Template:Lognormal distribution Likelihood ratio confidence bounds - Revision history

Richard House: Redirected page to The Lognormal Distribution#Likelihood Ratio Confidence Bounds

2012-08-13T06:06:38Z

Redirected page to The Lognormal Distribution#Likelihood Ratio Confidence Bounds

← Older revision		Revision as of 06:06, 13 August 2012
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	~~===Likelihood Ratio Confidence Bounds===~~		#REDIRECT [[The_Lognormal_Distribution#Likelihood_Ratio_Confidence_Bounds]]

	~~====Bounds on Parameters====~~
	~~As covered in Chapter~~ [[~~Parameter Estimation~~]]~~, the likelihood confidence bounds are calculated by finding values for <math>{{\theta }_{1}}</math> and <math>{{\theta }_{2}}</math> that satisfy:~~

	~~::<math>-2\cdot \text{ln}\left( \frac{L({{\theta }_{1}},{{\theta }_{2}})}{L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})} \right)=\chi _{\alpha ;1}^{2}</math>~~

	~~This equation can be rewritten as:~~

	~~::<math>L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}</math>~~

	~~For complete data, the likelihood formula for the normal distribution is given by:~~

	::<math>L({\mu }',{{\sigma' }})=\underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};{\mu }',{{\sigma' }})=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma' }}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-{\mu }'}{{{\sigma'}}} \right)}^{2}}}}</math>

	where the <math>{{x}_{i}}</math> values represent the original time-to-failure data. For a given value of <math>\alpha </math> , values for <math>{\mu }'</math> and <math>{{\sigma' }}</math> can be found which represent the maximum and minimum values that satisfy likelihood ratio equation. These represent the confidence bounds for the parameters at a confidence level <math>\delta ,</math> where <math>\alpha =\delta </math> for two-sided bounds and <math>\alpha =2\delta -1</math> for one-sided.


	~~'''Example 5:'''~~
	~~{{Example: Lognormal Distribution Likelihood Ratio Bound (Parameters)}}~~

	~~====Bounds on Time and Reliability====~~
	In order to calculate the bounds on a time estimate for a given reliability, or on a reliability estimate for a given time, the likelihood function needs to be rewritten in terms of one parameter and time/reliability, so that the maximum and minimum values of the time can be observed as the parameter is varied. This can be accomplished by substituting a form of the normal reliability equation into the likelihood function. The normal reliability equation can be written as:

	~~::<math>R=1-\Phi \left( \frac{\text{ln}(t)-{\mu }'}{{{\sigma'}}} \right)</math>~~

	~~This can be rearranged to the form:~~

	~~::<math>{\mu }'=\text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R)</math>~~

	where <math>{{\Phi }^{-1}}</math> is the inverse standard normal. This equation can now be substituted into likelihood function to produce a likelihood equation in terms of <math>{{\sigma'}},</math> <math>t</math> and <math>R</math>:

	::<math>L({{\sigma'}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma'}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma'}}} \right)}^{2}}}}</math>

	The unknown variable <math>t/R</math> depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then <math>R</math> is a known constant and <math>t</math> is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown variable. Either way, the above equation can be used to solve the likelihood ratio equation for the values of interest.


	~~'''Example 6:'''~~
	~~{{Example: Lognormal Distribution Likelihood Ratio Bound (Time)}}~~



	~~'''Example 7:'''~~
	~~{{Example: Lognormal Distribution Likelihood Ratio Bound (Reliability)}}~~

Harry Guo: /* Bounds on Time and Reliability */

2012-02-13T23:23:03Z

Bounds on Time and Reliability

← Older revision		Revision as of 23:23, 13 February 2012
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	The unknown variable <math>t/R</math> depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then <math>R</math> is a known constant and <math>t</math> is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown variable. Either way, the above equation can be used to solve the likelihood ratio equation for the values of interest.		The unknown variable <math>t/R</math> depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then <math>R</math> is a known constant and <math>t</math> is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown variable. Either way, the above equation can be used to solve the likelihood ratio equation for the values of interest.


	'''Example 6:'''		'''Example 6:'''
	{{Example: Lognormal Distribution Likelihood Ratio Bound (Time)}}		{{Example: Lognormal Distribution Likelihood Ratio Bound (Time)}}



	'''Example 7:'''		'''Example 7:'''
	{{Example: Lognormal Distribution Likelihood Ratio Bound (Reliability)}}		{{Example: Lognormal Distribution Likelihood Ratio Bound (Reliability)}}

Harry Guo: /* Bounds on Time and Reliability */

2012-02-13T23:22:33Z

Bounds on Time and Reliability

← Older revision		Revision as of 23:22, 13 February 2012
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	::<math>{\mu }'=\text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R)</math>		::<math>{\mu }'=\text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R)</math>

	where <math>{{\Phi }^{-1}}</math> is the inverse standard normal. This equation can now be substituted into likelihood function to produce a likelihood equation in terms of <math>{{\sigma'}},</math> <math>t</math> and <math>R~~\ \ :~~</math>		where <math>{{\Phi }^{-1}}</math> is the inverse standard normal. This equation can now be substituted into likelihood function to produce a likelihood equation in terms of <math>{{\sigma'}},</math> <math>t</math> and <math>R</math>:

	::<math>L({{\sigma'}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma'}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma'}}} \right)}^{2}}}}</math>		::<math>L({{\sigma'}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma'}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma'}}} \right)}^{2}}}}</math>

Harry Guo: /* Bounds on Time and Reliability */

2012-02-13T23:22:12Z

Bounds on Time and Reliability

← Older revision		Revision as of 23:22, 13 February 2012
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	::<math>L({{\sigma'}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma'}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma'}}} \right)}^{2}}}}</math>		::<math>L({{\sigma'}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma'}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma'}}} \right)}^{2}}}}</math>

	The unknown variable <math>t/R</math> depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then <math>R</math> is a known constant and <math>t</math> is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown variable. Either way, ~~Eqn. (lognormliketr)~~ can be used to solve the likelihood ratio equation for the values of interest.		The unknown variable <math>t/R</math> depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then <math>R</math> is a known constant and <math>t</math> is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown variable. Either way, the above equation can be used to solve the likelihood ratio equation for the values of interest.

	'''Example 6:'''		'''Example 6:'''

Harry Guo: /* Bounds on Time and Reliability */

2012-02-13T23:21:44Z

Bounds on Time and Reliability

← Older revision		Revision as of 23:21, 13 February 2012
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	In order to calculate the bounds on a time estimate for a given reliability, or on a reliability estimate for a given time, the likelihood function needs to be rewritten in terms of one parameter and time/reliability, so that the maximum and minimum values of the time can be observed as the parameter is varied. This can be accomplished by substituting a form of the normal reliability equation into the likelihood function. The normal reliability equation can be written as:		In order to calculate the bounds on a time estimate for a given reliability, or on a reliability estimate for a given time, the likelihood function needs to be rewritten in terms of one parameter and time/reliability, so that the maximum and minimum values of the time can be observed as the parameter is varied. This can be accomplished by substituting a form of the normal reliability equation into the likelihood function. The normal reliability equation can be written as:

	::<math>R=1-\Phi \left( \frac{\text{ln}(t)-{\mu }'}{{{\sigma ~~}_{{{T}~~'}}}} \right)</math>		::<math>R=1-\Phi \left( \frac{\text{ln}(t)-{\mu }'}{{{\sigma'}}} \right)</math>

	This can be rearranged to the form:		This can be rearranged to the form:

	::<math>{\mu }'=\text{ln}(t)-{{\sigma ~~}_{{{T}~~'}}}\cdot {{\Phi }^{-1}}(1-R)</math>		::<math>{\mu }'=\text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R)</math>

	where <math>{{\Phi }^{-1}}</math> is the inverse standard normal. This equation can now be substituted into ~~Eqn. (lognormlikelihood)~~ to produce a likelihood equation in terms of <math>{{\sigma ~~}_{{{T}~~'}}},</math> <math>t</math> and <math>R\ \ :</math>		where <math>{{\Phi }^{-1}}</math> is the inverse standard normal. This equation can now be substituted into likelihood function to produce a likelihood equation in terms of <math>{{\sigma'}},</math> <math>t</math> and <math>R\ \ :</math>

	::<math>L({{\sigma ~~}_{{{T}~~'}}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma ~~}_{{{T}~~'}}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma ~~}_{{{T}~~'}}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma ~~}_{{{T}~~'}}}} \right)}^{2}}}}</math>		::<math>L({{\sigma'}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma'}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma'}}} \right)}^{2}}}}</math>

	The unknown variable <math>t/R</math> depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then <math>R</math> is a known constant and <math>t</math> is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown variable. Either way, Eqn. (lognormliketr) can be used to solve ~~Eqn. (lratio3)~~ for the values of interest.		The unknown variable <math>t/R</math> depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then <math>R</math> is a known constant and <math>t</math> is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown variable. Either way, Eqn. (lognormliketr) can be used to solve the likelihood ratio equation for the values of interest.

	'''Example 6:'''		'''Example 6:'''

Harry Guo: /* Bounds on Parameters */

2012-02-13T23:13:47Z

Bounds on Parameters

← Older revision		Revision as of 23:13, 13 February 2012
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	====Bounds on Parameters====		====Bounds on Parameters====
	As covered in Chapter [[Parameter Estimation]], the likelihood confidence bounds are calculated by finding values for <math>{{\theta }_{1}}</math> and <math>{{\theta }_{2}}</math> that satisfy:		As covered in Chapter [[Parameter Estimation]], the likelihood confidence bounds are calculated by finding values for <math>{{\theta }_{1}}</math> and <math>{{\theta }_{2}}</math> that satisfy:


	::<math>-2\cdot \text{ln}\left( \frac{L({{\theta }_{1}},{{\theta }_{2}})}{L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})} \right)=\chi _{\alpha ;1}^{2}</math>		::<math>-2\cdot \text{ln}\left( \frac{L({{\theta }_{1}},{{\theta }_{2}})}{L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})} \right)=\chi _{\alpha ;1}^{2}</math>

Harry Guo: /* Likelihood Ratio Confidence Bounds */

2012-02-13T23:13:29Z

Likelihood Ratio Confidence Bounds

← Older revision		Revision as of 23:13, 13 February 2012
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	====Bounds on Parameters====		====Bounds on Parameters====
	As covered in Chapter 5, the likelihood confidence bounds are calculated by finding values for <math>{{\theta }_{1}}</math> and <math>{{\theta }_{2}}</math> that satisfy:		As covered in Chapter [[Parameter Estimation]], the likelihood confidence bounds are calculated by finding values for <math>{{\theta }_{1}}</math> and <math>{{\theta }_{2}}</math> that satisfy:


Line 8:		Line 8:

	This equation can be rewritten as:		This equation can be rewritten as:


	::<math>L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}</math>		::<math>L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}</math>
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	For complete data, the likelihood formula for the normal distribution is given by:		For complete data, the likelihood formula for the normal distribution is given by:

			::<math>L({\mu }',{{\sigma' }})=\underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};{\mu }',{{\sigma' }})=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma' }}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-{\mu }'}{{{\sigma'}}} \right)}^{2}}}}</math>

	::<math>L({\mu }',{{\sigma }_{{{T}'}}})=\underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};{\mu }',{{\sigma }_{{{T}'}}})=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma }_{{{T}'}}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		where the <math>{{x}_{i}}</math> values represent the original time-to-failure data. For a given value of <math>\alpha </math> , values for <math>{\mu }'</math> and <math>{{\sigma' }}</math> can be found which represent the maximum and minimum values that satisfy likelihood ratio equation. These represent the confidence bounds for the parameters at a confidence level <math>\delta ,</math> where <math>\alpha =\delta </math> for two-sided bounds and <math>\alpha =2\delta -1</math> for one-sided.

	where the <math>{{x}_{i}}</math> values represent the original time-to-failure data. For a given value of <math>\alpha </math> , values for <math>{\mu }'</math> and <math>{{\sigma ~~}_{{{T}~~'}}}</math> can be found which represent the maximum and minimum values that satisfy ~~Eqn. (lratio3)~~. These represent the confidence bounds for the parameters at a confidence level <math>\delta ,</math> where <math>\alpha =\delta </math> for two-sided bounds and <math>\alpha =2\delta -1</math> for one-sided.

Harry Guo at 18:52, 13 February 2012

2012-02-13T18:52:43Z

@@ Line 41: / Line 41: @@
 {{Example: Lognormal Distribution Likelihood Ratio Bound (Time)}}
+'''Example 7:'''
-<center><math>\begin{matrix}
+{{Example: Lognormal Distribution Likelihood Ratio Bound (Reliability)}}

Harry Guo at 18:50, 13 February 2012

2012-02-13T18:50:45Z

@@ Line 38: / Line 38: @@
 The unknown variable  <math>t/R</math>  depends on what type of bounds are being determined.  If one is trying to determine the bounds on time for a given reliability, then  <math>R</math>  is a known constant and  <math>t</math>  is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then  <math>t</math>  is a known constant and  <math>R</math>  is the unknown variable. Either way, Eqn. (lognormliketr) can be used to solve Eqn. (lratio3) for the values of interest.
-====Example 6====
+'''Example 6:'''
-For the data given in Example 5, determine the two-sided 75% confidence bounds on the time estimate for a reliability of 80%.  The ML estimate for the time at  <math>R(t)=80%</math>  is 55.718.
+{{Example: Lognormal Distribution Likelihood Ratio Bound (Time)}}
 ====Example 7====

Harry Guo at 18:48, 13 February 2012

2012-02-13T18:48:39Z

@@ Line 19: / Line 19: @@
 where the  <math>{{x}_{i}}</math>  values represent the original time-to-failure data.  For a given value of  <math>\alpha </math> , values for  <math>{\mu }'</math>  and  <math>{{\sigma }_{{{T}'}}}</math>  can be found which represent the maximum and minimum values that satisfy Eqn. (lratio3). These represent the confidence bounds for the parameters at a confidence level  <math>\delta ,</math>  where  <math>\alpha =\delta </math>  for two-sided bounds and  <math>\alpha =2\delta -1</math>  for one-sided.
-<center><math>\begin{align}
+'''Example 5:'''
-   L({{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_{{{T}'}}})= & \underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};{{\widehat{\mu }}^{\prime }},{{\widehat{\sigma }}_{{{T}'}}}), \\
+{{Example: Lognormal Distribution Likelihood Ratio Bound (Parameters)}}
 ====Bounds on Time and Reliability====