91Ó°ÊÓ

Consider a random sample of size \(n\) from a continuous distribution having median 0 so that the probability of any one observation being positive is \(.5\). Disregarding the signs of the observations, rank them from smallest to largest in absolute value, and let \(W=\) the sum of the ranks of the observations having positive signs. For example, if the observations are \(-.3,+.7,+2.1\), and \(-2.5\), then the ranks of positive observations are 2 and 3 , so \(W=5\). In Chapter \(15, W\) will be called Wilcoxon's signed-rank statistic. \(W\) can be represented as follows: where the \(Y_{i}\) 's are independent Bernoulli rv's, each with \(p=.5\left(Y_{l}=1\right.\) corresponds to the observation with rank \(i\) being positive). a. Determine \(E\left(Y_{i}\right)\) and then \(E(W)\) using the equation for \(W\). [Hint: The first \(n\) positive integers sum to \(n(n+1) / 2\).] b. Determine \(V\left(Y_{j}\right)\) and then \(V(W)\). [Hint: The sum of the squares of the first \(n\) positive integers can be expressed as \(n(n+1)(2 n+1) / 6 .]\)

Step by step solution

01

Define the Variables

We have a random variable \(W\) which is the sum of the ranks of observations that have positive signs. The indicator variables \(Y_i\) denote whether the \(i\)-th ranked observation is positive (1 if positive, 0 otherwise).

02

Calculate Expected Value of Y_i

Given that each \(Y_i\) follows a Bernoulli distribution with \(p = 0.5\), the expected value \(E(Y_i)\) is simply \(p\). Therefore, \(E(Y_i) = 0.5\).

03

Derive the Expected Value of W

The rank of a positive observation is multiplied by its corresponding \(Y_i\). Thus, \(W = \sum_{i=1}^{n} i Y_i\). Now, calculate \(E(W)\) using linearity of expectation: \[E(W) = \sum_{i=1}^{n} i E(Y_i) = \sum_{i=1}^{n} i \times 0.5 = 0.5 \cdot \frac{n(n+1)}{2} = \frac{n(n+1)}{4}.\]

04

Calculate Variance of Y_i

Since \(Y_i\) is a Bernoulli random variable, its variance is \(V(Y_i) = p(1-p) = 0.5(0.5) = 0.25\).

05

Derive the Variance of W

The variance of \(W\), when \(W = \sum_{i=1}^{n} i Y_i\), is \(V(W) = \sum_{i=1}^{n} i^2 V(Y_i)\) because the \(Y_i\)'s are independent. Hence: \[ V(W) = \sum_{i=1}^{n} i^2 \times 0.25 = 0.25 \cdot \frac{n(n+1)(2n+1)}{6} = \frac{n(n+1)(2n+1)}{24}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bernoulli Distribution

A Bernoulli distribution is one of the simplest probability distributions, and it models a random experiment with exactly two possible outcomes. These outcomes are often referred to as "success" and "failure", where the probability of success is denoted by the parameter \( p \). For instance, if we flip a coin, the experiment results in heads (success) with probability \( p \) and tails (failure) with probability \( 1-p \).
In the context of the Wilcoxon Signed-Rank Test, each observation's sign is treated as a Bernoulli random variable \( Y_i \), where a positive observation is a "success" with probability \( p = 0.5 \). This means each observation has an equal chance of being positive or negative, making the distribution fair based on this hypothesis.
The Bernoulli distribution is critical for analyzing the expected outcomes and variability of any situation where there are only two possible states. It allows us to model situations efficiently and helps in calculating expected values and variances.

Expected Value

The expected value, or mean, of a random variable gives us a measure of its center or average. In simpler terms, it tells us what value we would expect to see if we repeated an experiment an infinite number of times. For a Bernoulli random variable \( Y_i \) with parameter \( p = 0.5 \), the expected value is the probability of success, which is \( E(Y_i) = 0.5 \).
This concept is utilized in the Wilcoxon Signed-Rank Test to calculate the expected value of \( W \). The random variable \( W \) is the sum of the ranks of positive observations: \( W = \sum_{i=1}^{n} i Y_i \). With the property of the expectation operator being linear, we can calculate the expected value of \( W \) as:
\[ E(W) = \sum_{i=1}^{n} i \times 0.5 = 0.5 \cdot \frac{n(n+1)}{2}. \]
Thus, \( E(W) \) is given by \( \frac{n(n+1)}{4} \), providing an average or expected sum of ranks for positive observations.

Variance

Variance is a measure of how spread out the values of a random variable are around their mean, or expected value. The variance of a Bernoulli random variable \( Y_i \), with success probability \( p \), is calculated as \( V(Y_i) = p(1-p) \). Given \( p = 0.5 \), the variance \( V(Y_i) \) becomes \( 0.25 \).
Understanding variance is important for evaluating the stability or reliability of a measurement. In the Wilcoxon Signed-Rank Test, we compute the variance of the statistic \( W \), where \( W = \sum_{i=1}^{n} i Y_i \). The independence of the \( Y_i \)'s allows us to sum their variances, adjusted by their ranks, as:
\[ V(W) = \sum_{i=1}^{n} i^2 \times 0.25 = 0.25 \cdot \frac{n(n+1)(2n+1)}{6}. \]
Therefore, the variance of \( W \) is \( \frac{n(n+1)(2n+1)}{24} \), giving insight into the dispersion of the rank sum of positive observations.

Random Sample

A random sample is a subset of a population selected in such a way that each member has an equal chance of being chosen. In statistics, it's essential for ensuring that our results are not biased and are representative of the whole population.
For the Wilcoxon Signed-Rank Test, we consider a random sample of observations from a continuous distribution. Each observation is treated independently, with the median set to 0, and each has an equal probability of being positive or negative.
Random sampling ensures the validity and reliability of statistical tests. It allows us to infer results about a larger population based on the analysis of the sample, keeping the results unbiased and generalizable.

Rank Statistics

Rank statistics are non-parametric and involve arranging data into an order and analyzing the ranks rather than the raw data. This technique is particularly useful when the data do not follow a normal distribution.
In the given scenario, we consider the absolute values of observations, rank them, and use their ranks for further analysis. Specifically, we sum the ranks of positive observations to compute the statistic \( W \) in the Wilcoxon Signed-Rank Test.
Rank statistics have several advantages:

• They are distribution-free, meaning they don't assume a specific distribution form.
• They are robust against outliers and non-normal data.
• Their analysis focuses on the order rather than the actual values, making them widely applicable.

This approach simplifies complex datasets into easily interpretable results, providing a straightforward method for hypothesis testing.