Template:Simple actuarial method - Revision history

Kate Racaza: Redirected page to Non-Parametric Life Data Analysis

2012-06-29T07:35:15Z

Redirected page to Non-Parametric Life Data Analysis

← Older revision		Revision as of 07:35, 29 June 2012
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	~~== Actuarial~~-~~Simple Method ==~~		#REDIRECT [[Non-Parametric Life Data Analysis]]

	The actuarial-simple method is an easy-to-use form of non-parametric data analysis that can be used for multiple censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <span class="texhtml">''r''<sub>''j''</sub>,</span> versus the number of operating units in that time period, <span class="texhtml">''n''<sub>''j''</sub></span> . The equation for the reliability estimator for the standard actuarial method is given by:

	~~::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m</math>~~

	~~where:~~

	~~::<math>\begin{align}~~
	~~& m= \text{the total number of intervals} \\~~
	~~& n= \text{the total number of units}~~
	~~\end{align}</math>~~

	~~The variable <span class="texhtml">''n''<sub>''i''</sub></span> is defined by:~~

	~~::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>~~

	~~where:~~

	~~::<math>\begin{align}~~
	~~& {{r}_{j}}= \text{the number of failures in interval }j \\~~
	~~& {{s}_{j}}= \text{the number of suspensions in interval }j~~
	~~\end{align}</math>~~

	~~<br>'''Example 2:''' {{Example: Simple-Actuarial Example}}~~

Kate Racaza at 22:13, 9 March 2012

2012-03-09T22:13:22Z

← Older revision		Revision as of 22:13, 9 March 2012
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	==Simple ~~Actuarial~~ Method==		== Actuarial-Simple Method ==
	The simple ~~actuarial~~ method is an easy-to-use form of ~~nonparametric~~ data analysis that can be used for ~~multiply~~ censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <~~math~~>{{r~~}_{~~j}},</~~math~~> versus the number of operating units in that time period, <~~math~~>{{n~~}_{~~j}}</~~math~~> . The equation for the reliability estimator for the standard actuarial method is given by:
			The actuarial-simple method is an easy-to-use form of non-parametric data analysis that can be used for multiple censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <span class="texhtml">''r''<sub>''j''</sub>,</span> versus the number of operating units in that time period, <span class="texhtml">''n''<sub>''j''</sub></span> . The equation for the reliability estimator for the standard actuarial method is given by:

	::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m</math>		::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m</math>
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	\end{align}</math>		\end{align}</math>

	The variable <~~math~~>{{n~~}_{~~i}}</~~math~~> is defined by:		The variable <span class="texhtml">''n''<sub>''i''</sub></span> is defined by:

	::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>		::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>
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	\end{align}</math>		\end{align}</math>

			<br>'''Example 2:''' {{Example: Simple-Actuarial Example}}
	'''Example 2:'''
	{{Example: Simple-Actuarial Example}}

Harry Guo: /* Simple Actuarial Method */

2012-02-20T20:46:15Z

Simple Actuarial Method

← Older revision		Revision as of 20:46, 20 February 2012
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	::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m</math>		::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m</math>

	:where:		where:

	::<math>\begin{align}		::<math>\begin{align}
	& m= & \text{the total number of intervals} \\		& m= \text{the total number of intervals} \\
	& n= & \text{the total number of units}		& n= \text{the total number of units}
	\end{align}</math>		\end{align}</math>

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	::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>		::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>

	:where:		where:

	::<math>\begin{align}		::<math>\begin{align}
	& {{r}_{j}}= & \text{the number of failures in interval }j \\		& {{r}_{j}}= \text{the number of failures in interval }j \\
	& {{s}_{j}}= & \text{the number of suspensions in interval }j		& {{s}_{j}}= \text{the number of suspensions in interval }j
	\end{align}</math>		\end{align}</math>

Harry Guo: /* Simple Actuarial Method */

2012-02-20T19:04:50Z

Simple Actuarial Method

← Older revision		Revision as of 19:04, 20 February 2012
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	& {{s}_{j}}= & \text{the number of suspensions in interval }j		& {{s}_{j}}= & \text{the number of suspensions in interval }j
	\end{align}</math>		\end{align}</math>


	'''Example 2:'''		'''Example 2:'''
	{{Example: Simple-Actuarial Example}}		{{Example: Simple-Actuarial Example}}

Harry Guo: /* Simple Actuarial Method */

2012-02-20T19:01:33Z

Simple Actuarial Method

@@ Line 22: / Line 22: @@
 \end{align}</math>
+'''Example 2:'''
-====Example 10====
+{{Example: Simple-Actuarial Example}}

Harry Guo at 18:48, 20 February 2012

2012-02-20T18:48:42Z

← Older revision		Revision as of 18:48, 20 February 2012
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	===Simple Actuarial Method===		==Simple Actuarial Method==
	The simple actuarial method is an easy-to-use form of nonparametric data analysis that can be used for multiply censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <math>{{r}_{j}},</math> versus the number of operating units in that time period, <math>{{n}_{j}}</math> . The equation for the reliability estimator for the standard actuarial method is given by:		The simple actuarial method is an easy-to-use form of nonparametric data analysis that can be used for multiply censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <math>{{r}_{j}},</math> versus the number of operating units in that time period, <math>{{n}_{j}}</math> . The equation for the reliability estimator for the standard actuarial method is given by:

Nicolette Young: Created page with '===Simple Actuarial Method=== The simple actuarial method is an easy-to-use form of nonparametric data analysis that can be used for multiply censored data that are arranged in i…'

2012-01-05T00:02:08Z

Created page with '===Simple Actuarial Method=== The simple actuarial method is an easy-to-use form of nonparametric data analysis that can be used for multiply censored data that are arranged in i…'

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===Simple Actuarial Method===
The simple actuarial method is an easy-to-use form of nonparametric data analysis that can be used for multiply censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <math>{{r}_{j}},</math> versus the number of operating units in that time period, <math>{{n}_{j}}</math> . The equation for the reliability estimator for the standard actuarial method is given by:

::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m</math>

:where:

::<math>\begin{align}
& m= & \text{the total number of intervals} \\
& n= & \text{the total number of units}
\end{align}</math>

The variable <math>{{n}_{i}}</math> is defined by:

::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>

:where:

::<math>\begin{align}
& {{r}_{j}}= & \text{the number of failures in interval }j \\
& {{s}_{j}}= & \text{the number of suspensions in interval }j
\end{align}</math>

====Example 10====
A group of 55 units are put on a life test during which the units are evaluated every 50 hours, with the following results:

<center><math>\begin{matrix}
Start & End & Number of & Number of \\
Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} \\
0 & 50 & 2 & 4 \\
50 & 100 & 0 & 5 \\
100 & 150 & 2 & 2 \\
150 & 200 & 3 & 5 \\
200 & 250 & 2 & 1 \\
250 & 300 & 1 & 2 \\
300 & 350 & 2 & 1 \\
350 & 400 & 3 & 3 \\
400 & 450 & 3 & 4 \\
450 & 500 & 1 & 2 \\
500 & 550 & 2 & 1 \\
550 & 600 & 1 & 0 \\
600 & 650 & 2 & 1 \\
\end{matrix}</math></center>

=====Solution to Example 10=====
The reliability estimates for the simple actuarial method can be obtained by expanding the data table to include terms used in calculation of the reliability estimates for Eqn. (simpact):

<center><math>\begin{matrix}
Start & End & Number of & Number of & Available & {} & {} \\
Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, {{n}_{i}} & 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} & \mathop{}_{}^{}1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \\
0 & 50 & 2 & 4 & 55 & 0.964 & 0.964 \\
50 & 100 & 0 & 5 & 49 & 1.000 & 0.964 \\
100 & 150 & 2 & 2 & 44 & 0.955 & 0.920 \\
150 & 200 & 3 & 5 & 40 & 0.925 & 0.851 \\
200 & 250 & 2 & 1 & 32 & 0.938 & 0.798 \\
250 & 300 & 1 & 2 & 29 & 0.966 & 0.770 \\
300 & 350 & 2 & 1 & 26 & 0.923 & 0.711 \\
350 & 400 & 3 & 3 & 23 & 0.870 & 0.618 \\
400 & 450 & 3 & 4 & 17 & 0.824 & 0.509 \\
450 & 500 & 1 & 2 & 10 & 0.900 & 0.458 \\
500 & 550 & 2 & 1 & 7 & 0.714 & 0.327 \\
550 & 600 & 1 & 0 & 4 & 0.750 & 0.245 \\
600 & 650 & 2 & 1 & 3 & 0.333 & 0.082 \\
\end{matrix}</math></center>

As can be determined from the preceding table, the reliability estimates for the failure times are:

<center><math>\begin{matrix}
Failure Period & Reliability \\
End Time & Estimate \\
50 & 96.4% \\
150 & 92.0% \\
200 & 85.1% \\
250 & 79.8% \\
300 & 77.0% \\
350 & 71.1% \\
400 & 61.8% \\
450 & 50.9% \\
500 & 45.8% \\
550 & 32.7% \\
600 & 24.5% \\
650 & 8.2% \\
\end{matrix}</math></center>