# Is every estimator a sufficient statistic

parameter estimationstatistical-inferencestatistics

It is clear that not any sufficient statistic (s.s.) makes a good estimator (since a monotonic transform of a s.s. is still a s.s.). But is a "good" estimator of the parameter always a s.s.? If not, under what conditions is this the case?

I gather that the dimension of the s.s. may vary with the size of the data (e.g. Cauchy distribution), so perhaps we have to restrict ourselves to the exponential family of distributions for a start?

I will leave open what "good" means, e.g. it could be unbiasedness or another condition under which this result holds.

The notion of sufficiency is relevant to estimation because sufficiency is often a desirable property of an estimator. However, it is not the only desirable property. Unbiasedness is also desirable, for example; however, an unbiased statistic need not be sufficient; e.g., $$X_1, \ldots, X_n$$ are iid random variables drawn from a parametric distribution with finite mean $$\mu$$ and variance $$\sigma^2$$; the sample mean $$\bar X$$ is an estimator of $$\mu$$ but so is $$X_1$$. Both are unbiased for $$\mu$$, but the latter is not sufficient for $$\mu$$.
Consistency is also a desirable property, but here again we can easily construct consistent but insufficient statistics; e.g., $$(X_1 + \cdots + X_{n-1})/(n-1)$$, the mean of the sample that omits the last observation, is consistent and unbiased but not sufficient because it has discarded information about $$\mu$$ that was present in the original sample.