Template:Fisher Matrix Confidence Bounds - Revision history

Richard House: Redirected page to Confidence Bounds#Fisher Matrix Confidence Bounds

2012-08-12T23:32:19Z

Redirected page to Confidence Bounds#Fisher Matrix Confidence Bounds

Lisa Hacker: /* Approximate Estimates of the Mean and Variance of a Function */

2012-03-30T22:43:56Z

Approximate Estimates of the Mean and Variance of a Function

← Older revision		Revision as of 22:43, 30 March 2012
Line 49:		Line 49:
	\end{align}</math>		\end{align}</math>

	In the equation above, the first summation is for complete data, the second summation is for right censored data, and the third summation is for interval or left censored data~~. For more information on these data types, see [[Life Data Classification]]~~. <br>		In the equation above, the first summation is for ''complete data'', the second summation is for ''right censored data'' and the third summation is for ''interval or left censored data''. <br>

	Then the Fisher information matrix is given by:		Then the Fisher information matrix is given by:

Lisa Hacker: /* Approximate Estimates of the Mean and Variance of a Function */

2012-03-30T22:06:10Z

Approximate Estimates of the Mean and Variance of a Function

← Older revision		Revision as of 22:06, 30 March 2012
Line 49:		Line 49:
	\end{align}</math>		\end{align}</math>

	In the equation above, the first summation is for complete data, the second summation is for right censored data, and the third summation is for interval or left censored data. For more information on these data types, see ~~Chapter~~ [[Life Data Classification]]. <br>Then the Fisher information matrix is given by:		In the equation above, the first summation is for complete data, the second summation is for right censored data, and the third summation is for interval or left censored data. For more information on these data types, see [[Life Data Classification]]. <br>

			Then the Fisher information matrix is given by:

	::<math>{{F}_{0}}=\left[ \begin{matrix}		::<math>{{F}_{0}}=\left[ \begin{matrix}

Kate Racaza at 18:01, 8 March 2012

2012-03-08T18:01:02Z

Show changes

Harry Guo: /* Confidence Bounds on Reliability (Type 2) */

2012-02-13T15:44:20Z

Confidence Bounds on Reliability (Type 2)

← Older revision		Revision as of 15:44, 13 February 2012
Line 161:		Line 161:
	::<math>\widehat{R}(T)={{e}^{-\widehat{\lambda }\cdot T}}</math>		::<math>\widehat{R}(T)={{e}^{-\widehat{\lambda }\cdot T}}</math>

	<br> Reliability bounds (Type 2) return the confidence bounds by determining the confidence intervals around <math>\widehat{\lambda }</math> and substituting these values into ~~Eqn. (cbr)~~. The bounds on <math>\widehat{\lambda }</math> were determined using ~~Eqns. (cblmu) and (cblml)~~, with its variance obtained from ~~Eqn. (Fisher2)~~. Once again, the procedure is more complicated for distributions with more than one parameter.		<br> Reliability bounds (Type 2) return the confidence bounds by determining the confidence intervals around <math>\widehat{\lambda }</math> and substituting these values into the above equation. The bounds on <math>\widehat{\lambda }</math> were determined using the method for the bounds on parameters, with its variance obtained from the Fisher Matrix. Once again, the procedure is more complicated for distributions with more than one parameter.

Harry Guo: /* Confidence Bounds on Time (Type 1) */

2012-02-13T15:43:07Z

Confidence Bounds on Time (Type 1)

← Older revision		Revision as of 15:43, 13 February 2012
Line 151:		Line 151:
	\end{align}</math>		\end{align}</math>

	Bounds on time (Type 1) return the confidence bounds around this time value by determining the confidence intervals around <math>\widehat{\lambda }</math> and substituting these values into ~~Eqn. (cb)~~. The bounds on <math>\widehat{\lambda }</math> were determined using ~~Eqns. (cblmu) and (cblml)~~, with its variance obtained from ~~Eqn. (Fisher2)~~. Note that the procedure is slightly more complicated for distributions with more than one parameter.		Bounds on time (Type 1) return the confidence bounds around this time value by determining the confidence intervals around <math>\widehat{\lambda }</math> and substituting these values into the above equation. The bounds on <math>\widehat{\lambda }</math> were determined using the method for the bounds on parameters, with its variance obtained from the Fisher Matrix. Note that the procedure is slightly more complicated for distributions with more than one parameter.

	<br>		<br>

Harry Guo: /* Approximate Confidence Intervals on the Parameters */

2012-02-11T00:07:07Z

Approximate Confidence Intervals on the Parameters

← Older revision		Revision as of 00:07, 11 February 2012
Line 141:		Line 141:
	\end{align}</math>		\end{align}</math>

	<br> The same procedure can be extended for the case of a two or more parameter distribution. Lloyd and Lipow [24] further elaborate on this procedure.		<br> The same procedure can be extended for the case of a two or more parameter distribution. Lloyd and Lipow [[Appendix: Weibull References\|[24]]] further elaborate on this procedure.

	=== Confidence Bounds on Time (Type 1) ===		=== Confidence Bounds on Time (Type 1) ===

Harry Guo: /* Approximate Confidence Intervals on the Parameters */

2012-02-11T00:05:44Z

Approximate Confidence Intervals on the Parameters

← Older revision		Revision as of 00:05, 11 February 2012
Line 103:		Line 103:
	::<math>P\left( {{C}_{1}}<\theta <{{C}_{2}} \right)=\delta </math>		::<math>P\left( {{C}_{1}}<\theta <{{C}_{2}} \right)=\delta </math>

	<br> From ~~Eqn. (e729)~~:		<br> From the above equation:

	<br>		<br>
Line 115:		Line 115:
	::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi \left( {{K}_{\alpha }} \right)</math>		::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi \left( {{K}_{\alpha }} \right)</math>

	<br> Now by simplifying ~~Eqn. (e731)~~, one can obtain the approximate two-sided confidence bounds on the parameter <math>\theta ,</math> at a confidence level <math>\delta ,</math> or:		<br> Now by simplifying the equation for the confidence level, one can obtain the approximate two-sided confidence bounds on the parameter <math>\theta ,</math> at a confidence level <math>\delta ,</math> or:

	::<math>\left( \widehat{\theta }-{{K}_{\tfrac{1-\delta }{2}}}\cdot \sqrt{Var\left( \widehat{\theta } \right)}<\theta <\widehat{\theta }+{{K}_{\tfrac{1-\delta }{2}}}\cdot \sqrt{Var\left( \widehat{\theta } \right)} \right)</math>		::<math>\left( \widehat{\theta }-{{K}_{\tfrac{1-\delta }{2}}}\cdot \sqrt{Var\left( \widehat{\theta } \right)}<\theta <\widehat{\theta }+{{K}_{\tfrac{1-\delta }{2}}}\cdot \sqrt{Var\left( \widehat{\theta } \right)} \right)</math>

Harry Guo: /* Approximate Estimates of the Mean and Variance of a Function */

2012-02-11T00:04:31Z

Approximate Estimates of the Mean and Variance of a Function

← Older revision		Revision as of 00:04, 11 February 2012
Line 37:		Line 37:
	</math>		</math>

	<br> Note that the derivatives of ~~Eqn. (var)~~ are evaluated at <math>{{\widehat{\theta }}_{1}}={{\theta }_{1}}</math> and <math>{{\widehat{\theta }}_{2}}={{\theta }_{2}},</math> where E <math>\left( {{\widehat{\theta }}_{1}} \right)\simeq {{\theta }_{1}}</math> and E <math>\left( {{\widehat{\theta }}_{2}} \right)\simeq {{\theta }_{2}}.</math> <br> <br>		<br> Note that the derivatives of the above equation are evaluated at <math>{{\widehat{\theta }}_{1}}={{\theta }_{1}}</math> and <math>{{\widehat{\theta }}_{2}}={{\theta }_{2}},</math> where E <math>\left( {{\widehat{\theta }}_{1}} \right)\simeq {{\theta }_{1}}</math> and E <math>\left( {{\widehat{\theta }}_{2}} \right)\simeq {{\theta }_{2}}.</math> <br> <br>

	'''Parameter Variance and Covariance Determination'''		'''Parameter Variance and Covariance Determination'''
Line 79:		Line 79:
	\end{matrix} \right]}^{-1}}</math>		\end{matrix} \right]}^{-1}}</math>

	<br> Then the variance of a function (<math>Var(G)</math>) can be estimated using ~~Eqn. (var)~~. Values for the variance and covariance of the parameters are obtained from ~~Eqn. (Fisher2)~~. Once they have been obtained, the approximate confidence bounds on the function are given as:		<br> Then the variance of a function (<math>Var(G)</math>) can be estimated using equation for the variance. Values for the variance and covariance of the parameters are obtained from Fisher Matrix equation. Once they have been obtained, the approximate confidence bounds on the function are given as:

	::<math>C{{B}_{R}}=E(G)\pm {{z}_{\alpha }}\sqrt{Var(G)}</math>		::<math>C{{B}_{R}}=E(G)\pm {{z}_{\alpha }}\sqrt{Var(G)}</math>

Harry Guo: /* Approximate Estimates of the Mean and Variance of a Function */

2012-02-10T23:42:19Z

Approximate Estimates of the Mean and Variance of a Function

← Older revision		Revision as of 23:42, 10 February 2012
Line 49:		Line 49:
	\end{align}</math>		\end{align}</math>

	In the equation above, the first summation is for complete data, the second summation is for right censored data, and the third summation is for interval or left censored data. For more information on these data types, see [[Life Data Classification ~~\| Chapter 5~~]]. <br> Then the Fisher information matrix is given by:		In the equation above, the first summation is for complete data, the second summation is for right censored data, and the third summation is for interval or left censored data. For more information on these data types, see Chapter [[Life Data Classification]]. <br> Then the Fisher information matrix is given by:

	::<math>{{F}_{0}}=\left[ \begin{matrix}		::<math>{{F}_{0}}=\left[ \begin{matrix}