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 \( \boxed{(D)} \).
Three villages P, Q, and R are located in such a way that the distance PQ = 13 km, QR = 14 km, and RP = 15 km, as shown in the figure. A straight road joins Q and R. It is proposed to connect P to this road QR by constructing another road. What is the minimum possible length (in km) of this connecting road?
Note: The figure shown is representative.
For the clock shown in the figure, if
O = O Q S Z P R T, and
X = X Z P W Y O Q,
then which one among the given options is most appropriate for P?
“His life was divided between the books, his friends, and long walks. A solitary man, he worked at all hours without much method, and probably courted his fatal illness in this way. To his own name there is not much to show; but such was his liberality that he was continually helping others, and fruits of his erudition are widely scattered, and have gone to increase many a comparative stranger’s reputation.” (From E.V. Lucas’s “A Funeral”)
Based only on the information provided in the above passage, which one of the following statements is true?