On the conditional increments of degradation processes

Recently, Wang and Xu (2010) developed an efficient EM algorithm for the semiparametric inference of the inverse Gaussian (IG) process. In the presence of missing degradation data, the algorithm needs to compute the mean of the IG increment during some time interval [_s1,_s2]$[s_1,s_2]$ conditional on that the process is tied down on two points before _s1$s_1$ and after _s2$s_2$, respectively. This study derives the conditional distribution of this increment and gives the expectation of the increment and of the reciprocal of the increment. The results simplify the implementation of the EM algorithm for the IG process. In a similar vein, we further derive distributions for the conditional increments of the Wiener process and the Gamma process, respectively.