure rate <math>\hat{\lambda }</math> the upper (<math>{{\lambda }_{U}}</math>) and lower (<math>{{\lambda }_{L}}</math>) bounds are …'

New page

====Bounds on the Parameters====

For the failure rate <math>\hat{\lambda }</math> the upper (<math>{{\lambda }_{U}}</math>) and lower (<math>{{\lambda }_{L}}</math>) bounds are estimated by [30]:

::<math>\begin{align}
& {{\lambda }_{U}}= & \hat{\lambda }\cdot {{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}} \\
& & \\
& {{\lambda }_{L}}= & \frac{\hat{\lambda }}{{{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}}}
\end{align}</math>

where <math>{{K}_{\alpha }}</math> is defined by:

::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>

If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds.
The variance of <math>\hat{\lambda },</math> <math>Var(\hat{\lambda }),</math> is estimated from the Fisher matrix, as follows:

::<math>Var(\hat{\lambda })={{\left( -\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \right)}^{-1}}</math>

where <math>\Lambda </math> is the log-likelihood function of the exponential distribution, described in Appendix C.

Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the <math>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{T}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e. <math>Var(\hat{\gamma })=0.</math> (See the discussion in Appendix C for more information.)

Template:Bounds on the Parameters FMB ED - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

Lisa Hacker: /* Bounds on the Parameters */

Lisa Hacker: /* Bounds on the Parameters */

Harry Guo: /* Bounds on the Parameters */

Nicolette Young: Created page with '====Bounds on the Parameters==== For the failure rate \hat{\lambda } the upper ({{\lambda }_{U}}) and lower ({{\lambda }_{L}}) bounds are …'

Nicolette Young: Created page with '====Bounds on the Parameters==== For the failure rate $\hat{\lambda }$ the upper ( ${{\lambda }_{U}}$ ) and lower ( ${{\lambda }_{L}}$ ) bounds are …'