Template:Example: Lognormal Distribution RRX - Revision history

Lisa Hacker: Redirected page to The Lognormal Distribution

2012-08-13T03:29:42Z

Redirected page to The Lognormal Distribution

← Older revision		Revision as of 03:29, 13 August 2012
Line 1:		Line 1:
	~~'''Lognormal Distribution RRX Example'''~~		#REDIRECT [[The Lognormal Distribution]]

	Using the data of [[Lognormal Example 2 Data\|Example 2]] and assuming a lognormal distribution, estimate the parameters and estimate the correlation coefficient, <math>\rho </math> , using rank regression on X.

	~~'''Solution'''~~

	~~Table 2 constructed in [[Lognormal Example 2 Data\|Example 2]] applies to this example as well. Using the values in this table we get:~~

	~~::<math>\begin{align}~~
	& \hat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,t_{i}^{\prime }{{y}_{i}}-\tfrac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,t_{i}^{\prime }\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,y_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}} \right)}^{2}}}{14}} \\
	~~& & \\~~
	~~& \widehat{b}= & \frac{10.4473-(49.2220)(0)/14}{11.3646-{{(0)}^{2}}/14}~~
	~~\end{align}</math>~~

	~~or:~~

	~~::<math>\widehat{b}=0.9193</math>~~

	~~and:~~

	::<math>\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,t_{i}^{\prime }}{14}-\widehat{b}\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}</math>

	~~or:~~

	~~::<math>\widehat{a}=\frac{49.2220}{14}-(0.9193)\frac{(0)}{14}=3.5159</math>~~

	~~Therefore:~~

	~~::<math>{\sigma'}=\widehat{b}=0.9193</math>~~

	~~and:~~

	~~::<math>{\mu }'=\frac{\widehat{a}}{\widehat{b}}{\sigma'}=\frac{3.5159}{0.9193}\cdot 0.9193=3.5159</math>~~

	~~Using for Mean and Standard Deviation we get:~~

	~~::<math>\overline{T}=\mu =51.3393\text{ hours}</math>~~

	~~and:~~

	~~::<math>{\sigma'}=59.1682\text{ hours}.</math>~~

	~~The correlation coefficient is found using the equation in~~ [[The ~~Correlation Coefficient Calculation\|previous section]]:~~

	~~::<math>\widehat{\rho }=0.9754.</math>~~

	Note that the regression on Y analysis is not necessarily the same as the regression on X. The only time when the results of the two regression types are the same (i.e., will yield the same equation for a line) is when the data lie perfectly on a line.

	~~Using Weibull++ , with the Rank Regression on X option, the results are:~~

	~~[[Image:~~Lognormal Distribution ~~Example 3 Data and Result.png\|center\|550px\|~~ ]]

Richard House at 04:45, 8 August 2012

2012-08-08T04:45:12Z

← Older revision		Revision as of 04:45, 8 August 2012
Line 40:		Line 40:

	::<math>{\sigma'}=59.1682\text{ hours}.</math>		::<math>{\sigma'}=59.1682\text{ hours}.</math>


	The correlation coefficient is found using the equation in [[The Correlation Coefficient Calculation\|previous section]]:		The correlation coefficient is found using the equation in [[The Correlation Coefficient Calculation\|previous section]]:
Line 46:		Line 45:
	::<math>\widehat{\rho }=0.9754.</math>		::<math>\widehat{\rho }=0.9754.</math>

	Note that the regression on Y analysis is not necessarily the same as the regression on X. The only time when the results of the two regression types are the same (i.e. will yield the same equation for a line) is when the data lie perfectly on a line.		Note that the regression on Y analysis is not necessarily the same as the regression on X. The only time when the results of the two regression types are the same (i.e., will yield the same equation for a line) is when the data lie perfectly on a line.

	Using Weibull++ , with the Rank Regression on X option, the results are:		Using Weibull++ , with the Rank Regression on X option, the results are:

	[[Image:Lognormal Distribution Example 3 Data and Result.png~~\|thumb~~\|center\|~~250px~~\| ]]		[[Image:Lognormal Distribution Example 3 Data and Result.png\|center\|550px\| ]]

Nicolette Young at 21:20, 25 April 2012

2012-04-25T21:20:44Z

← Older revision		Revision as of 21:20, 25 April 2012
Line 50:		Line 50:
	Using Weibull++ , with the Rank Regression on X option, the results are:		Using Weibull++ , with the Rank Regression on X option, the results are:

	[[Image:Lognormal Distribution Example 3 Data and Result.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Lognormal Distribution Example 3 Data and Result.png\|thumb\|center\|250px\| ]]

Harry Guo at 17:00, 2 March 2012

2012-03-02T17:00:34Z

← Older revision		Revision as of 17:00, 2 March 2012
Line 33:		Line 33:
	::<math>{\mu }'=\frac{\widehat{a}}{\widehat{b}}{\sigma'}=\frac{3.5159}{0.9193}\cdot 0.9193=3.5159</math>		::<math>{\mu }'=\frac{\widehat{a}}{\widehat{b}}{\sigma'}=\frac{3.5159}{0.9193}\cdot 0.9193=3.5159</math>

	Using ~~Eqns. (mean)~~ and ~~(sdv)~~ we get:		Using for Mean and Standard Deviation we get:

	::<math>\overline{T}=\mu =51.3393\text{ hours}</math>		::<math>\overline{T}=\mu =51.3393\text{ hours}</math>

Harry Guo at 16:59, 2 March 2012

2012-03-02T16:59:47Z

← Older revision		Revision as of 16:59, 2 March 2012
Line 50:		Line 50:
	Using Weibull++ , with the Rank Regression on X option, the results are:		Using Weibull++ , with the Rank Regression on X option, the results are:

	[[Image:Lognormal Distribution Example 3 Data and Result\|thumb\|center\|400px\| ]]		[[Image:Lognormal Distribution Example 3 Data and Result.png\|thumb\|center\|400px\| ]]

Harry Guo at 16:58, 2 March 2012

2012-03-02T16:58:57Z

← Older revision		Revision as of 16:58, 2 March 2012
Line 5:		Line 5:
	'''Solution'''		'''Solution'''

	Table 9.2 constructed in [[Lognormal Example 2 Data\|Example 2]] applies to this example as well. Using the values in this table we get:		Table 2 constructed in [[Lognormal Example 2 Data\|Example 2]] applies to this example as well. Using the values in this table we get:

	::<math>\begin{align}		::<math>\begin{align}
Line 50:		Line 50:
	Using Weibull++ , with the Rank Regression on X option, the results are:		Using Weibull++ , with the Rank Regression on X option, the results are:

	[[Image:~~5folio.png~~\|thumb\|center\|400px\| ]]		[[Image:Lognormal Distribution Example 3 Data and Result\|thumb\|center\|400px\| ]]

Harry Guo at 21:48, 13 February 2012

2012-02-13T21:48:01Z

← Older revision		Revision as of 21:48, 13 February 2012
Line 8:		Line 8:

	::<math>\begin{align}		::<math>\begin{align}
	& \hat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{\prime }{{y}_{i}}-\tfrac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{\prime }\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,y_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}} \right)}^{2}}}{14}} \\		& \hat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,t_{i}^{\prime }{{y}_{i}}-\tfrac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,t_{i}^{\prime }\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,y_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}} \right)}^{2}}}{14}} \\
	& & \\		& & \\
	& \widehat{b}= & \frac{10.4473-(49.2220)(0)/14}{11.3646-{{(0)}^{2}}/14}		& \widehat{b}= & \frac{10.4473-(49.2220)(0)/14}{11.3646-{{(0)}^{2}}/14}
Line 19:		Line 19:
	and:		and:

	::<math>\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{\prime }}{14}-\widehat{b}\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}</math>		::<math>\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,t_{i}^{\prime }}{14}-\widehat{b}\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}</math>

	or:		or:
Line 25:		Line 25:
	::<math>\widehat{a}=\frac{49.2220}{14}-(0.9193)\frac{(0)}{14}=3.5159</math>		::<math>\widehat{a}=\frac{49.2220}{14}-(0.9193)\frac{(0)}{14}=3.5159</math>

	Therefore~~, from Eqn. (blnx)~~:		Therefore:

	::<math>{{\sigma ~~}_{{{T}~~'}}}=\widehat{b}=0.9193</math>		::<math>{\sigma'}=\widehat{b}=0.9193</math>

	and ~~from Eqn. (alnx)~~:		and:

	::<math>{\mu }'=\frac{\widehat{a}}{\widehat{b}}{{\sigma ~~}_{{{T}~~'}}}=\frac{3.5159}{0.9193}\cdot 0.9193=3.5159</math>		::<math>{\mu }'=\frac{\widehat{a}}{\widehat{b}}{\sigma'}=\frac{3.5159}{0.9193}\cdot 0.9193=3.5159</math>

	Using Eqns. (mean) and (sdv) we get:		Using Eqns. (mean) and (sdv) we get:
Line 39:		Line 39:
	and:		and:

	::<math>{{\sigma ~~}_{T}~~}=59.1682\text{ hours}.</math>		::<math>{\sigma'}=59.1682\text{ hours}.</math>


	The correlation coefficient is found using ~~Eqn. (RHOln)~~:		The correlation coefficient is found using the equation in [[The Correlation Coefficient Calculation\|previous section]]:

	::<math>\widehat{\rho }=0.9754.</math>		::<math>\widehat{\rho }=0.9754.</math>

Harry Guo at 21:40, 13 February 2012

2012-02-13T21:40:09Z

← Older revision		Revision as of 21:40, 13 February 2012
Line 5:		Line 5:
	'''Solution'''		'''Solution'''

	Table 9.2 constructed in Example 2 applies to this example as well. Using the values in this table we get:		Table 9.2 constructed in [[Lognormal Example 2 Data\|Example 2]] applies to this example as well. Using the values in this table we get:

	::<math>\begin{align}		::<math>\begin{align}

Harry Guo at 21:38, 13 February 2012

2012-02-13T21:38:39Z

← Older revision		Revision as of 21:38, 13 February 2012
Line 1:		Line 1:
	'''Lognormal Distribution RRX Example'''		'''Lognormal Distribution RRX Example'''

	Using the data of Example 2 and assuming a lognormal distribution, estimate the parameters and estimate the correlation coefficient, <math>\rho </math> , using rank regression on X.		Using the data of [[Lognormal Example 2 Data\|Example 2]] and assuming a lognormal distribution, estimate the parameters and estimate the correlation coefficient, <math>\rho </math> , using rank regression on X.

	'''Solution'''		'''Solution'''

Harry Guo at 18:18, 13 February 2012

2012-02-13T18:18:27Z

← Older revision		Revision as of 18:18, 13 February 2012
Line 4:		Line 4:

	'''Solution'''		'''Solution'''

	Table 9.2 constructed in Example 2 applies to this example as well. Using the values in this table we get:		Table 9.2 constructed in Example 2 applies to this example as well. Using the values in this table we get: