Semi-Sufficiency in Accelerated Life Testing

Extremely long testing time causes a well known problem in reliability analysis. The most successful method of treating it is the application of accelerating stress. S. S is a vector of physical effects like temperature, voltage or pressure. The lifetime of an object put under stress S is described by a stochastic quantity TS. The distribution belongs to a family parameterized by θ (S), the vector of statistical parameters given stress S. In practice the relation between these parameters and the stress components is often known to be of some functional form ψ with unknown physical parameters c=(c1 ,…,cn) (1) $$ {\rm{\theta }}({\rm{S}}) = {\rm{\psi }}({\rm{S,c}}) $$ All knowledge about these physical parameters c before the experiment is put into the prior density π(c). Now we can make stochastically independent observations on m different stress levels Si i=1(1)m. On each of these we get a sample of size ki. Using the life time densities f(.) we get the likelihood function l(c;D) given the data D=(tij, i=1(1)m, j=1(1)ki) (2) $$ 1({\rm{c;D}}) = \mathop \Pi \limits_{{\rm{i}} = 1}^{\rm{m}} \mathop {{\Pi ^{\rm{i}}}}\limits_{{\rm{j}} = 1}^{\rm{k}} {\rm{f}}({{\rm{t}}_{{\rm{ij}}}}|{{\rm{s}}_{\rm{i}}}{\rm{,c}}). $$ Calculating the posterior density π(c|D) via Bayes theorem (3) $$ \prod ({\rm{c|D}}) \propto \prod ({\rm{c}})1({\rm{c;D}}) $$ we can make estimates about the physical parameters c, the statistical parameters given the usual stress Su (4) $$ {\rm{\theta }}({{\rm{S}}_{\rm{u}}}) = {\rm{\psi }}({{\rm{S}}_{\rm{u}}}{\rm{,c}}) $$ or the predictive density f(t|Su) of an object under usual stress Su.