Fisher Matrix Confidence Bounds - Revision history

Steve Sharp: /* Confidence Bounds on Time (Type 1) */

2011-06-30T18:59:37Z

Confidence Bounds on Time (Type 1)

← Older revision		Revision as of 18:59, 30 June 2011
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	Eqn. (cl2) can be solved for <math>{{T}_{U}}(R)</math>, where:		Eqn. (cl2) can be solved for <math>{{T}_{U}}(R)</math>, where:
	#<math>CL</math> is confidence level,		::#<math>CL</math> is confidence level,
	#<math>\varphi (\beta )</math> is the prior <math>pdf</math> of the parameter <math>\beta </math>. For non-informative prior distribution, <math>\varphi (\beta )=\tfrac{1}{\beta }.</math>		::#<math>\varphi (\beta )</math> is the prior <math>pdf</math> of the parameter <math>\beta </math>. For non-informative prior distribution, <math>\varphi (\beta )=\tfrac{1}{\beta }.</math>
	#<math>\varphi (\eta )</math> is the prior <math>pdf</math> of the parameter <math>\eta .</math>. For non-informative prior distribution, <math>\varphi (\eta )=\tfrac{1}{\eta }.</math>		::#<math>\varphi (\eta )</math> is the prior <math>pdf</math> of the parameter <math>\eta .</math>. For non-informative prior distribution, <math>\varphi (\eta )=\tfrac{1}{\eta }.</math>
	#<math>L(\bullet )</math> is the likelihood function.		::#<math>L(\bullet )</math> is the likelihood function.
	The same method can be used to get the one-sided lower bound of <math>T(R)</math> from:		The same method can be used to get the one-sided lower bound of <math>T(R)</math> from:

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	Using the same method for the one-sided bounds, <math>{{T}_{U}}(R)</math> and <math>{{T}_{L}}(R)</math> can be solved.		Using the same method for the one-sided bounds, <math>{{T}_{U}}(R)</math> and <math>{{T}_{L}}(R)</math> can be solved.

	====Confidence Bounds on Reliability (Type 2)====		====Confidence Bounds on Reliability (Type 2)====
	For a given failure time distribution and a given time <math>T</math>, <math>R(T)</math> is a function of <math>T</math> and the distribution parameters. To illustrate the procedure for obtaining confidence bounds, the two-parameter Weibull distribution is used as an example. Bounds, for the case of other distributions, can be obtained in similar fashion. For example, for two parameter Weibull distribution:		For a given failure time distribution and a given time <math>T</math>, <math>R(T)</math> is a function of <math>T</math> and the distribution parameters. To illustrate the procedure for obtaining confidence bounds, the two-parameter Weibull distribution is used as an example. Bounds, for the case of other distributions, can be obtained in similar fashion. For example, for two parameter Weibull distribution:

Steve Sharp: /* Bayesian Confidence Bounds */

2011-06-30T18:59:08Z

Bayesian Confidence Bounds

← Older revision		Revision as of 18:59, 30 June 2011
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	where:		where:
	#<math>CL</math> is confidence level		::#<math>CL</math> is confidence level
	#<math>\theta </math> is the parameter vector		::#<math>\theta </math> is the parameter vector
	#<math>L(\bullet )</math> is the likelihood function		::#<math>L(\bullet )</math> is the likelihood function
	#<math>\varphi (\theta )</math> is the prior <math>pdf</math> of the parameter vector <math>\theta </math>		::#<math>\varphi (\theta )</math> is the prior <math>pdf</math> of the parameter vector <math>\theta </math>
	#<math>\varsigma </math> is the range of <math>\theta </math>		::#<math>\varsigma </math> is the range of <math>\theta </math>
	#<math>\xi </math> is the range in which <math>\theta </math> changes from <math>\Psi (T,R)</math> till <math>{\theta }'s</math> maximum value or from <math>{\theta }'s</math> minimum value till <math>\Psi (T,R)</math>		::#<math>\xi </math> is the range in which <math>\theta </math> changes from <math>\Psi (T,R)</math> till <math>{\theta }'s</math> maximum value or from <math>{\theta }'s</math> minimum value till <math>\Psi (T,R)</math>
	#<math>\Psi (T,R)</math> is function such that if <math>T</math> is given then the bounds are calculated for <math>R</math> and if <math>R</math> is given, then he bounds are calculated for <math>T</math>.		::#<math>\Psi (T,R)</math> is function such that if <math>T</math> is given then the bounds are calculated for <math>R</math> and if <math>R</math> is given, then he bounds are calculated for <math>T</math>.
	If <math>T</math> is given, then from Eqn. (BayesCL) and <math>\Psi </math> and for a given <math>CL,</math> the bounds on <math>R</math> are calculated.		If <math>T</math> is given, then from Eqn. (BayesCL) and <math>\Psi </math> and for a given <math>CL,</math> the bounds on <math>R</math> are calculated.
	If <math>R</math> is given, then from Eqn. (BayesCL) and <math>\Psi </math> and for a given <math>CL,</math> the bounds on <math>T</math> are calculated.		If <math>R</math> is given, then from Eqn. (BayesCL) and <math>\Psi </math> and for a given <math>CL,</math> the bounds on <math>T</math> are calculated.

Steve Sharp: /* Bayesian Confidence Bounds */

2011-06-30T18:58:30Z

Bayesian Confidence Bounds

← Older revision		Revision as of 18:58, 30 June 2011
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	where:		where:
	#<math>f(\theta \|Data)</math> is the <math>posterior</math> <math>pdf</math> of <math>\theta </math>		::#<math>f(\theta \|Data)</math> is the <math>posterior</math> <math>pdf</math> of <math>\theta </math>
	#<math>\theta </math> is the parameter vector of the chosen distribution (i.e. Weibull, lognormal, etc.)		::#<math>\theta </math> is the parameter vector of the chosen distribution (i.e. Weibull, lognormal, etc.)
	#<math>L(\bullet )</math> is the likelihood function		::#<math>L(\bullet )</math> is the likelihood function
	#<math>\varphi (\theta )</math> is the <math>prior</math> <math>pdf</math> of the parameter vector <math>\theta </math>		::#<math>\varphi (\theta )</math> is the <math>prior</math> <math>pdf</math> of the parameter vector <math>\theta </math>
	#<math>\varsigma </math> is the range of <math>\theta </math>.		::#<math>\varsigma </math> is the range of <math>\theta </math>.
	In other words, the prior knowledge is provided in the form of the prior <math>pdf</math> of the parameters, which in turn is combined with the sample data in order to obtain the posterior <math>pdf.</math> Different forms of prior information exist, such as past data, expert opinion or non-informative (refer to Chapter 3). It can be seen from Eqn. (BayesRule) that we are now dealing with distributions of parameters rather than single value parameters. For example, consider a one-parameter distribution with a positive parameter <math>{{\theta }_{1}}</math>. Given a set of sample data, and a prior distribution for <math>{{\theta }_{1}},</math> <math>\varphi ({{\theta }_{1}}),</math> Eqn. (BayesRule) can be written as:		In other words, the prior knowledge is provided in the form of the prior <math>pdf</math> of the parameters, which in turn is combined with the sample data in order to obtain the posterior <math>pdf.</math> Different forms of prior information exist, such as past data, expert opinion or non-informative (refer to Chapter 3). It can be seen from Eqn. (BayesRule) that we are now dealing with distributions of parameters rather than single value parameters. For example, consider a one-parameter distribution with a positive parameter <math>{{\theta }_{1}}</math>. Given a set of sample data, and a prior distribution for <math>{{\theta }_{1}},</math> <math>\varphi ({{\theta }_{1}}),</math> Eqn. (BayesRule) can be written as:

Steve Sharp: /* Introduction */

2011-06-30T18:57:43Z

Introduction

← Older revision		Revision as of 18:57, 30 June 2011
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	where:		where:
	#<math>L(\theta )</math> is the likelihood function for the unknown parameter vector <math>\theta </math>		::#<math>L(\theta )</math> is the likelihood function for the unknown parameter vector <math>\theta </math>
	#<math>L(\widehat{\theta })</math> is the likelihood function calculated at the estimated vector <math>\widehat{\theta }</math>		::#<math>L(\widehat{\theta })</math> is the likelihood function calculated at the estimated vector <math>\widehat{\theta }</math>
	#<math>\chi _{\alpha ;k}^{2}</math> is the chi-squared statistic with probability <math>\alpha </math> and <math>k</math> degrees of freedom, where <math>k</math> is the number of quantities jointly estimated		::#<math>\chi _{\alpha ;k}^{2}</math> is the chi-squared statistic with probability <math>\alpha </math> and <math>k</math> degrees of freedom, where <math>k</math> is the number of quantities jointly estimated
	If <math>\delta </math> is the confidence level, then <math>\alpha =\delta </math> for two-sided bounds and <math>\alpha =(2\delta -1)</math> for one-sided. Recall from Chapter 3 that if <math>x</math> is a continuous random variable with <math>pdf</math>:		If <math>\delta </math> is the confidence level, then <math>\alpha =\delta </math> for two-sided bounds and <math>\alpha =(2\delta -1)</math> for one-sided. Recall from Chapter 3 that if <math>x</math> is a continuous random variable with <math>pdf</math>:

Steve Sharp at 16:19, 29 June 2011

2011-06-29T16:19:37Z

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Steve Sharp: Created page with 'This section presents an overview of the theory on obtaining approximate confidence bounds on suspended (multiply censored) data. The methodology used is the so-called Fisher mat…'

2011-06-28T00:04:25Z

Created page with 'This section presents an overview of the theory on obtaining approximate confidence bounds on suspended (multiply censored) data. The methodology used is the so-called Fisher mat…'

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