Recursive Z-statistic

In respones to: Consider a standard Z-statistic used in hypothesis testing. One of the variables needed to compute the Z-statistic is the number of observations. The problem is that with each additional observation one has to recompute the Z-statistic from scratch. It seems like there is no recursive formulation, e.g. a representation such asZ(n) = Z(n-1) + new piece of information. Is there perhaps an approximate recursive formulation? Any other thoughts?Thanks.

An example hypothesis test is:

$H_0:$ $\mu ={\mu }_0$

$H_1:$ $\mu \ne {\mu }_0$

We reject this hypothesis if $\overline{x}$ is either greater than or lower than a critical value.Assuming the critical values do not change all you have to update is $Z_0$.

The test statistic is:

$$Z_0=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$$

Assuming you know $\sigma$, when you get a new variable $X_{n+1}$ you can update $\overline{x}$using $n$, $\overline{X}$, and $X_{n+1}$, then recalculate $Z_0$.

Now if you do not know $\sigma$, and your sample size is large enough to use the Normal distribution, you haveto update your sample variance, $S^2$. If your sample size is not large enough and you are using the t-distribution thenyour critical values will change when $n$ changes.

To do update $S$ without recalculating, you should keep running totals of ${\sum }_i{X}_i$ and ${\sum }_i{X}_i^2$,so you can update $S$ using the computation formula for the sample variance.