Sequential detection of a change in a normal mean when the initial value is unknown
Let \(x_ 1,x_ 2,..\). be independent normal variables with variance 1 and E \(x_ n=\mu_ 0\) for \(n\leq \nu\) whereas E \(x_ n=\mu_ 0+\mu\) for \(n>\nu\), where \(\mu_ 0\) and \(\mu\) are unknown and of \(\nu\) it is only given that \(\nu \geq \nu_ 0\), a given nonnegative integer. The hypothesis \(\nu =\infty\) (meaning E \(x_ n=\mu_ 0\) for all n) is to be tested against the alternative \(\nu <\infty\). For any sequential procedure with stopping time T it is desirable to have \(E_{\infty}T\) large and \(E_{\nu}(T-\nu | T>\nu)\) small in some sense. At first it is assumed that \(\mu >0\). Three procedures are considered; all three are based on the maximal invariant \(y_ 2,y_ 3,...\), with \(y_ n=x_ n-x_ 1\), under the group of transformations \(x_ n\to x_ n+c\), \(n=1,2,...\), and the three corresponding stopping times are denoted N, \(\hat T,\) and \(\tau\), respectively. In the first procedure \(\delta >0\) and \(B>0\) are chosen and N is the least integer \(n>\nu_ 0\) for which \(R_ n(\delta)\geq B\), where \(R_ n(\delta)\) is the sum over i from \(\nu_ 0\) to n-1 of the likelihood ratio statistics (based on \(y_ 2,y_ 2,...)\) for testing \(\nu =\infty\) versus \(\nu =i\), \(\mu =\delta.\) In the second procedure \(R_ n(\delta)\) is replaced by \(\int R_ n(\delta)G(d\delta)\) with G a prior on \(\delta\). The third procedure is a CUSUM test based on the (invariant) sequence \(z_ 1,z_ 2,...\), where \(z_ n=(n/(n+1))^{1/2}(x_{n+1}-\bar x_ n).\) Two-sided analogues of these procedures are also considered in order to deal with arbitrary \(\mu \neq 0\). Asymptotic results are derived for \(E_{\infty}(T-\nu_ 0)\) as \(B\to \infty\) and for \(E_{\nu}(T-\nu | T>\nu)\) as \(B\to \infty\), \(\nu \to \infty\) under some restrictions, where \(T=N\) or \(\hat T.\) Monte Carlo experiments indicate that the asymptotic formulas provide reasonably good approximations. These experiments are also used to compare the three procedures. Their relative merits depend on the values of \(\mu\), \(\delta\), and \(\nu_ 0\), and no simple conclusions can be drawn.
scientific article; zbMATH DE number 4211317