# Completeness of a statistic in a truncated distribution

Suppose a random sample $$x_1,\dots, x_n$$ (i.i.d.) from a random variable $$X$$ defined over $$(\Omega,\mathcal{F},P)$$ whose probability density function is $$f(x_1,\dots, x_n;\theta)$$ and $$T(x_1,\dots, x_n)$$ is a complete statistic for this family. Let $$f_A(x_1,\dots, x_n;\theta)$$ be the probability density function associated with $$P$$ when we truncate it over $$A$$, a subset of the sample space $$\Omega$$, given by $$f_A(x_1,\dots, x_n;\theta) = \frac{f(x_1,\dots, x_n;\theta)I_{A}(x_1,\dots, x_n)}{P_\theta (A)}$$ such that $$I_{A}(x_1,\dots, x_n)$$ is the indicator function that is 1 when every point belongs to $$A$$ and 0 if at least one isn't and $$P_\theta(A)$$ is the probability of set $$A$$ that acts as a normalizing constant. Show that $$T(x_1,\dots, x_n)$$ is also complete for the truncated family.

I was studying mathematical statistics and found this problem. The problem set also had a exercise about showing that if $$T$$ is sufficient for a distribution then it is sufficient for the truncated one that is easily solved by using Fisher-Neyman factorization. I tried to show it through the definition of completeness, $$\mathbb{E}_\theta(g(t))0 \xrightarrow{} P(g(t)=0)=1$$, but this takes me nowhere as we know nothing about the pdf of $$T$$.

Consulting the Theory of Point Estimation (Lehmann and Casella, 1999) from which this exercise is taken

the second question contains the term in addition, which means that $$T$$ is sufficient.

If $$T$$ is incomplete and sufficient for the family of truncated distributions, there exists a non-zero function $$g$$ such that the expectation of $$g(T)$$ is zero for all $$f_A(\cdot;\theta)$$'s. This implies $$\mathbb E_\theta[g(T(X_1,\ldots,X_n)\mathbb I_A(X_1,\ldots,X_n)]=0\quad\forall\theta$$ for the untruncated distributions.

Hint 1: Apply the law of the total expectation to the above to derive a function of $$T$$ with expectation zero

Hint 2: Show that $$g$$ is not constant.

• Thanks for the answer! The first part is very clear to me and also the logic of hint 2, however i don't get it how should i condition using the law of the total expectations. Would it be on $X_1,\dots,X_n \in A$? – Sergio Andrade Nov 12 '20 at 11:53
• You have to solve Hint 1. The term inside the expectation need be turned into a function of $T$ only. – Xi'an Nov 12 '20 at 12:05