Introduction to Confidence Bounds - Revision history

Lisa Hacker at 21:49, 18 September 2023

2023-09-18T21:49:53Z

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	<noinclude>{{Navigation box}}[[Category: Shared Articles]]		<noinclude>{{Navigation box}}[[Category: Shared Articles]]
	''This article also appears in the [~~[Confidence Bounds\|~~Life ~~Data Analysis Reference]~~] and [~~[Appendix D~~: ~~Confidence Bounds\|~~Accelerated ~~Life Testing Data Analysis Reference]~~] books.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference] and [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference] books.'' </noinclude>

	== What Are Confidence Bounds?==		== What Are Confidence Bounds?==

M Spivey: /* Approximate Estimates of the Mean and Variance of a Function */

2023-03-22T22:51:56Z

Approximate Estimates of the Mean and Variance of a Function

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	::<math>\begin{align}		::<math>\begin{align}
	Var( G( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}}))= &{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})^2}_{{\widehat{\theta_{1}}}={\theta_{1}}}Var(\widehat{\theta_{1}})+{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})^2}_{{\widehat{\theta_{2}}}={\theta_{2}}}Var(\widehat{\theta_{2}})\\		Var( G( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}}))= &{\left(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}}\right)^2}_{{\widehat{\theta_{1}}}={\theta_{1}}}Var(\widehat{\theta_{1}})+{\left(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}}\right)^2}_{{\widehat{\theta_{2}}}={\theta_{2}}}Var(\widehat{\theta_{2}})\\
	& +2{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})}_{{\widehat{\theta_{1}}}={\theta_{1}}}{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})}_{{\widehat{\theta_{2}}}={\theta_{2}}}Cov(\widehat{\theta_{1}},\widehat{\theta_{2}}) \\		& +2{\left(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}}\right)}_{{\widehat{\theta_{1}}}={\theta_{1}}}{\left(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}}\right)}_{{\widehat{\theta_{2}}}={\theta_{2}}}Cov(\widehat{\theta_{1}},\widehat{\theta_{2}}) \\
	& +O(\frac{1}{n^{\tfrac{3}{2}}})		& +O\left(\frac{1}{n^{\tfrac{3}{2}}}\right)
	\end{align}		\end{align}
	\,\!</math>		\,\!</math>

Sharon Honecker: /* Approximate Estimates of the Mean and Variance of a Function */ Removed squares from partial derivatives in covariance term of variance of function of two parameters.

2016-12-08T20:04:15Z

Approximate Estimates of the Mean and Variance of a Function: Removed squares from partial derivatives in covariance term of variance of function of two parameters.

← Older revision		Revision as of 20:04, 8 December 2016
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	::<math>\begin{align}		::<math>\begin{align}
	Var( G( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}}))= &{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})^2}_{{\widehat{\theta_{1}}}={\theta_{1}}}Var(\widehat{\theta_{1}})+{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})^2}_{{\widehat{\theta_{2}}}={\theta_{2}}}Var(\widehat{\theta_{2}})\\		Var( G( {{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}}))= &{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})^2}_{{\widehat{\theta_{1}}}={\theta_{1}}}Var(\widehat{\theta_{1}})+{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})^2}_{{\widehat{\theta_{2}}}={\theta_{2}}}Var(\widehat{\theta_{2}})\\
	& +2{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})^2}_{{\widehat{\theta_{1}}}={\theta_{1}}}{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})^2}_{{\widehat{\theta_{2}}}={\theta_{2}}}Cov(\widehat{\theta_{1}},\widehat{\theta_{2}}) \\		& +2{(\frac{\partial G}{\partial {{\widehat{\theta }}_{1}}})}_{{\widehat{\theta_{1}}}={\theta_{1}}}{(\frac{\partial G}{\partial {{\widehat{\theta }}_{2}}})}_{{\widehat{\theta_{2}}}={\theta_{2}}}Cov(\widehat{\theta_{1}},\widehat{\theta_{2}}) \\
	& +O(\frac{1}{n^{\tfrac{3}{2}}})		& +O(\frac{1}{n^{\tfrac{3}{2}}})
	\end{align}		\end{align}

Nicolette Young: /* Likelihood Ratio Confidence Bounds */

2015-12-21T17:52:52Z

Likelihood Ratio Confidence Bounds

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	The maximum likelihood estimators (MLE) of <math>{{\theta}_{1}},{{\theta}_{2}},...,{{\theta}_{k}}\,\!</math> are <math>k\,\!</math> are obtained by maximizing <math>L.\,\!</math> These are represented by the <math>L(\widehat{\theta })\,\!</math> term in the denominator of the ratio in the likelihood ratio equation. Since the values of the data points are known, and the values of the parameter estimates <math>\widehat{\theta }\,\!</math> have been calculated using MLE methods, the only unknown term in the likelihood ratio equation is the <math>L(\theta)\,\!</math> term in the numerator of the ratio. It remains to find the values of the unknown parameter vector <math>\theta\,\!</math> that satisfy the likelihood ratio equation. For distributions that have two parameters, the values of these two parameters can be varied in order to satisfy the likelihood ratio equation. The values of the parameters that satisfy this equation will change based on the desired confidence level <math>\delta;\,\!</math> but at a given value of <math>\delta\,\!</math> there is only a certain region of values for <math>{{\theta}_{1}}\,\!</math> and <math>{{\theta}_{2}}\,\!</math> for which the likelihood ratio equation holds true. This region can be represented graphically as a contour plot, an example of which is given in the following graphic.		The maximum likelihood estimators (MLE) of <math>{{\theta}_{1}},{{\theta}_{2}},...,{{\theta}_{k}}\,\!</math> are <math>k\,\!</math> are obtained by maximizing <math>L.\,\!</math> These are represented by the <math>L(\widehat{\theta })\,\!</math> term in the denominator of the ratio in the likelihood ratio equation. Since the values of the data points are known, and the values of the parameter estimates <math>\widehat{\theta }\,\!</math> have been calculated using MLE methods, the only unknown term in the likelihood ratio equation is the <math>L(\theta)\,\!</math> term in the numerator of the ratio. It remains to find the values of the unknown parameter vector <math>\theta\,\!</math> that satisfy the likelihood ratio equation. For distributions that have two parameters, the values of these two parameters can be varied in order to satisfy the likelihood ratio equation. The values of the parameters that satisfy this equation will change based on the desired confidence level <math>\delta;\,\!</math> but at a given value of <math>\delta\,\!</math> there is only a certain region of values for <math>{{\theta}_{1}}\,\!</math> and <math>{{\theta}_{2}}\,\!</math> for which the likelihood ratio equation holds true. This region can be represented graphically as a contour plot, an example of which is given in the following graphic.

	[[Image:Examplecontourplot.png\|center\|~~350px~~]]		[[Image:Examplecontourplot.png\|center\|450px]]

	The region of the contour plot essentially represents a cross-section of the likelihood function surface that satisfies the conditions of the likelihood ratio equation.		The region of the contour plot essentially represents a cross-section of the likelihood function surface that satisfies the conditions of the likelihood ratio equation.
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	Contour plots can be used for comparing data sets. Consider two data sets, one for an old product design and another for a new design. The engineer would like to determine if the two designs are significantly different and at what confidence. By plotting the contour plots of each data set in an overlay plot (the same distribution must be fitted to each data set), one can determine the confidence at which the two sets are significantly different. If, for example, there is no overlap (i.e., the two plots do not intersect) between the two 90% contours, then the two data sets are significantly different with a 90% confidence. If the two 95% contours overlap, then the two designs are NOT significantly different at the 95% confidence level. An example of non-intersecting contours is shown next. For details on comparing data sets, see [[Comparing Life Data Sets]].		Contour plots can be used for comparing data sets. Consider two data sets, one for an old product design and another for a new design. The engineer would like to determine if the two designs are significantly different and at what confidence. By plotting the contour plots of each data set in an overlay plot (the same distribution must be fitted to each data set), one can determine the confidence at which the two sets are significantly different. If, for example, there is no overlap (i.e., the two plots do not intersect) between the two 90% contours, then the two data sets are significantly different with a 90% confidence. If the two 95% contours overlap, then the two designs are NOT significantly different at the 95% confidence level. An example of non-intersecting contours is shown next. For details on comparing data sets, see [[Comparing Life Data Sets]].

	<br>[[Image:Contourplot.png\|center\|~~350px~~]]		<br>[[Image:Contourplot.png\|center\|450px]]

Nicolette Young: /* One-Sided and Two-Sided Confidence Bounds */

2015-12-21T17:52:12Z

One-Sided and Two-Sided Confidence Bounds

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	'''Two-Sided Bounds'''		'''Two-Sided Bounds'''

	[[Image:two sided bounds.png\|center\|~~250px~~]]		[[Image:two sided bounds.png\|center\|350px]]

	When we use two-sided confidence bounds (or intervals), we are looking at a closed interval where a certain percentage of the population is likely to lie. That is, we determine the values, or bounds, between which lies a specified percentage of the population. For example, when dealing with 90% two-sided confidence bounds of <math>(X,Y)\,\!</math>, we are saying that 90% of the population lies between <math>X\,\!</math> and <math>Y\,\!</math> with 5% less than <math>X\,\!</math> and 5% greater than <math>Y\,\!</math>.		When we use two-sided confidence bounds (or intervals), we are looking at a closed interval where a certain percentage of the population is likely to lie. That is, we determine the values, or bounds, between which lies a specified percentage of the population. For example, when dealing with 90% two-sided confidence bounds of <math>(X,Y)\,\!</math>, we are saying that 90% of the population lies between <math>X\,\!</math> and <math>Y\,\!</math> with 5% less than <math>X\,\!</math> and 5% greater than <math>Y\,\!</math>.
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	One-sided confidence bounds are essentially an open-ended version of two-sided bounds. A one-sided bound defines the point where a certain percentage of the population is either higher or lower than the defined point. This means that there are two types of one-sided bounds: upper and lower. An upper one-sided bound defines a point that a certain percentage of the population is less than. Conversely, a lower one-sided bound defines a point that a specified percentage of the population is greater than.		One-sided confidence bounds are essentially an open-ended version of two-sided bounds. A one-sided bound defines the point where a certain percentage of the population is either higher or lower than the defined point. This means that there are two types of one-sided bounds: upper and lower. An upper one-sided bound defines a point that a certain percentage of the population is less than. Conversely, a lower one-sided bound defines a point that a specified percentage of the population is greater than.

	[[Image:one sided bounds.png\|center\|~~250px~~]]		[[Image:one sided bounds.png\|center\|350px]]

	For example, if <math>X\,\!</math> is a 95% upper one-sided bound, this would imply that 95% of the population is less than <math>X\,\!</math>. If <math>X\,\!</math> is a 95% lower one-sided bound, this would indicate that 95% of the population is greater than <math>X\,\!</math>. Care must be taken to differentiate between one- and two-sided confidence bounds, as these bounds can take on identical values at different percentage levels. For example, in the figures above, we see bounds on a hypothetical distribution. Assuming that this is the same distribution in all of the figures, we see that <math>X\,\!</math> marks the spot below which 5% of the distribution's population lies. Similarly, <math>Y\,\!</math> represents the point above which 5% of the population lies. Therefore, <math>X\,\!</math> and <math>Y\,\!</math> represent the 90% two-sided bounds, since 90% of the population lies between the two points. However, <math>X\,\!</math> also represents the lower one-sided 95% confidence bound, since 95% of the population lies above that point; and <math>Y\,\!</math> represents the upper one-sided 95% confidence bound, since 95% of the population is below <math>Y\,\!</math>. It is important to be sure of the type of bounds you are dealing with, particularly as both one-sided bounds can be displayed simultaneously in Weibull++. In Weibull++, we use upper to represent the higher limit and lower to represent the lower limit, regardless of their position, but based on the value of the results. So if obtaining the confidence bounds on the reliability, we would identify the lower value of reliability as the lower limit and the higher value of reliability as the higher limit. If obtaining the confidence bounds on probability of failure we will again identify the lower numeric value for the probability of failure as the lower limit and the higher value as the higher limit.		For example, if <math>X\,\!</math> is a 95% upper one-sided bound, this would imply that 95% of the population is less than <math>X\,\!</math>. If <math>X\,\!</math> is a 95% lower one-sided bound, this would indicate that 95% of the population is greater than <math>X\,\!</math>. Care must be taken to differentiate between one- and two-sided confidence bounds, as these bounds can take on identical values at different percentage levels. For example, in the figures above, we see bounds on a hypothetical distribution. Assuming that this is the same distribution in all of the figures, we see that <math>X\,\!</math> marks the spot below which 5% of the distribution's population lies. Similarly, <math>Y\,\!</math> represents the point above which 5% of the population lies. Therefore, <math>X\,\!</math> and <math>Y\,\!</math> represent the 90% two-sided bounds, since 90% of the population lies between the two points. However, <math>X\,\!</math> also represents the lower one-sided 95% confidence bound, since 95% of the population lies above that point; and <math>Y\,\!</math> represents the upper one-sided 95% confidence bound, since 95% of the population is below <math>Y\,\!</math>. It is important to be sure of the type of bounds you are dealing with, particularly as both one-sided bounds can be displayed simultaneously in Weibull++. In Weibull++, we use upper to represent the higher limit and lower to represent the lower limit, regardless of their position, but based on the value of the results. So if obtaining the confidence bounds on the reliability, we would identify the lower value of reliability as the lower limit and the higher value of reliability as the higher limit. If obtaining the confidence bounds on probability of failure we will again identify the lower numeric value for the probability of failure as the lower limit and the higher value as the higher limit.

Richard House at 18:28, 24 September 2012

2012-09-24T18:28:19Z

Show changes

Richard House: /* Beta Binomial Confidence Bounds */

2012-09-04T20:50:46Z

Beta Binomial Confidence Bounds

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	Keep in mind that one must be careful to select the appropriate values for <math>P\,\!</math> based on the type of confidence bounds desired. For example, if two-sided 80% confidence bounds are to be calculated, one must solve the equation twice (once with <math>P = 0.1\,\!</math> and once with <math>P = 0.9\,\!</math>) in order to place the bounds around 80% of the population. <br><br>Using this methodology, the appropriate ranks are obtained and plotted based on the desired confidence level. These points are then joined by a smooth curve to obtain the corresponding confidence bound.		Keep in mind that one must be careful to select the appropriate values for <math>P\,\!</math> based on the type of confidence bounds desired. For example, if two-sided 80% confidence bounds are to be calculated, one must solve the equation twice (once with <math>P = 0.1\,\!</math> and once with <math>P = 0.9\,\!</math>) in order to place the bounds around 80% of the population. <br><br>Using this methodology, the appropriate ranks are obtained and plotted based on the desired confidence level. These points are then joined by a smooth curve to obtain the corresponding confidence bound.

	In Weibull++, this non-parametric methodology is used only when plotting bounds on the mixed Weibull distribution. Full details on this methodology can be found in [[Appendix:_Life_Data_Analysis_References\|~~Kececioglu~~ [20]]]. These binomial equations can again be transformed using the beta and F distributions, thus the name beta binomial confidence bounds.		In Weibull++, this non-parametric methodology is used only when plotting bounds on the mixed Weibull distribution. Full details on this methodology can be found in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[20]]]. These binomial equations can again be transformed using the beta and F distributions, thus the name beta binomial confidence bounds.

	== Likelihood Ratio Confidence Bounds ==		== Likelihood Ratio Confidence Bounds ==

Richard House at 20:50, 4 September 2012

2012-09-04T20:50:15Z

← Older revision		Revision as of 20:50, 4 September 2012
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	Keep in mind that one must be careful to select the appropriate values for <math>P\,\!</math> based on the type of confidence bounds desired. For example, if two-sided 80% confidence bounds are to be calculated, one must solve the equation twice (once with <math>P = 0.1\,\!</math> and once with <math>P = 0.9\,\!</math>) in order to place the bounds around 80% of the population. <br><br>Using this methodology, the appropriate ranks are obtained and plotted based on the desired confidence level. These points are then joined by a smooth curve to obtain the corresponding confidence bound.		Keep in mind that one must be careful to select the appropriate values for <math>P\,\!</math> based on the type of confidence bounds desired. For example, if two-sided 80% confidence bounds are to be calculated, one must solve the equation twice (once with <math>P = 0.1\,\!</math> and once with <math>P = 0.9\,\!</math>) in order to place the bounds around 80% of the population. <br><br>Using this methodology, the appropriate ranks are obtained and plotted based on the desired confidence level. These points are then joined by a smooth curve to obtain the corresponding confidence bound.

	In Weibull++, this non-parametric methodology is used only when plotting bounds on the mixed Weibull distribution. Full details on this methodology can be found in [[Appendix D: ~~Weibull References~~\|Kececioglu [20]]]. These binomial equations can again be transformed using the beta and F distributions, thus the name beta binomial confidence bounds.		In Weibull++, this non-parametric methodology is used only when plotting bounds on the mixed Weibull distribution. Full details on this methodology can be found in [[Appendix:_Life_Data_Analysis_References\|Kececioglu [20]]]. These binomial equations can again be transformed using the beta and F distributions, thus the name beta binomial confidence bounds.

	== Likelihood Ratio Confidence Bounds ==		== Likelihood Ratio Confidence Bounds ==

Richard House: /* Approximate Confidence Intervals on the Parameters */

2012-09-04T20:28:03Z

Approximate Confidence Intervals on the Parameters

← Older revision		Revision as of 20:28, 4 September 2012
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	\end{align}</math>		\end{align}</math>

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

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

Richard House: /* Fisher Matrix Confidence Bounds */

2012-09-04T20:27:33Z

Fisher Matrix Confidence Bounds

← Older revision		Revision as of 20:27, 4 September 2012
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	== Fisher Matrix Confidence Bounds ==		== Fisher Matrix Confidence Bounds ==
	This section presents an overview of the theory on obtaining approximate confidence bounds on suspended (multiple censored) data. The methodology used is the so-called Fisher matrix bounds (FM), described in Nelson [[~~Appendix_D~~:~~_Weibull_References~~\|[30]]] and Lloyd and Lipow [[~~Appendix_D~~:~~_Weibull_References~~\|[24]]]. These bounds are employed in most other commercial statistical applications. In general, these bounds tend to be more optimistic than the non-parametric rank based bounds. This may be a concern, particularly when dealing with small sample sizes. Some statisticians feel that the Fisher matrix bounds are too optimistic when dealing with small sample sizes and prefer to use other techniques for calculating confidence bounds, such as the likelihood ratio bounds. <br>		This section presents an overview of the theory on obtaining approximate confidence bounds on suspended (multiple censored) data. The methodology used is the so-called Fisher matrix bounds (FM), described in Nelson [[Appendix:_Life_Data_Analysis_References\|[30]]] and Lloyd and Lipow [[Appendix:_Life_Data_Analysis_References\|[24]]]. These bounds are employed in most other commercial statistical applications. In general, these bounds tend to be more optimistic than the non-parametric rank based bounds. This may be a concern, particularly when dealing with small sample sizes. Some statisticians feel that the Fisher matrix bounds are too optimistic when dealing with small sample sizes and prefer to use other techniques for calculating confidence bounds, such as the likelihood ratio bounds. <br>

	=== Approximate Estimates of the Mean and Variance of a Function ===		=== Approximate Estimates of the Mean and Variance of a Function ===