Template:Example: Likelihood Ratio Bounds on Parameters - Revision history

Nicolette Young at 17:53, 21 December 2015

2015-12-21T17:53:38Z

← Older revision		Revision as of 17:53, 21 December 2015
Line 37:		Line 37:
	These data are represented graphically in the following contour plot:		These data are represented graphically in the following contour plot:

	[[Image:weibull parameter contour plot.png\|center\|~~475px~~]]		[[Image:weibull parameter contour plot.png\|center\|425px]]

	(Note that this plot is generated with degrees of freedom <math>k = 1\,\!</math>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <math>k = 2\,\!</math>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <math>\beta\,\!</math> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>\eta\,\!</math> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <math>\eta\,\!</math>, we can perform the same procedure as before, but finding the two values of <math>\eta\,\!</math> that correspond with a given value of <math>\beta\,\!</math> Using this method, we find that the 90% confidence limits on <math>\eta\,\!</math> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.		(Note that this plot is generated with degrees of freedom <math>k = 1\,\!</math>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <math>k = 2\,\!</math>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <math>\beta\,\!</math> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>\eta\,\!</math> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <math>\eta\,\!</math>, we can perform the same procedure as before, but finding the two values of <math>\eta\,\!</math> that correspond with a given value of <math>\beta\,\!</math> Using this method, we find that the 90% confidence limits on <math>\eta\,\!</math> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.

	Note that the points where <math>\beta\,\!</math> are maximized and minimized do not necessarily correspond with the points where <math>\eta\,\!</math> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.		Note that the points where <math>\beta\,\!</math> are maximized and minimized do not necessarily correspond with the points where <math>\eta\,\!</math> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.

Nicolette Young at 17:53, 21 December 2015

2015-12-21T17:53:20Z

← Older revision		Revision as of 17:53, 21 December 2015
Line 37:		Line 37:
	These data are represented graphically in the following contour plot:		These data are represented graphically in the following contour plot:

	[[Image:weibull parameter contour plot.png\|center\|~~375px~~]]		[[Image:weibull parameter contour plot.png\|center\|475px]]

	(Note that this plot is generated with degrees of freedom <math>k = 1\,\!</math>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <math>k = 2\,\!</math>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <math>\beta\,\!</math> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>\eta\,\!</math> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <math>\eta\,\!</math>, we can perform the same procedure as before, but finding the two values of <math>\eta\,\!</math> that correspond with a given value of <math>\beta\,\!</math> Using this method, we find that the 90% confidence limits on <math>\eta\,\!</math> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.		(Note that this plot is generated with degrees of freedom <math>k = 1\,\!</math>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <math>k = 2\,\!</math>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <math>\beta\,\!</math> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>\eta\,\!</math> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <math>\eta\,\!</math>, we can perform the same procedure as before, but finding the two values of <math>\eta\,\!</math> that correspond with a given value of <math>\beta\,\!</math> Using this method, we find that the 90% confidence limits on <math>\eta\,\!</math> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.

	Note that the points where <math>\beta\,\!</math> are maximized and minimized do not necessarily correspond with the points where <math>\eta\,\!</math> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.		Note that the points where <math>\beta\,\!</math> are maximized and minimized do not necessarily correspond with the points where <math>\eta\,\!</math> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.

Richard House at 18:31, 24 September 2012

2012-09-24T18:31:27Z

← Older revision		Revision as of 18:31, 24 September 2012
Line 1:		Line 1:
	'''Likelihood Ratio Bounds on Parameters'''		'''Likelihood Ratio Bounds on Parameters'''

	Five units were put on a reliability test and experienced failures at 10, 20, 30, 40 and 50 hours. Assuming a Weibull distribution, the MLE parameter estimates are calculated to be <math>\widehat{\beta }=2.2938</math> and <math>\widehat{\eta }=33.9428.</math> Calculate the 90% two-sided confidence bounds on these parameters using the likelihood ratio method.		Five units were put on a reliability test and experienced failures at 10, 20, 30, 40 and 50 hours. Assuming a Weibull distribution, the MLE parameter estimates are calculated to be <math>\widehat{\beta }=2.2938\,\!</math> and <math>\widehat{\eta }=33.9428.\,\!</math> Calculate the 90% two-sided confidence bounds on these parameters using the likelihood ratio method.

	'''Solution'''		'''Solution'''
Line 13:		Line 13:
	\\		\\
	L(\widehat{\beta },\widehat{\eta })= & 1.714714\times {{10}^{-9}}		L(\widehat{\beta },\widehat{\eta })= & 1.714714\times {{10}^{-9}}
	\end{align}</math>		\end{align}\,\!</math>

	where <math>{{x}_{i}}\,\!</math> are the original time-to-failure data points. We can now rearrange the likelihood ratio equation to the form:		where <math>{{x}_{i}}\,\!</math> are the original time-to-failure data points. We can now rearrange the likelihood ratio equation to the form:

	::<math>L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\~~alpha~~ ;1}^{2}}{2}}}=0</math>		::<math>L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}=0\,\!</math>

	Since our specified confidence level, <math>\delta\,\!</math>, is 90%, we can calculate the value of the chi-squared statistic, <math>\chi _{0.9;1}^{2}=2.705543.\,\!</math> We then substitute this information into the equation:		Since our specified confidence level, <math>\delta\,\!</math>, is 90%, we can calculate the value of the chi-squared statistic, <math>\chi _{0.9;1}^{2}=2.705543.\,\!</math> We then substitute this information into the equation:

	::<math>\begin{align}		::<math>\begin{align}
	L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\~~alpha~~ ;1}^{2}}{2}}}= & 0 \\		L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}= & 0 \\
	\\		\\
	L(\beta ,\eta )-1.714714\times {{10}^{-9}}\cdot {{e}^{\tfrac{-2.705543}{2}}}= & 0 \\		L(\beta ,\eta )-1.714714\times {{10}^{-9}}\cdot {{e}^{\tfrac{-2.705543}{2}}}= & 0 \\
	\\		\\
	L(\beta ,\eta )-4.432926\cdot {{10}^{-10}}= & 0		L(\beta ,\eta )-4.432926\cdot {{10}^{-10}}= & 0
	\end{align}</math>		\end{align}\,\!</math>

	The next step is to find the set of values of <math>\beta\,\!</math> and <math>\eta\,\!</math> that satisfy this equation, or find the values of <math>\beta\,\!</math> and <math>\eta\,\!</math> such that <math>L(\beta ,\eta )=4.432926\cdot {{10}^{-10}}.\,\!</math>		The next step is to find the set of values of <math>\beta\,\!</math> and <math>\eta\,\!</math> that satisfy this equation, or find the values of <math>\beta\,\!</math> and <math>\eta\,\!</math> such that <math>L(\beta ,\eta )=4.432926\cdot {{10}^{-10}}.\,\!</math>

Richard House at 04:07, 23 August 2012

2012-08-23T04:07:56Z

← Older revision		Revision as of 04:07, 23 August 2012
Line 15:		Line 15:
	\end{align}</math>		\end{align}</math>

	where <~~span class="texhtml"~~>''x~~''<sub>''~~i~~''</sub>~~</~~span~~> are the original time-to-failure data points. We can now rearrange the likelihood ratio equation to the form:		where <math>{{x}_{i}}\,\!</math> are the original time-to-failure data points. We can now rearrange the likelihood ratio equation to the form:

	::<math>L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}=0</math>		::<math>L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}=0</math>

	Since our specified confidence level, <~~span class="texhtml"~~>δ</~~span~~>, is 90%, we can calculate the value of the chi-squared statistic, <math>\chi _{0.9;1}^{2}=2.705543.</math> We then substitute this information into the equation:		Since our specified confidence level, <math>\delta\,\!</math>, is 90%, we can calculate the value of the chi-squared statistic, <math>\chi _{0.9;1}^{2}=2.705543.\,\!</math> We then substitute this information into the equation:

	::<math>\begin{align}		::<math>\begin{align}
Line 29:		Line 29:
	\end{align}</math>		\end{align}</math>

	The next step is to find the set of values of <~~span class="texhtml"~~>β</~~span~~> and <~~span class="texhtml"~~>η</~~span~~> that satisfy this equation, or find the values of <~~span class="texhtml"~~>β</~~span~~> and <~~span class="texhtml"~~>η</~~span~~> such that <math>L(\beta ,\eta )=4.432926\cdot {{10}^{-10}}.</math>		The next step is to find the set of values of <math>\beta\,\!</math> and <math>\eta\,\!</math> that satisfy this equation, or find the values of <math>\beta\,\!</math> and <math>\eta\,\!</math> such that <math>L(\beta ,\eta )=4.432926\cdot {{10}^{-10}}.\,\!</math>

	The solution is an iterative process that requires setting the value of <~~span class="texhtml"~~>β</~~span~~> and finding the appropriate values of <~~span class="texhtml"~~>η</~~span~~>, and vice versa. The following table gives values of <~~span class="texhtml"~~>β</~~span~~> based on given values of <~~span class="texhtml"~~>η</~~span~~>.		The solution is an iterative process that requires setting the value of <math>\beta\,\!</math> and finding the appropriate values of <math>\eta\,\!</math>, and vice versa. The following table gives values of <math>\beta\,\!</math> based on given values of <math>\eta\,\!</math>.

	[[Image:Confidencechart1.png\|center\|300px]]		[[Image:Confidencechart1.png\|center\|300px]]
Line 39:		Line 39:
	[[Image:weibull parameter contour plot.png\|center\|375px]]		[[Image:weibull parameter contour plot.png\|center\|375px]]

	(Note that this plot is generated with degrees of freedom <~~span class="texhtml"~~>''k'' = 1</~~span~~>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <~~span class="texhtml"~~>''k'' = 2</~~span~~>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <~~span class="texhtml"~~>β</~~span~~> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <~~span class="texhtml"~~>η</~~span~~> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <~~span class="texhtml"~~>η</~~span~~>, we can perform the same procedure as before, but finding the two values of <~~span class="texhtml"~~>η</~~span~~> that correspond with a given value of <~~span class="texhtml"~~>β.</~~span~~> Using this method, we find that the 90% confidence limits on <~~span class="texhtml"~~>η</~~span~~> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.		(Note that this plot is generated with degrees of freedom <math>k = 1\,\!</math>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <math>k = 2\,\!</math>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <math>\beta\,\!</math> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>\eta\,\!</math> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <math>\eta\,\!</math>, we can perform the same procedure as before, but finding the two values of <math>\eta\,\!</math> that correspond with a given value of <math>\beta\,\!</math> Using this method, we find that the 90% confidence limits on <math>\eta\,\!</math> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.

	Note that the points where <~~span class="texhtml"~~>β</~~span~~> are maximized and minimized do not necessarily correspond with the points where <~~span class="texhtml"~~>η</~~span~~> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.		Note that the points where <math>\beta\,\!</math> are maximized and minimized do not necessarily correspond with the points where <math>\eta\,\!</math> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.

Richard House at 07:33, 6 August 2012

2012-08-06T07:33:46Z

← Older revision		Revision as of 07:33, 6 August 2012
Line 3:		Line 3:
	Five units were put on a reliability test and experienced failures at 10, 20, 30, 40 and 50 hours. Assuming a Weibull distribution, the MLE parameter estimates are calculated to be <math>\widehat{\beta }=2.2938</math> and <math>\widehat{\eta }=33.9428.</math> Calculate the 90% two-sided confidence bounds on these parameters using the likelihood ratio method.		Five units were put on a reliability test and experienced failures at 10, 20, 30, 40 and 50 hours. Assuming a Weibull distribution, the MLE parameter estimates are calculated to be <math>\widehat{\beta }=2.2938</math> and <math>\widehat{\eta }=33.9428.</math> Calculate the 90% two-sided confidence bounds on these parameters using the likelihood ratio method.

	~~<br>~~'''Solution'''		'''Solution'''

	The first step is to calculate the likelihood function for the parameter estimates:		The first step is to calculate the likelihood function for the parameter estimates:
Line 15:		Line 15:
	\end{align}</math>		\end{align}</math>

	~~<br>~~where <span class="texhtml">''x''<sub>''i''</sub></span> are the original time-to-failure data points. We can now rearrange the likelihood ratio equation to the form:		where <span class="texhtml">''x''<sub>''i''</sub></span> are the original time-to-failure data points. We can now rearrange the likelihood ratio equation to the form:

	~~<br>~~

	::<math>L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}=0</math>		::<math>L(\beta ,\eta )-L(\widehat{\beta },\widehat{\eta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}=0</math>

	~~<br>~~Since our specified confidence level, <span class="texhtml">δ</span>, is 90%, we can calculate the value of the chi-squared statistic, <math>\chi _{0.9;1}^{2}=2.705543.</math> We then substitute this information into the equation:		Since our specified confidence level, <span class="texhtml">δ</span>, is 90%, we can calculate the value of the chi-squared statistic, <math>\chi _{0.9;1}^{2}=2.705543.</math> We then substitute this information into the equation:

	::<math>\begin{align}		::<math>\begin{align}
Line 31:		Line 29:
	\end{align}</math>		\end{align}</math>

	~~<br>~~The next step is to find the set of values of <span class="texhtml">β</span> and <span class="texhtml">η</span> that satisfy this equation, or find the values of <span class="texhtml">β</span> and <span class="texhtml">η</span> such that <math>L(\beta ,\eta )=4.432926\cdot {{10}^{-10}}.</math>		The next step is to find the set of values of <span class="texhtml">β</span> and <span class="texhtml">η</span> that satisfy this equation, or find the values of <span class="texhtml">β</span> and <span class="texhtml">η</span> such that <math>L(\beta ,\eta )=4.432926\cdot {{10}^{-10}}.</math>

	The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.		The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.

	~~<br>~~[[Image:Confidencechart1.png\|center\|300px]]		[[Image:Confidencechart1.png\|center\|300px]]

	~~<br>~~These data are represented graphically in the following contour plot:		These data are represented graphically in the following contour plot:

	~~<br>~~[[Image:weibull parameter contour plot.png\|center\|375px]]		[[Image:weibull parameter contour plot.png\|center\|375px]]

	~~<br>~~(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.		(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.

	~~<br>~~Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.		Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.

Chris Kahn at 13:49, 25 June 2012

2012-06-25T13:49:42Z

← Older revision		Revision as of 13:49, 25 June 2012
Line 35:		Line 35:
	The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.		The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.

	<br>[[Image:Confidencechart1.png\|center\|~~200px~~]]		<br>[[Image:Confidencechart1.png\|center\|300px]]

	<br>These data are represented graphically in the following contour plot:		<br>These data are represented graphically in the following contour plot:

	<br>[[Image:weibull parameter contour plot.png\|center\|~~250px~~]]		<br>[[Image:weibull parameter contour plot.png\|center\|375px]]

	<br>(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.		<br>(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.

	<br>Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.		<br>Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.

Nicolette Young at 17:48, 25 April 2012

2012-04-25T17:48:03Z

← Older revision		Revision as of 17:48, 25 April 2012
Line 35:		Line 35:
	The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.		The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.

	<br>[[Image:Confidencechart1.png\|center\|~~400px~~]]		<br>[[Image:Confidencechart1.png\|center\|200px]]

	<br>These data are represented graphically in the following contour plot:		<br>These data are represented graphically in the following contour plot:

	<br>[[Image:weibull parameter contour plot.png\|center\|~~350px~~]]		<br>[[Image:weibull parameter contour plot.png\|center\|250px]]

	<br>(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.		<br>(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.

	<br>Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.		<br>Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.

Nicolette Young at 18:41, 2 April 2012

2012-04-02T18:41:07Z

← Older revision		Revision as of 18:41, 2 April 2012
Line 35:		Line 35:
	The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.		The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.

	<br>[[Image:Confidencechart1.png\|center\|~~300px~~]]		<br>[[Image:Confidencechart1.png\|center\|400px]]

	<br>These data are represented graphically in the following contour plot:		<br>These data are represented graphically in the following contour plot:

Nikki Helms at 18:28, 2 April 2012

2012-04-02T18:28:33Z

← Older revision		Revision as of 18:28, 2 April 2012
Line 35:		Line 35:
	The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.		The solution is an iterative process that requires setting the value of <span class="texhtml">β</span> and finding the appropriate values of <span class="texhtml">η</span>, and vice versa. The following table gives values of <span class="texhtml">β</span> based on given values of <span class="texhtml">η</span>.

	<br>[[Image:Confidencechart1.~~gif\|thumb~~\|center\|300px]]		<br>[[Image:Confidencechart1.png\|center\|300px]]

	<br>These data are represented graphically in the following contour plot:		<br>These data are represented graphically in the following contour plot:

Nicolette Young at 16:24, 13 March 2012

2012-03-13T16:24:05Z

← Older revision		Revision as of 16:24, 13 March 2012
Line 39:		Line 39:
	<br>These data are represented graphically in the following contour plot:		<br>These data are represented graphically in the following contour plot:

	<br>[[Image:weibull parameter contour plot.png~~\|thumb~~\|center\|350px]]		<br>[[Image:weibull parameter contour plot.png\|center\|350px]]

	<br>(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.		<br>(Note that this plot is generated with degrees of freedom <span class="texhtml">''k'' = 1</span>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <span class="texhtml">''k'' = 2</span>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <span class="texhtml">β</span> is 1.142, while the highest is 3.950. These represent the two-sided 90% confidence limits on this parameter. Since solutions for the equation do not exist for values of <span class="texhtml">η</span> below 23 or above 50, these can be considered the 90% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <span class="texhtml">η</span>, we can perform the same procedure as before, but finding the two values of <span class="texhtml">η</span> that correspond with a given value of <span class="texhtml">β.</span> Using this method, we find that the 90% confidence limits on <span class="texhtml">η</span> are 22.474 and 49.967, which are close to the initial estimates of 23 and 50.

	<br>Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.		<br>Note that the points where <span class="texhtml">β</span> are maximized and minimized do not necessarily correspond with the points where <span class="texhtml">η</span> are maximized and minimized. This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.