Template:Laplace trend test rsa

Laplace Trend Test

The Laplace Trend Test tests the hypothesis that a trend does not exist within the data. The Laplace Trend test is applicable to the following data types: Multiple Systems-Concurrent Operating Times, Repairable and Fleet. The Laplace Trend Test can determine whether the system is deteriorating, improving, or if there is no trend at all. Calculate the test statistic, [math]\displaystyle{ U }[/math] , using the following equation:

[math]\displaystyle{ U=\frac{\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}}{N}-\tfrac{T}{2}}{T\sqrt{\tfrac{1}{12N}}} }[/math]

where:

• [math]\displaystyle{ T }[/math] = total operating time (termination time)

• [math]\displaystyle{ {{X}_{i}} }[/math] = age of the system at the [math]\displaystyle{ {{i}^{th}} }[/math] successive failure

• [math]\displaystyle{ N }[/math] = total number of failures

The test statistic [math]\displaystyle{ U }[/math] is approximately a standard normal random variable. The critical value is read from the Standard Normal tables with a given significance level, [math]\displaystyle{ \alpha }[/math] .

Example
Consider once again the data in Table B.1. Check for a trend within System 1 assuming a significance level of 0.10. Calculate the test statistic [math]\displaystyle{ U }[/math] for System 1 using Eqn. (Utatistic).

[math]\displaystyle{ U=-2.6121 }[/math]

From the Standard Normal tables with a significance level of 0.10, the critical value is equal to 1.645. If [math]\displaystyle{ -1.645\lt U\lt 1.645 }[/math] then we would fail to reject the hypothesis of no trend. However, since [math]\displaystyle{ U\lt -1.645 }[/math] then an improving trend exists within System 1.
If [math]\displaystyle{ U\gt 1.645 }[/math] then a deteriorating trend would exist.