Parameter Estimation - Revision history

Lisa Hacker: Reverted edits by Lisa Hacker (talk) to last revision by M Spivey

2023-03-10T01:14:20Z

Reverted edits by Lisa Hacker (talk) to last revision by M Spivey

← Older revision		Revision as of 01:14, 10 March 2023
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	This same methodology can be applied to other distributions with ''cdf'' equations that can be linearized. Different probability papers exist for each distribution, because different distributions have different ''cdf'' equations. ReliaSoft's software tools automatically create these plots for you. Special scales on these plots allow you to derive the parameter estimates directly from the plots, similar to the way <math>\beta\,\!</math> and <math>\eta\,\!</math> were obtained from the Weibull probability plot. The following example demonstrates the method again, this time using the 1-parameter exponential distribution.		This same methodology can be applied to other distributions with ''cdf'' equations that can be linearized. Different probability papers exist for each distribution, because different distributions have different ''cdf'' equations. ReliaSoft's software tools automatically create these plots for you. Special scales on these plots allow you to derive the parameter estimates directly from the plots, similar to the way <math>\beta\,\!</math> and <math>\eta\,\!</math> were obtained from the Weibull probability plot. The following example demonstrates the method again, this time using the 1-parameter exponential distribution.

			{{:Probability Plotting Example}}

	== Comments on the Probability Plotting Method ==		== Comments on the Probability Plotting Method ==

Lisa Hacker at 01:12, 10 March 2023

2023-03-10T01:12:39Z

← Older revision		Revision as of 01:12, 10 March 2023
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	This same methodology can be applied to other distributions with ''cdf'' equations that can be linearized. Different probability papers exist for each distribution, because different distributions have different ''cdf'' equations. ReliaSoft's software tools automatically create these plots for you. Special scales on these plots allow you to derive the parameter estimates directly from the plots, similar to the way <math>\beta\,\!</math> and <math>\eta\,\!</math> were obtained from the Weibull probability plot. The following example demonstrates the method again, this time using the 1-parameter exponential distribution.		This same methodology can be applied to other distributions with ''cdf'' equations that can be linearized. Different probability papers exist for each distribution, because different distributions have different ''cdf'' equations. ReliaSoft's software tools automatically create these plots for you. Special scales on these plots allow you to derive the parameter estimates directly from the plots, similar to the way <math>\beta\,\!</math> and <math>\eta\,\!</math> were obtained from the Weibull probability plot. The following example demonstrates the method again, this time using the 1-parameter exponential distribution.

	~~{{:Probability Plotting Example}}~~

	== Comments on the Probability Plotting Method ==		== Comments on the Probability Plotting Method ==

M Spivey: /* Linearizing the Unreliability Function */

2022-12-17T00:07:10Z

Linearizing the Unreliability Function

← Older revision		Revision as of 00:07, 17 December 2022
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	and an intercept of:		and an intercept of:

	::<math>b=-\beta \cdot ln(\eta)\,\!</math>		::<math>b=-\beta \cdot \ln(\eta)\,\!</math>

	==Constructing the Paper==		==Constructing the Paper==

Sharon Honecker: /* Linearizing the Unreliability Function */ fixed missing ln in linearized equation

2016-11-03T23:05:04Z

Linearizing the Unreliability Function: fixed missing ln in linearized equation

← Older revision		Revision as of 23:05, 3 November 2016
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	\ln (1-Q(t))=& -\left(\tfrac{t}{\eta}\right)^{\beta} \\		\ln (1-Q(t))=& -\left(\tfrac{t}{\eta}\right)^{\beta} \\
	\ln ( -\ln (1-Q(t)))= & \beta \left(\ln \left( \frac{t}{\eta }\right)\right) \\		\ln ( -\ln (1-Q(t)))= & \beta \left(\ln \left( \frac{t}{\eta }\right)\right) \\
	\ln \left( \ln \left( \frac{1}{1-Q(t)}\right) \right) = & \beta\ln{ t} -\beta(\eta ) \\		\ln \left( \ln \left( \frac{1}{1-Q(t)}\right) \right) = & \beta\ln{ t} -\beta\ln{\eta} \\
	\end{align}\,\!</math>		\end{align}\,\!</math>

Nicolette Young at 21:44, 21 December 2015

2015-12-21T21:44:32Z

← Older revision		Revision as of 21:44, 21 December 2015
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	The range of <math>\hat \rho \,\!</math> is <math>-1\le \hat{\rho }\le 1\,\!</math>.		The range of <math>\hat \rho \,\!</math> is <math>-1\le \hat{\rho }\le 1\,\!</math>.

	[[Image:correlationcoeffficient.png\|center\|~~500px~~]]		[[Image:correlationcoeffficient.png\|center\|600px]]

	The closer the value is to <math>\pm 1\,\!</math>, the better the linear fit. Note that +1 indicates a perfect fit (the paired values (<math>{{x}_{i}},{{y}_{i}}\,\!</math>) lie on a straight line) with a positive slope, while -1 indicates a perfect fit with a negative slope. A correlation coefficient value of zero would indicate that the data are randomly scattered and have no pattern or correlation in relation to the regression line model.		The closer the value is to <math>\pm 1\,\!</math>, the better the linear fit. Note that +1 indicates a perfect fit (the paired values (<math>{{x}_{i}},{{y}_{i}}\,\!</math>) lie on a straight line) with a positive slope, while -1 indicates a perfect fit with a negative slope. A correlation coefficient value of zero would indicate that the data are randomly scattered and have no pattern or correlation in relation to the regression line model.

Nicolette Young: /* Constructing the Paper */

2015-12-21T20:28:18Z

Constructing the Paper

← Older revision		Revision as of 20:28, 21 December 2015
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	Such papers have been created by different vendors and are called ''probability plotting papers''. ReliaSoft's reliability engineering resource website at www.weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].		Such papers have been created by different vendors and are called ''probability plotting papers''. ReliaSoft's reliability engineering resource website at www.weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].

	[[Image:WeibullPaper2C.png\|center\|~~500px~~]]		[[Image:WeibullPaper2C.png\|center\|450px]]

	To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.		To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.

	[[Image:different_weibull_paper.png\|center\|~~500px~~]]		[[Image:different_weibull_paper.png\|center\|450px]]

	This paper is constructed based on the mentioned y and x transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.		This paper is constructed based on the mentioned y and x transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.

Nicolette Young: /* Constructing the Paper */

2015-12-21T20:26:43Z

Constructing the Paper

← Older revision		Revision as of 20:26, 21 December 2015
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	Such papers have been created by different vendors and are called ''probability plotting papers''. ReliaSoft's reliability engineering resource website at www.weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].		Such papers have been created by different vendors and are called ''probability plotting papers''. ReliaSoft's reliability engineering resource website at www.weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].

	[[Image:WeibullPaper2C.png\|center\|~~400px~~]]		[[Image:WeibullPaper2C.png\|center\|500px]]

	To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.		To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.

	[[Image:different_weibull_paper.png\|center\|~~400px~~]]		[[Image:different_weibull_paper.png\|center\|500px]]

	This paper is constructed based on the mentioned y and x transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.		This paper is constructed based on the mentioned y and x transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.

Kate Racaza: /* Shortfalls of the Rank Adjustment Method */

2015-08-20T22:26:36Z

Shortfalls of the Rank Adjustment Method

← Older revision		Revision as of 22:26, 20 August 2015
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	::<math>\begin{array}{*{35}{l}}		::<math>\begin{array}{*{35}{l}}
	\widehat{\beta }= & \text{0}\text{.81} \\		\widehat{\beta }= & \text{0}\text{.81} \\
	\widehat{\eta }= & \text{11,~~417~~ hr} \\		\widehat{\eta }= & \text{11,400 hr} \\
	\end{array}		\end{array}
	\,\!</math>		\,\!</math>
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	::<math>\begin{array}{*{35}{l}}		::<math>\begin{array}{*{35}{l}}
	\widehat{\beta }= & \text{1~~}\text{~~.33} \\		\widehat{\beta }= & \text{1.33} \\
	\widehat{\eta }= & \text{6,~~900~~ hr} \\		\widehat{\eta }= & \text{6,920 hr} \\
	\end{array}\,\!</math>		\end{array}\,\!</math>

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	::<math>\begin{array}{*{35}{l}}		::<math>\begin{array}{*{35}{l}}
	\widehat{\beta }= & \text{0}\text{.~~9337~~} \\		\widehat{\beta }= & \text{0}\text{.93} \\
	\widehat{\eta }= & \text{21,~~348~~ hr} \\		\widehat{\eta }= & \text{21,300 hr} \\
	\end{array}\,\!</math>		\end{array}\,\!</math>

Harry Guo: /* Shortfalls of the Rank Adjustment Method */

2015-02-25T20:56:40Z

Shortfalls of the Rank Adjustment Method

← Older revision		Revision as of 20:56, 25 February 2015
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	As we can see, there is a sizable difference in the results of the two sets calculated using MLE and the results using regression with the SRM. The results for both cases are identical when using the regression estimation technique with SRM, as SRM considers only the positions of the suspensions. The MLE results are quite different for the two cases, with the second case having a much larger value of <math>\eta \,\!</math>, which is due to the higher values of the suspension times in Case 2. This is because the maximum likelihood technique, unlike rank regression with SRM, considers the values of the suspensions when estimating the parameters. This is illustrated in the [[Parameter_Estimation#Maximum_Likelihood_Estimation_.28MLE.29\|discussion of MLE]] given below.		As we can see, there is a sizable difference in the results of the two sets calculated using MLE and the results using regression with the SRM. The results for both cases are identical when using the regression estimation technique with SRM, as SRM considers only the positions of the suspensions. The MLE results are quite different for the two cases, with the second case having a much larger value of <math>\eta \,\!</math>, which is due to the higher values of the suspension times in Case 2. This is because the maximum likelihood technique, unlike rank regression with SRM, considers the values of the suspensions when estimating the parameters. This is illustrated in the [[Parameter_Estimation#Maximum_Likelihood_Estimation_.28MLE.29\|discussion of MLE]] given below.

	One alternative to improve the regression method is to use the following ReliaSoft ~~Rank~~ Method (RRM) to calculate the rank. RRM does consider the effect of the censoring time.		One alternative to improve the regression method is to use the following ReliaSoft Ranking Method (RRM) to calculate the rank. RRM does consider the effect of the censoring time.

	== ReliaSoft's Ranking Method (RRM) for Interval Censored Data==		== ReliaSoft's Ranking Method (RRM) for Interval Censored Data==

Harry Guo: /* Shortfalls of the Rank Adjustment Method */

2015-02-25T20:55:43Z

Shortfalls of the Rank Adjustment Method

← Older revision		Revision as of 20:55, 25 February 2015
Line 531:		Line 531:
	As we can see, there is a sizable difference in the results of the two sets calculated using MLE and the results using regression with the SRM. The results for both cases are identical when using the regression estimation technique with SRM, as SRM considers only the positions of the suspensions. The MLE results are quite different for the two cases, with the second case having a much larger value of <math>\eta \,\!</math>, which is due to the higher values of the suspension times in Case 2. This is because the maximum likelihood technique, unlike rank regression with SRM, considers the values of the suspensions when estimating the parameters. This is illustrated in the [[Parameter_Estimation#Maximum_Likelihood_Estimation_.28MLE.29\|discussion of MLE]] given below.		As we can see, there is a sizable difference in the results of the two sets calculated using MLE and the results using regression with the SRM. The results for both cases are identical when using the regression estimation technique with SRM, as SRM considers only the positions of the suspensions. The MLE results are quite different for the two cases, with the second case having a much larger value of <math>\eta \,\!</math>, which is due to the higher values of the suspension times in Case 2. This is because the maximum likelihood technique, unlike rank regression with SRM, considers the values of the suspensions when estimating the parameters. This is illustrated in the [[Parameter_Estimation#Maximum_Likelihood_Estimation_.28MLE.29\|discussion of MLE]] given below.

	~~The~~ following ReliaSoft Rank Method (RRM) ~~can~~ consider the effect of time ~~for censored data~~.		One alternative to improve the regression method is to use the following ReliaSoft Rank Method (RRM) to calculate the rank. RRM does consider the effect of the censoring time.

	== ReliaSoft's Ranking Method (RRM) for Interval Censored Data==		== ReliaSoft's Ranking Method (RRM) for Interval Censored Data==