Statistics for Maintenance Modelling: Repairable Systems

# Testing for a Trend - The Laplace Test¤

The process in which the ROCOF varies with time may be described as a NHPP with (t). It is important to test whether (t) is time dependent or not. Under the assumption of constant (t), the times of failures (not between failures),
T1, T2, T3, ... Tn, are ordered uniform variables over (0, T0] with mean T0/2 and variance T02/12. Let t1, t2, t3, ... tn and t0 denote their realisations and by the central limit theorem, the statistic ti/n has an approximate normal distribution with mean t0/2 and variance t02 /(12n), so the statistic:

U =

=

is approximately N(0,1), and may be used to test the null hypothesis of no trend, that is 1 = 0 or 1 = 1 in 1(t) and 2(t) respectively. The equation we've generated here is commonly called the Laplace test.

The central limit theorem says that the distribution of any reasonably well behaved function of a sample mean tends to the normal distribution as the sample size increases to infinity.