Significant change in estimate with new observation? Suppose I have some yearly observations (e.g. rates) $x_1$,$x_2$...,$x_{t-1}$. I estimate some parameter of interest as $\bar{x}_{t-1}=\frac{1}{t-1}\sum_i{x_i}$.
Now at year $t$, I get a new observation $x_t$, so that I can compute $\bar{x}_{t}$. 


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*How can I "decide" if I need to update my estimate from $\bar{x}_{t-1}$ to $\bar{x}_{t}$?

*I get new observations for subsequent years, how can I "decide" to update my estimate or not?

 A: Your estimate of $\bar{x}_{t-1}$ is an estimate of a different quantity than your estimate of $\bar{x}_t$. So, you should always update, but recognize you are estimating something else each time.
Let's make this more concrete. Suppose your $x_i$ are average interest rates on a particular type of investment. Then $\bar{x}_t$ is the average interest rate over the previous $t$ years. $\bar{x}_{t-1}$ is the average over years $1$ to $t-1$. 
A: I have never used it for this kind of problem, but I think what you are asking for is the information lost by modelling the value of interest, say $\bar{x}$, by using either $\bar{X}_t$ or $\bar{X}_{t-1}$. If that's the case, my first guess would be to use the Akaike's Information Criterion (AIC) (or the AICc if you have a low number of samples) in order to estimate the (relative) information loss of using $\bar{X}_t$ or $\bar{X}_{t-1}$: if the value of the AIC for both models is less than 2, then both models are indistinguishable, and it makes no sense to update the model. 
The only "problem" of using the AIC is that you need a model for the likelihood; however, if you have a relatively large number of observations, you can rely on the Central Limit Theorem if you are willing to put constraints on the stochastic process that makes up the noise.
A: Related to Peter Flom's point, what is the population you are trying to estimate the parameter in? 
If you expect x to be similarly reflected in all years (and assume year to year differences are just random error), then you could just do a power calculation to find out the number of observations needed to estimate a mean value at a given level of precision. 
A: $\bar{X}_t = \frac{(t-1)}{t}\bar{X}_{t-1} + \frac{1}{t}X_t$ so I suppose you could say that if $\frac{1}{t}(X_t - \bar{X}_{t-1}) > \epsilon$, then update.  You can make similar formulas for more than one year. 
