How do I know if a sufficient statistic is also complete?

Question

For example, for an i.i.d. sample of random variables $X_i$ distributed according to a normal distribution, I found a sufficient statistic—the sample mean. How do I know if this is also complete? Thank you.

Answer 1

I am sure you know the formal definition of completeness, so I will try to offer some intuition on the subject.

Informally a statistic $T(X)$ is complete if two different parameters $\theta$ of the distribution of X, cannot give rise to the same distribution for $T(X)$.

The distribution of sample mean $\bar X$ is $N(\mu,\sigma^2/n)$, which is different if you change either $\mu$ or $\sigma^2$. So by the above intuition, it is complete.

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1. Consider $1_{X_1<0}$. The distribution of this is $Bernoulli$ with $p_{\mu,\sigma}=\int_{-\infty}^0 \frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2/2\sigma^2} dx$, which can be made same for different choice of $\mu$ and $\sigma$. Hence it is not complete.

2. $\bar X^2$ is not complete either because it has same distribution for $\mu$ and $-\mu$. Note also that it would have been complete, if it was known that $\mu \in [0,\infty)$.

3. $X_1-X_2$ is not complete. Because its distribution is $N(0,2\sigma^2)$, and is same for different $\mu$'s.

4. $X_1+2X_2$ is however, complete. Because its distribution $N(3\mu,5\sigma^2)$ cannot be same for different combinations of $\mu$ and $\sigma$.

I hope that helps. Examples above are assuming that the variance $\sigma^2$ is also a parameter.