Question

Let \(Y_{1}, \ldots, Y_{n} \stackrel{\mathrm{iid}}{\sim} N\left(\mu, \sigma^{2}\right)\) with \(\sigma^{2}\) known. Show that \(\left(Y_{1}-\bar{Y}, \ldots, Y_{n}-\bar{Y}\right)\) is distribution constant, and deduce that \(\bar{Y}\) and \(\sum\left(Y_{j}-\bar{Y}\right)^{2}\) are independent.

Short answer

\(\bar{Y}\) and \(\sum (Y_j - \bar{Y})^2\) are independent due to orthogonal transformations.

Step-by-step solution

Express Individual Deviations from the Mean

Start by expressing the deviations of each sample from the sample mean \( \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i \). So each deviation is: \( Y_i - \bar{Y} \). These deviations essentially measure how far each data point is from their average.

Understand the Distribution of Deviations

Note that the sum of the deviations from the mean is zero:\[ \sum_{i=1}^{n} (Y_i - \bar{Y}) = \sum_{i=1}^{n} Y_i - n \bar{Y} = n\bar{Y} - n\bar{Y} = 0. \] This property indicates that once the mean \( \bar{Y} \) is known, the deviations \( Y_1 - \bar{Y}, \ldots, Y_n - \bar{Y} \) are not dependent on \( \mu \), and thus their distribution is centered around zero, suggesting a constant distribution characteristic.

Analyze the Independence of \( \bar{Y} \)

The sample mean \( \bar{Y} \) is the average of the normally distributed i.i.d variables, hence \( \bar{Y} \sim N(\mu, \sigma^2/n) \). Since \( \bar{Y} \) is essentially the component in the deviations \( Y_i - \bar{Y} \) whose sum is zero, it indicates independence from any variation in \( \sum (Y_i - \bar{Y})^2 \).

Deduce Independence using Orthogonal Transformation

Transform to orthogonal coordinates, where the first coordinate is \( \bar{Y} \) capturing total information about the mean, and remaining \( n-1 \) coordinates as \( Z_i = Y_i - \bar{Y} \). These coordinates are orthogonal transformations, separating the mean calculation from sum of squares calculation \( \sum (Y_i - \bar{Y})^2 \). Such transformations ensure that \( \bar{Y} \) and \( \sum (Y_i - \bar{Y})^2 \) are independent since they describe independent modes of variance in the data.

Key concepts

Deviation from the Mean

In a set of data, deviation from the mean helps us to understand how individual data points differ from the average value. For a group of random variables that are independent and identically distributed (i.i.d) with a normal distribution, like \(Y_1, Y_2, \ldots, Y_n\), we often look at each one’s deviation: \(Y_i - \bar{Y}\). This expression illustrates how far each \(Y_i\) is from the sample mean \(\bar{Y}\).Calculating deviations involves these steps:

• First, find the sample mean \(\bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i\).
• Then, subtract \(\bar{Y}\) from each individual value \(Y_i\) to get \(Y_i - \bar{Y}\).

This computation helps in identifying if the data is spread out or clustered closely around the mean.An important property of these deviations is that their sum equals zero: \(\sum_{i=1}^{n} (Y_i - \bar{Y}) = 0\). This balance indicates that the deviations are symmetrically dispersed about the mean, meaning there are equal amounts of data above and below it.

Orthogonal Transformation

Orthogonal transformation is a technique used in statistics to simplify complex data into uncorrelated components. In our given scenario involving normal distributions, orthogonal transformations help us to decompose the data into the sample mean and deviations from the mean.Here's how it works:

• First, the sample mean \(\bar{Y}\) acts as an axis representing the average value of the distributions.
• Next, each deviation \(Y_i - \bar{Y}\) forms new axes (coordinates), which are "orthogonal" to the mean. This implies they are independent of it.

By organizing our data using such transformations, each new coordinate, or axis, represents a unique and separate source of variability. As a result, the transformations simplify into a form where the mean and the sum of squares of deviations \(\sum (Y_i - \bar{Y})^2\) are statistically independent.This independence is crucial as it enables separate analysis of the central tendency and the spread (or variance) of the data without them influencing each other.

Sample Mean

The sample mean, represented as \(\bar{Y}\), is a core concept in statistics. When dealing with random variables that are i.i.d and normally distributed, the sample mean gives us a concise measure of the average value for this set.To calculate it, follow these steps:

• Add up all the values from your sample data: \(Y_1 + Y_2 + \ldots + Y_n\).
• Divide by the number of data points \(n\) to find the average: \(\bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i\).

The sample mean is normally distributed itself, with the distribution \(N(\mu, \sigma^2/n)\). This distribution characteristic ensures that as sample size \(n\) increases, the sample mean becomes a more accurate estimate of the population mean \(\mu\).Furthermore, in the orthogonal transformation discussed earlier, \(\bar{Y}\) becomes an independent parameter, separate from the sum of squared deviations. This independence aids significantly in statistical inference, such as hypothesis testing, because it implies that knowledge of \(\bar{Y}\) alone will not provide information about the spread of the data.