# Is the sample mean complete sufficient for the Expectation when $X \sim F$ where $F$ is some symmetric distribution?

Let $$X_1,X_2... X_n$$ be iid $$\sim F$$ where $$F$$ is any symmetric continuous distribution and let $$\mid E(X)\mid<\infty$$.

Is $$\bar{x}$$ complete sufficient for $$E(x)=\int{xf(x)dx}$$?

Assume that all parameters of $$F$$ except $$E(x)=\mu$$ are known and that the support of $$X$$ does not depend on $$\mu$$

• I only ever see sufficiency defined for parameters that fully characterise the distribution. Here E(x) does not characterise F, and one could see the shape of F as another "parameter"; certainly it is additional information required to characterise F. In such a situation I only see joint sufficient statistics defined, for all characterising parameters together. I haven't come across a definition of sufficiency that covers one parameter only, if that parameter doesn't characterise the distribution. No such definition may exist, in which case the concept of sufficiency doesn't apply. – Lewian Dec 3 '19 at 16:16
• I made some changes. Suppose E(x) fully characterizes the distribution (it can have other parameters but all of these are assumed known). Also, let the support of $X$ not depend on $\mu$ to remove uniform – Marj Dec 3 '19 at 16:41
• @jbowman: Do you mean Uniform(0,a) or something? Uniform(0,1) is a single distribution and doesn't have a free parameter. – Lewian Dec 3 '19 at 16:49
• @Lewian - oh, right, of course. Thanks! – jbowman Dec 3 '19 at 16:50
• @Marj: Chances are one can still construct a counterexample based on the uniform. Take a mixture of 0.9*Uniform(0,2a)+0.1*something else symmetric about a with support on the full real line. Still the mean doesn't hold all information for estimating a. – Lewian Dec 3 '19 at 16:51

Consider the Laplace distribution with known scale. It is symmetric, continuous, and has support equal to the real line. The unknown location parameter $$\mu$$ is equal to the expected value. Yet the sample mean is not the sufficient statistic for $$\mu$$; the sufficient statistic in this case is the entire sample.