Question

Let \( T \) be a complete and sufficient statistic for a family \( \mathcal{P} \) of distributions and let \( U \) be a sufficient statistic for \( \mathcal{P} \). If \( P_f(T \geq 0) = 1 \) for all \( f \in \mathcal{P} \), then which one of the following options is NOT necessarily correct?

• A. \( T^2 \) is a complete statistic for \( \mathcal{P} \)
• B. \( T^2 \) is a minimal sufficient statistic for \( \mathcal{P} \)
• C. \( T \) is a function of \( U \)
• D. \( U \) is a function of \( T \)

Hint:

For sufficient and complete statistics, be cautious in assuming that one statistic must be a function of another. A complete and sufficient statistic often contains all the information about the parameter, but it does not imply that other sufficient statistics are functions of it.

Answer 1

The Correct Option is D

Step 1: Understanding the properties of complete and sufficient statistics.
A statistic \( T \) is {complete} if for any measurable function \( g \), \( E[g(T)] = 0 \) implies \( P(g(T) = 0) = 1 \).
A statistic \( T \) is {sufficient} for a family of distributions if the conditional distribution of the sample given \( T \) does not depend on the parameter \( f \). The fact that \( T \) is complete and sufficient means that it contains all the information about the parameter \( f \). Moreover, if \( U \) is a sufficient statistic, \( T \) may or may not be a function of \( U \). 
Step 2: Analyzing the options.
Option (A): \( T^2 \) is still a complete statistic because completeness is preserved under one-to-one transformations.
Option (B): \( T^2 \) is a minimal sufficient statistic because \( T \) is minimal, and any one-to-one transformation of a minimal sufficient statistic remains minimal.
Option (C): \( T \) being a function of \( U \) is not necessarily true. While \( T \) is complete and sufficient, it is not guaranteed to be a function of another sufficient statistic \( U \).
Option (D): \( U \) being a function of \( T \) is not necessarily true. Since \( T \) is complete and sufficient, and \( U \) is merely sufficient, \( U \) does not have to be a function of \( T \). Thus, the correct answer is \( \boxed{(D)} \).