On the discrimination problem for a class of stochastic processes with ordered first-passage times (Q2739985)

Let \(\{X(s), s\geq 0\}\) be a continuous-time stochastic process defined on \([0,r),\;r>0\), or on the set of integers \(\{0,1,\ldots,K\}\) and let \(X(0)\) be distributed according to one of two preassigned distributions whose choice has been made via a Bernoulli trial with parameter \(p\). Furthermore, the family of first-passage times of \(\{X(s), s\geq 0\}\) through the state \(0\) is increasing in the initial state according to the likelihood ratio order. That means that if \(X\) and \(Y\) are absolutely continuous (discrete) random variables with probability densities (distributions) \(f_{X}(t)\) and \(f_{Y}(t)\), respectively, \(X\) is said to be smaller than \(Y\) in the likelihood ratio order if \(f_{X}(t)/f_{Y}(t)\) is non-increasing in \(t\) over the union of the supports of \(X\) and \(Y\).NEWLINENEWLINENEWLINEThe following Bayes discrimination problem is solved: to select the distribution of \(X(0)\) by knowing that the first-passage time through \(0\) for a sample path of \(\{X(s), s\geq 0\}\) is known to be \(t\).