The Exponential Distribution - Revision history

Lisa Hacker at 00:23, 9 March 2023

2023-03-09T00:23:36Z

M Spivey: /* Example: LR Bounds on Reliability */

2022-11-14T20:38:30Z

Example: LR Bounds on Reliability

← Older revision		Revision as of 20:38, 14 November 2022
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	'''Solution'''		'''Solution'''

	In this example, we are trying to determine the 85% two-sided confidence bounds on the reliability estimate of 50.881%. This is accomplished by substituting <math>t=50\,\!</math> and <math>\alpha =0.85\,\!</math> into the likelihood ratio bound equation. It now remains to find the values of <math>R\,\!</math> which satisfy this equation. Since there is only one parameter, there are only two values of <math>t\,\!</math> that will satisfy the equation. These values represent the <math>\delta =85%\,\!</math> two-sided confidence limits of the reliability estimate <math>\hat{R}\,\!</math>. For our problem, the confidence limits are:		In this example, we are trying to determine the 85% two-sided confidence bounds on the reliability estimate of 50.881%. This is accomplished by substituting <math>t=50\,\!</math> and <math>\alpha =0.85\,\!</math> into the likelihood ratio bound equation. It now remains to find the values of <math>R\,\!</math> which satisfy this equation. Since there is only one parameter, there are only two values of <math>R\,\!</math> that will satisfy the equation. These values represent the <math>\delta =85%\,\!</math> two-sided confidence limits of the reliability estimate <math>\hat{R}\,\!</math>. For our problem, the confidence limits are:

	::<math>{{\hat{R}}_{t=50}}=(29.861%,71.794%)\,\!</math>		::<math>{{\hat{R}}_{t=50}}=(29.861\%,71.794\%)\,\!</math>.

	===Bayesian Confidence Bounds===		===Bayesian Confidence Bounds===

M Spivey: /* Bounds on the Parameters */

2022-11-12T00:19:52Z

Bounds on the Parameters

← Older revision		Revision as of 00:19, 12 November 2022
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	where <math>\Lambda \,\!</math> is the log-likelihood function of the exponential distribution, described in [[Appendix:_Log-Likelihood_Equations\|Appendix D]].		where <math>\Lambda \,\!</math> is the log-likelihood function of the exponential distribution, described in [[Appendix:_Log-Likelihood_Equations\|Appendix D]].

	Note that, for fixed <math>\lambda \,\!</math>, the log-likelihood function is increasing in <math>\gamma</math>. This means that the MLE solution for <math>\gamma</math> cannot be found by setting <math>\tfrac{d \Lambda}{d \gamma}</math> to zero. (Since <math>\Lambda</math> is increasing in <math>\gamma</math>, <math>\tfrac{d \Lambda}{d \gamma}</math> can never be zero.) Instead, the MLE solution for <math>\gamma</math> is simply its largest possible value allowed by the sample; namely, <math>\gamma = t_1</math>, the first failure time. The MLE solution for <math>\lambda</math> is found in the usual fashion by setting <math>\tfrac{d \Lambda}{d \lambda}</math> to zero and solving. Weibull++ treats <math>\gamma</math> as a constant when computing bounds; i.e., <math>Var(\hat{\gamma}) = 0</math>. (See the discussion in Appendix D for more information.)		Note that, for fixed <math>\lambda \,\!</math>, the log-likelihood function is increasing in <math>\gamma</math>. This means that the MLE solution for <math>\gamma</math> cannot be found by setting <math>\tfrac{\partial \Lambda}{\partial \gamma}</math> to zero. (Since <math>\Lambda</math> is increasing in <math>\gamma</math>, <math>\tfrac{\partial \Lambda}{\partial \gamma}</math> can never be zero.) Instead, the MLE solution for <math>\gamma</math> is simply its largest possible value allowed by the sample; namely, <math>\gamma = t_1</math>, the first failure time. The MLE solution for <math>\lambda</math> is found in the usual fashion by setting <math>\tfrac{\partial \Lambda}{\partial \lambda}</math> to zero and solving. Weibull++ treats <math>\gamma</math> as a constant when computing bounds; i.e., <math>Var(\hat{\gamma}) = 0</math>. (See the discussion in Appendix D for more information.)

	====Bounds on Reliability====		====Bounds on Reliability====

M Spivey: /* Bounds on the Parameters */

2022-11-12T00:14:40Z

Bounds on the Parameters

← Older revision		Revision as of 00:14, 12 November 2022
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	where <math>\Lambda \,\!</math> is the log-likelihood function of the exponential distribution, described in [[Appendix:_Log-Likelihood_Equations\|Appendix D]].		where <math>\Lambda \,\!</math> is the log-likelihood function of the exponential distribution, described in [[Appendix:_Log-Likelihood_Equations\|Appendix D]].

	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:_Log-Likelihood_Equations\|~~Appendix D]] for more information.)		Note that, for fixed <math>\lambda \,\!</math>, the log-likelihood function is increasing in <math>\gamma</math>. This means that the MLE solution for <math>\gamma</math> cannot be found by setting <math>\tfrac{d \Lambda}{d \gamma}</math> to zero. (Since <math>\Lambda</math> is increasing in <math>\gamma</math>, <math>\tfrac{d \Lambda}{d \gamma}</math> can never be zero.) Instead, the MLE solution for <math>\gamma</math> is simply its largest possible value allowed by the sample; namely, <math>\gamma = t_1</math>, the first failure time. The MLE solution for <math>\lambda</math> is found in the usual fashion by setting <math>\tfrac{d \Lambda}{d \lambda}</math> to zero and solving. Weibull++ treats <math>\gamma</math> as a constant when computing bounds; i.e., <math>Var(\hat{\gamma}) = 0</math>. (See the discussion in Appendix D for more information.)

	====Bounds on Reliability====		====Bounds on Reliability====

Sharon Honecker: /* The 2-Parameter Exponential Distribution */

2017-07-24T20:04:52Z

The 2-Parameter Exponential Distribution

← Older revision		Revision as of 20:04, 24 July 2017
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	The 2-parameter exponential ''pdf'' is given by:		The 2-parameter exponential ''pdf'' is given by:

	::<math>f(t)=\lambda {{e}^{-\lambda (t-\gamma )}},f(t)\ge 0,\lambda >0,t\ge ~~0\text{ or }~~\gamma \,\!</math>		::<math>f(t)=\lambda {{e}^{-\lambda (t-\gamma )}},f(t)\ge 0,\lambda >0,t\ge \gamma \,\!</math>

	where <math>\gamma \,\!</math> is the location parameter.		where <math>\gamma \,\!</math> is the location parameter.

Nicolette Young: /* MLE Example */

2015-12-16T17:15:26Z

MLE Example

← Older revision		Revision as of 17:15, 16 December 2015
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	The probability plot is:		The probability plot is:

	[[Image:Exponential Distribution Example 4 Plot.png\|center\|~~700px~~\|]]		[[Image:Exponential Distribution Example 4 Plot.png\|center\|650px\|]]

	==Confidence Bounds==		==Confidence Bounds==

Nicolette Young: /* MLE Example */

2015-12-16T17:14:52Z

MLE Example

← Older revision		Revision as of 17:14, 16 December 2015
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	The probability plot is:		The probability plot is:

	[[Image:Exponential Distribution Example 4 Plot.png\|center\|~~600px~~\|]]		[[Image:Exponential Distribution Example 4 Plot.png\|center\|700px\|]]

	==Confidence Bounds==		==Confidence Bounds==

Nicolette Young: /* MLE Example */

2015-12-16T17:14:32Z

MLE Example

← Older revision		Revision as of 17:14, 16 December 2015
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	Using Weibull++:		Using Weibull++:

	[[Image:Exponential Distribution Example 4 Data.png\|center\|~~550px~~\|]]		[[Image:Exponential Distribution Example 4 Data.png\|center\|700px\|]]

	The probability plot is:		The probability plot is:

	[[Image:Exponential Distribution Example 4 Plot.png\|center\|~~550px~~\|]]		[[Image:Exponential Distribution Example 4 Plot.png\|center\|600px\|]]

	==Confidence Bounds==		==Confidence Bounds==

Nicolette Young: /* RRX Example */

2015-12-16T17:14:10Z

RRX Example

← Older revision		Revision as of 17:14, 16 December 2015
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	[[Image:Exponential Distribution Example 3 Plot.png\|center\|~~700px~~\|]]		[[Image:Exponential Distribution Example 3 Plot.png\|center\|600px\|]]

	===Maximum Likelihood Estimation===		===Maximum Likelihood Estimation===

Nicolette Young: /* RRX Example */

2015-12-16T17:13:49Z

RRX Example

← Older revision		Revision as of 17:13, 16 December 2015
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	[[Image:Exponential Distribution Example 3 Data Folio.png\|center\|~~550px~~\|]]		[[Image:Exponential Distribution Example 3 Data Folio.png\|center\|700px\|]]


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	[[Image:Exponential Distribution Example 3 Plot.png\|center\|~~550px~~\|]]		[[Image:Exponential Distribution Example 3 Plot.png\|center\|700px\|]]

	===Maximum Likelihood Estimation===		===Maximum Likelihood Estimation===