Let $X_1, X_2, \ldots, X_n$ be independent random variables following a common continuous distribution $F$, which is symmetric about $0$. For $i = 1, 2, \ldots, n$, define$S_i = \begin{cases} 1 & ext{if } X_i 0 \ -1 & ext{if } X_i

Question

Let $X_1, X_2, \ldots, X_n$ be independent random variables following a common continuous distribution $F$, which is symmetric about $0$. For $i = 1, 2, \ldots, n$, define
$S_i = \begin{cases} 1 & \text{if } X_i > 0 \\ -1 & \text{if } X_i < 0 \text{ and} \\ 0 & \text{if } X_i = 0 \end{cases}$
$R_i = \text{rank of } |X_i| \text{ in the set } \{|X_1|, \cdots, |X_n|\}$.
Which of the following statements are correct?

Prepp Team · Accepted Answer

The problem asks about the properties of derived random variables $S_i$ (signs) and $R_i$ (ranks) from a set of independent and identically distributed (i.i.d.) continuous random variables $X_1, \ldots, X_n$ whose distribution $F$ is symmetric about 0.

Analyzing Statement A: Independence and Identical Distribution of Signs ($S_i$)

We need to determine if $S_1, S_2, \ldots, S_n$ are independent and identically distributed.

Identical Distribution: Since $X_i$ are identically distributed with a continuous distribution $F$ symmetric about 0, we have $P(X_i > 0) = P(X_i < 0) = 1/2$. The definition of $S_i$ is $S_i=1$ if $X_i > 0$ and $S_i=-1$ if $X_i < 0$ (since $P(X_i=0)=0$ for continuous distributions). Therefore, $P(S_i=1) = 1/2$ and $P(S_i=-1) = 1/2$. This holds for all $i$.
Independence: Since the original random variables $X_1, \ldots, X_n$ are independent, their signs $S_1, \ldots, S_n$ are also independent.

Thus, $S_1, S_2, \ldots, S_n$ are independent and identically distributed.

Conclusion: Statement A is correct.

Analyzing Statement B: Independence and Identical Distribution of Ranks ($R_i$)

We need to determine if $R_1, R_2, \ldots, R_n$ are independent and identically distributed.

$R_i$ is the rank of $|X_i|$ among $\{|X_1|, \ldots, |X_n|\}$.
The ranks are derived from the ordered values of $|X_1|, \ldots, |X_n|$.
While the random variables $|X_1|, \ldots, |X_n|$ are i.i.d. (since $X_i$ are i.i.d.), their ranks $(R_1, \ldots, R_n)$ are not independent. For instance, if $R_1 = n$, it implies $|X_1|$ is the largest, which restricts the possible values for $R_2$. The vector of ranks $(R_1, \ldots, R_n)$ follows a distribution that is uniform over all permutations of $(1, \ldots, n)$, which implies exchangeability but not independence.

Conclusion: Statement B is incorrect.

Analyzing Statement C: Independence of Sign Vector ($S$) and Rank Vector ($R$)

We need to determine if the vector $S = (S_1, \ldots, S_n)$ is independent of the vector $R = (R_1, \ldots, R_n)$.

The distribution of signs $S_i$ is independent of the distribution of magnitudes $|X_i|$.
The probability $P(S_i = s_i)$ is $1/2$ for $s_i = \pm 1$, regardless of the specific distribution $F$ (as long as it's symmetric about 0).
The joint distribution of ranks $(R_1, \ldots, R_n)$ for i.i.d. continuous random variables is uniform over the $n!$ possible permutations. This distribution does not depend on the specific form of $F$.
Crucially, due to the symmetry of $F$ about 0, the sign of $X_i$ provides no information about the rank of $|X_i|$. The probability $P(S_i = s_i, R_i = r_i)$ factors into $P(S_i = s_i) P(R_i = r_i)$. This independence extends to the vectors $S$ and $R$.

Conclusion: Statement C is correct.

Analyzing Statement D: Distribution of $T = \sum_{i=1}^n S_i R_i$

We need to determine if the distribution of the statistic $T = \sum_{i=1}^n S_i R_i$ depends on the functional form of $F$.

The statistic $T$ is a form of a signed rank statistic.
The distribution of $T$ depends on the joint distribution of the sign vector $S$ and the rank vector $R$.
From statement A, $S_i$ are i.i.d. with $P(S_i = \pm 1) = 1/2$. This distribution is independent of $F$.
From statement C, $S$ and $R$ are independent.
The distribution of the rank vector $R$ is uniform over permutations, which is also independent of the specific form of $F$.
Since the joint distribution of $(S, R)$ depends only on these distribution-free marginal distributions and their independence, the distribution of the statistic $T = \sum_{i=1}^n S_i R_i$ is independent of the functional form of $F$. This is a fundamental property exploited in non-parametric tests like the Wilcoxon signed-rank test.

Conclusion: Statement D is correct.