Template:Fisher Matrix Confidence Bounds ED - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

2012-08-10T06:55:54Z

Redirected page to The Exponential Distribution

← Older revision		Revision as of 06:55, 10 August 2012
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	~~===Fisher Matrix Bounds===~~		#REDIRECT [[The Exponential Distribution]]

	~~{{Bounds on the Parameters FMB ED}}~~


	~~{{Bounds on Reliability FMB ED}}~~

	~~{{Bounds on Time FMB ED}}~~

Nicolette Young: /* Bounds on Time */

2012-01-04T16:07:09Z

Bounds on Time

← Older revision		Revision as of 16:07, 4 January 2012
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	{{Bounds on Reliability FMB ED}}		{{Bounds on Reliability FMB ED}}

	~~====~~Bounds on Time~~====~~		{{Bounds on Time FMB ED}}

	~~The bounds around time for a given exponential percentile, or reliability value, are estimated by first solving the reliability equation with respect to time, or reliable life:~~


	~~::<math>\hat{T}=-\frac{1}{{\hat{\lambda }}}\cdot \ln (R)+\hat{\gamma }</math>~~


	~~The corresponding confidence bounds are estimated from:~~


	~~::<math>\begin{align}~~
	~~& {{T}_{U}}= & -\frac{1}{{{\lambda }_{L}}}\cdot \ln (R)+\hat{\gamma } \\~~
	~~& {{T}_{L}}= & -\frac{1~~}~~{{{\lambda }_{U}}}\cdot \ln (R)+\hat{\gamma }~~
	~~\end{align~~}~~</math>~~


	~~The same equations apply for the one-parameter exponential with <math>\gamma =0.</math>~~

Nicolette Young: /* Bounds on Reliability */

2012-01-04T16:06:38Z

Bounds on Reliability

← Older revision		Revision as of 16:06, 4 January 2012
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	~~====~~Bounds on Reliability~~====~~		{{Bounds on Reliability FMB ED}}

	~~The reliability of the two-parameter exponential distribution is:~~


	~~::<math>\hat{R}(T;\hat{\lambda })={{e}^{-\hat{\lambda }(T-\hat{\gamma })}}</math>~~


	~~The corresponding confidence bounds are estimated from:~~


	~~::<math>\begin{align}~~
	~~& {{R}_{L}}= & {{e}^{-{{\lambda }_{U}}(T-\hat{\gamma })~~}} \\
	~~& {{R}_{U}}= & {{e}^{-{{\lambda }_{L}}(T-\hat{\gamma })}}~~
	~~\end{align}</math>~~

	~~These equations hold true for the one-parameter exponential distribution, with <math>\gamma =0</math>.~~


	====Bounds on Time====		====Bounds on Time====

Nicolette Young: /* Fisher Matrix Bounds */

2012-01-04T16:05:49Z

Fisher Matrix Bounds

@@ Line 1: / Line 1: @@
 ===Fisher Matrix Bounds===
-====Bounds on the Parameters====
+{{Bounds on the Parameters FMB ED}}
 ====Bounds on Reliability====

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

2012-01-04T16:04:10Z

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

New page

===Fisher Matrix Bounds===

====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.)

====Bounds on Reliability====

The reliability of the two-parameter exponential distribution is:

::<math>\hat{R}(T;\hat{\lambda })={{e}^{-\hat{\lambda }(T-\hat{\gamma })}}</math>

The corresponding confidence bounds are estimated from:

::<math>\begin{align}
& {{R}_{L}}= & {{e}^{-{{\lambda }_{U}}(T-\hat{\gamma })}} \\
& {{R}_{U}}= & {{e}^{-{{\lambda }_{L}}(T-\hat{\gamma })}}
\end{align}</math>

These equations hold true for the one-parameter exponential distribution, with <math>\gamma =0</math>.

====Bounds on Time====

The bounds around time for a given exponential percentile, or reliability value, are estimated by first solving the reliability equation with respect to time, or reliable life:

::<math>\hat{T}=-\frac{1}{{\hat{\lambda }}}\cdot \ln (R)+\hat{\gamma }</math>

The corresponding confidence bounds are estimated from:

::<math>\begin{align}
& {{T}_{U}}= & -\frac{1}{{{\lambda }_{L}}}\cdot \ln (R)+\hat{\gamma } \\
& {{T}_{L}}= & -\frac{1}{{{\lambda }_{U}}}\cdot \ln (R)+\hat{\gamma }
\end{align}</math>

The same equations apply for the one-parameter exponential with <math>\gamma =0.</math>

Template:Fisher Matrix Confidence Bounds ED - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

Nicolette Young: /* Bounds on Time */

Nicolette Young: /* Bounds on Reliability */

Nicolette Young: /* Fisher Matrix Bounds */

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

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