Wilcoxon Signed-Rank Statistic Explained

This question concerns the Wilcoxon signed-rank statistic, $T^+$, derived from a random sample $X_1, X_2, X_3$ ($n=3$) from a distribution that possesses two key properties: it is continuous and symmetric about 0.

The statistic is defined as $T^+ = \sum_{i=1, X_i>0}^3 R_i$. Here, $R_i$ represents the rank of the absolute value $|X_i|$ when compared to the set of absolute values $\{|X_1|, |X_2|, |X_3|\}$.

Understanding the setup involves these points:

• Symmetry: Because the distribution is symmetric about 0, the probability of an observation being positive is equal to the probability of it being negative, i.e., $P(X_i > 0) = P(X_i < 0) = 1/2$. Also, $P(X_i = 0) = 0$ since the distribution is continuous.
• Ranks: The ranks $R_1, R_2, R_3$ assigned to the absolute values are a permutation of the set $\{1, 2, 3\}$. Since the distribution is continuous, all absolute values $|X_i|$ will be distinct with probability 1.
• Possible Outcomes: There are $2^n = 2^3 = 8$ possible sign combinations for the sample $(X_1, X_2, X_3)$. For each sign combination, there are $n! = 3! = 6$ ways the ranks $\{1, 2, 3\}$ can be assigned to the absolute values $|X_1|, |X_2|, |X_3|$. This gives a total of $8 \times 6 = 48$ possible, equally likely outcomes to consider.

Calculating $P(T^+ = 3)$

We need to determine the probability that $T^+ = 3$. The value $T^+=3$ can arise in two distinct scenarios based on the number of positive observations in the sample:

1. Scenario 1: One positive $X_i$. In this case, $T^+$ is simply the rank ($R_i$) of that single positive observation. For $T^+$ to equal 3, the rank of the positive observation must be 3 ($R_i = 3$).
2. Scenario 2: Two positive $X_i$. Here, $T^+$ is the sum of the ranks of the two positive observations. For $T^+$ to equal 3, the ranks of these two observations must be 1 and 2 (since $1+2=3$).

Let's count the number of outcomes corresponding to each scenario out of the 48 total outcomes:

• Counting for Scenario 1 (One positive $X_i$, Rank = 3):
  • If $X_1 > 0$ and $X_2, X_3 < 0$: $T^+ = R_1$. We need $R_1 = 3$. This happens if the rank permutation assigns rank 3 to $|X_1|$. There are 2 such permutations: $(R_1, R_2, R_3) = (3,1,2)$ or $(3,2,1)$.
  • If $X_2 > 0$ and $X_1, X_3 < 0$: $T^+ = R_2$. We need $R_2 = 3$. This happens if the rank permutation assigns rank 3 to $|X_2|$. There are 2 such permutations: $(1,3,2)$ or $(2,3,1)$.
  • If $X_3 > 0$ and $X_1, X_2 < 0$: $T^+ = R_3$. We need $R_3 = 3$. This happens if the rank permutation assigns rank 3 to $|X_3|$. There are 2 such permutations: $(1,2,3)$ or $(2,1,3)$.
  • Total outcomes for Scenario 1 = $2 + 2 + 2 = 6$.
• Counting for Scenario 2 (Two positive $X_i$, Ranks sum to 3):
  • If $X_1, X_2 > 0$ and $X_3 < 0$: $T^+ = R_1 + R_2$. We need $R_1 + R_2 = 3$, which implies $\{R_1, R_2\} = \{1, 2\}$. This happens if the rank permutation assigns ranks 1 and 2 to $|X_1|$ and $|X_2|$. There are 2 such permutations: $(1,2,3)$ or $(2,1,3)$.
  • If $X_1, X_3 > 0$ and $X_2 < 0$: $T^+ = R_1 + R_3$. We need $R_1 + R_3 = 3$, implying $\{R_1, R_3\} = \{1, 2\}$. There are 2 such permutations: $(1,3,2)$ or $(2,3,1)$.
  • If $X_2, X_3 > 0$ and $X_1 < 0$: $T^+ = R_2 + R_3$. We need $R_2 + R_3 = 3$, implying $\{R_2, R_3\} = \{1, 2\}$. There are 2 such permutations: $(3,1,2)$ or $(3,2,1)$.
  • Total outcomes for Scenario 2 = $2 + 2 + 2 = 6$.
• Total outcomes for $T^+=3$: Summing the outcomes from both scenarios gives $6 + 6 = 12$.

The probability $P(T^+ = 3)$ is calculated by dividing the number of favorable outcomes by the total number of outcomes:

$P(T^+ = 3) = \frac{\text{Number of outcomes where } T^+=3}{\text{Total number of outcomes}} = \frac{12}{48} = \frac{1}{4}$

Analyzing Wilcoxon Probabilities and Variance

We can determine the probability distribution of $T^+$ for $n=3$. The possible values $T^+$ can take are sums of subsets of $\{1, 2, 3\}$, weighted by the sign combinations.

Using these probabilities:

• Option 3 Check: $P(T^+ > 3) = P(T^+=4) + P(T^+=5) + P(T^+=6) = 1/8 + 1/8 + 1/8 = 3/8$. The option stated $5/8$, which is incorrect.
• Option 4 Check: $P(T^+ > 4) = P(T^+=5) + P(T^+=6) = 1/8 + 1/8 = 2/8 = 1/4$. The option stated $1/8$, which is incorrect.
• Option 2 Check: The theoretical variance for the Wilcoxon signed-rank statistic is given by the formula $\text{Var}(T^+) = \frac{n(n+1)(2n+1)}{12}$. For $n=3$, $\text{Var}(T^+) = \frac{3(3+1)(2 \cdot 3 + 1)}{12} = \frac{3 \times 4 \times 7}{12} = \frac{84}{12} = 7$. The option stated $7/2$, which is incorrect.

Based on the detailed calculation, the statement $P(T^+ = 3) = \frac{1}{4}$ is the only true statement among the options.