In this paper, we provide a stochastic ordering of the random vector T= X2/X1,2X3/ X1+X2),…, m-1 Xm/∑j="1m-1Xj) as the βj's change for any given 2⩽m⩽k.
It is reasonable to assume that, in a deterioration process, each X ( t ) is non-negative, and that the degree of deterioration increases with t (in a certain stochastic order).
The implication of the extra term in the stochastic order is that although the underlying deterministic order of the method may be high, the stochastic order is restricted by the noise term.
Based on this order relation, a generalized stochastic order has been introduced in [18].
For a given process ( X ( t ), t ≥ 0 ), we will use the notation X ( t ) ↑ ⋅ to indicate that X ( t ) is increasing in the ⋅ stochastic order, for all t ≥ 0 (and X ( t ) ↓ ⋅ if it is decreasing).
The random variable (X_{1}) is said to be smaller than (X_{2}) in the: (i) Stochastic order (denoted (X_{1}leq_{st}X_{2})) if (bar {F}_{1}(x leqoverline{F}_{2}(x)), for all (xgeq0).
Note that in a stochastic setting we judge order of accuracy through two mechanisms: strong order (where trajectories are compared with the true solutions) and weak order (where moments are compared).
Theoretically, the amount of noise in a stochastic simulation is of the order of, where n is the number of molecules [ 41].
As is usual in a stochastic analysis, concentrations and 0th/2nd order reaction rates have to be scaled by the factor V· A v, where V is the volume and A v denotes Avogadro's constant (approximately 6.02214179 × 1023 molecules-1).
Thus for a given coherent structure with known signature, the survival functions and mean time to failure of new systems can be easily calculated and comparisons can be done in terms of stochastic ordering, and mean time to failure ordering.
