Question

Let \(X_{n}\) and \(Y_{n}\) have a bivariate normal distribution with parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\) (free of \(n\) ) but \(\rho=1-1 / n\). Consider the conditional distribution of \(Y_{n}\), given \(X_{n}=x .\) Investigate the limit of this conditional distribution as \(n \rightarrow \infty\). What is the limiting distribution if \(\rho=-1+1 / n ?\) Reference to these facts is made in the Remark in Section \(2.4 .\)

Short answer

The limiting conditional distribution of \(Y_{n}\) given \(X_{n} = x\) as \(n\) tends to infinity for \(\rho = 1 - 1/n\) is a normal distribution with mean \(\mu_2 + \sigma_2 (\frac{x - \mu_1}{\sigma_1})\) and variance \(0\). If \(\rho = -1 + 1/n\), the limiting distribution is a normal distribution with mean \(\mu_2 - \sigma_2 (\frac{x - \mu_1}{\sigma_1})\) and variance \(0\).

Step-by-step solution

Find the Conditional Distribution

The conditional distribution of \(Y_{n}\) given \(X_{n} = x\) from a bivariate normal distribution is normally distributed where the mean equals \(\mu_2 + \rho \sigma_2 (\frac{x - \mu_1}{\sigma_1})\) and variance equals \(\sigma_2^2 (1 - \rho^2)\).

Compute Limit of Conditional Distribution as n tends to infinity

Replacing given values of \(\rho = 1 - 1/n\), the conditional mean equals \(\mu_2 + (1 - 1/n) \sigma_2 (\frac{x - \mu_1}{\sigma_1}) = \mu_2 + \sigma_2 (\frac{x - \mu_1}{\sigma_1}) - \frac{\sigma_2 (\frac{x - \mu_1}{\sigma_1})}{n}\) and the conditional variance equals \(\sigma_2^2(1-\rho^2) = \sigma_2^2 (1 - (1 - 1/n)^2) = \sigma_2^2 (1 - 1 + 2/n - 1/n^2) = \frac{2\sigma_2^2}{n} - \frac{\sigma_2^2}{n^2}\). As \(n\) tends to infinity, the term \(- \frac{\sigma_2 (\frac{x - \mu_1}{\sigma_1})}{n}\) tends to zero in mean and \(\frac{2\sigma_2^2}{n} - \frac{\sigma_2^2}{n^2}\) also tends to zero. Thus, the conditional limit of the distribution is a normal distribution with mean \(\mu_2 + \sigma_2 (\frac{x - \mu_1}{\sigma_1})\) and variance 0.

Compute Limiting Based on New Given Rho

Now consider \(\rho = -1 + 1/n\), resulting in a similar case except the sign of the mean term will be negative. The conditional mean now becomes \(\mu_2 + (-1 + 1/n) \sigma_2 (\frac{x - \mu_1}{\sigma_1})\) which tends to \(\mu_2 - \sigma_2 (\frac{x - \mu_1}{\sigma_1})\). The conditional variance remains the same as before which still tends to 0. Thus, the limiting distribution is a normal distribution with mean \(\mu_2 - \sigma_2 (\frac{x - \mu_1}{\sigma_1})\) and variance 0.

Key concepts

Conditional Distribution

The conditional distribution of a bivariate normal distribution, such as the one given in the exercise, is a concept where we are looking at the distribution of one variable, say \( Y_n \), given that we know the value of another variable, \( X_n \). When dealing with such distributions, it's essential to understand both the mean and the variance of the conditional distribution. For bivariate normals, these are calculated using specific formulas which involve parameters like \( \mu_1, \mu_2, \sigma_1, \sigma_2, \) and \( \rho \), the correlation coefficient.

• The conditional mean \( E(Y_n\mid X_n = x) \) is given by \( \mu_2 + \rho \sigma_2 \left(\frac{x - \mu_1}{\sigma_1}\right) \).
• The conditional variance is \( \sigma_2^2 (1 - \rho^2) \).

The exercise asks us to take a deeper look into this concept by investigating what happens to the conditional distribution as \( n \to \infty \) when the correlation is defined as a function of \( n \). This provides insight into how closely related \( X_n \) and \( Y_n \) become as the correlation approaches perfect correlation. Understanding conditional distributions helps us interpret statistical relationships in practical, real-world data.

Limiting Distribution

As \( n \) approaches infinity, the behavior of the conditional distribution changes significantly due to the expression for the correlation \( \rho = 1 - 1/n \) or \( \rho = -1 + 1/n \). These values of \( \rho \) imply that the correlation coefficient is tending towards 1 or -1, respectively, making \( X_n \) and \( Y_n \) strongly correlated.When \( \rho = 1 - 1/n \), the conditional mean simplifies because the term involving \( 1/n \) diminishes. Notably, the conditional variance tends to zero as \( n \) increases. In statistical terms, this implies that the conditional distribution becomes more concentrated around its mean, reflecting a near-perfect linear relationship.

• The limiting distribution, therefore, emerges as this situation progresses, showing that the conditional means converge to determinate values while their variances shrink to zero. Ultimately, this leads to behavior akin to one variable perfectly predicting the other.
• If \( \rho = -1 + 1/n \), the result is similar in that the variance still trends toward zero, but the orientation or sign of the mean is reversed, still indicating strong predictive capability between variables.

Understanding limiting distributions is crucial for analyzing how statistical models behave under idealized conditions, particularly concerning large sample or parameter conditions.

Correlation Coefficient

The correlation coefficient \( \rho \) is a measure that indicates the strength and direction of a linear relationship between two random variables. In the context of a bivariate normal distribution, \( \rho \) plays an essential role in determining both the mean and variance of the conditional distribution. It ranges from -1 to 1, where:

• A \( \rho \) close to 1 indicates a strong positive linear relationship.
• A \( \rho \) close to -1 signifies a strong negative linear relationship.
• A \( \rho \) around 0 suggests little to no linear relationship.

When \( \rho = 1 - 1/n \) or \( \rho = -1 + 1/n \), we are examining how changes in \( n \) influence the correlation, and as \( n \) becomes very large, \( \rho \) approaches 1 or -1. This results in strong deterministic relationships between variables. Specifically, in the exercise:

• As \( \rho \) approaches 1, variables additively predict each other with diminishing variability.
• Similarly, as \( \rho \) approaches -1, they predictively mirror each other in an inversely consistent pattern.

Understanding correlation coefficients within the bivariate framework is fundamental for modeling and interpreting data, particularly in applications involving prediction and regression analysis.