## Sequential detection of a change in a normal mean when the initial value is unknown.(English)Zbl 0732.62080

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.
Reviewer: R.A.Wijsman

### MSC:

 62L10 Sequential statistical analysis
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