How To Derive The Distribution Of $ \hat{\sigma}^2$

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Introduction

In the context of linear regression, the residual sum of squares (RSS) is a crucial component in estimating the variance of the error term. The formula for the estimated variance, denoted as $\hat{\sigma}^2$, is given by:

$$\hat{\sigma}^2 = \frac{1}{n - p - 1} ( \mathbf{Y} - \mathbf{X} \hat{\beta} )^\top ( \mathbf{Y} - \mathbf{X} \hat{\beta} )$$

where $\mathbf{Y}$ is the vector of response variables, $\mathbf{X}$ is the design matrix, $\hat{\beta}$ is the vector of estimated coefficients, $n$ is the number of observations, and $p$ is the number of predictors.

In this article, we will delve into the derivation of the distribution of $\hat{\sigma}^2$. This involves understanding the properties of the RSS and its relationship with the chi-squared distribution.

Properties of the Residual Sum of Squares

The RSS is defined as the sum of the squared differences between the observed response variables and the predicted values. Mathematically, it can be expressed as:

$$\text{RSS} = ( \mathbf{Y} - \mathbf{X} \hat{\beta} )^\top ( \mathbf{Y} - \mathbf{X} \hat{\beta} )$$

The RSS has several important properties that are essential in understanding its distribution. Firstly, the RSS is a quadratic form, which means it can be expressed as the product of a matrix and a vector. Specifically, the RSS can be written as:

$$\text{RSS} = \mathbf{e}^\top \mathbf{e}$$

where $\mathbf{e}$ is the vector of residuals.

Distribution of the Residual Sum of Squares

The distribution of the RSS is closely related to the chi-squared distribution. To establish this relationship, we need to consider the properties of the residuals. Specifically, we assume that the residuals are normally distributed with mean 0 and variance $\sigma^2$. This assumption is crucial in linear regression, as it allows us to make inferences about the population parameters.

Under this assumption, the distribution of the RSS can be shown to be:

$$\frac{\text{RSS}}{\sigma^2} \sim \chi^2_{n-p-1}$$

where $\chi^2_{n-p-1}$ denotes the chi-squared distribution with $n-p-1$ degrees of freedom.

Derivation of the Distribution of $\hat{\sigma}^2$

Now that we have established the distribution of the RSS, we can derive the distribution of $\hat{\sigma}^2$. Recall that $\hat{\sigma}^2$ is defined as:

$$\hat{\sigma}^2 = \frac{1}{n - p - 1} ( \mathbf{Y} - \mathbf{X} \hat{\beta} )^\top ( \mathbf{Y} - \mathbf{X} \hat{\beta} )$$

Using the properties of the RSS, we can rewrite $\hat{\sigma}^2$ as:

$$\hat{\sigma}^2 = \frac{\text{RSS}}{n - p - 1}$$

Since the RSS is distributed as $\sigma^2 \chi^2_{n-p-1}$, we can substitute this expression into the formula for $\hat{\sigma}^2$:

$$\hat{\sigma}^2 = \frac{\sigma^2 \chi^2_{n-p-1}}{n - p - 1}$$

This expression shows that $\hat{\sigma}^2$ is distributed as a scaled chi-squared distribution with $n-p-1$ degrees of freedom.

Conclusion

In this article, we have derived the distribution of $\hat{\sigma}^2$, which is a crucial component in linear regression. We have shown that $\hat{\sigma}^2$ is distributed as a scaled chi-squared distribution with $n-p-1$ degrees of freedom. This result has important implications for making inferences about the population parameters in linear regression.

References

• Hastie, T., Tibshirani, R., & Friedman, J. (2009). Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer.
• Seber, G. A. F. (1977). Linear Regression Analysis. Wiley.

Appendix

For readers who are interested in the technical details of the derivation, we provide a brief appendix below.

A.1 Derivation of the Distribution of the RSS

The derivation of the distribution of the RSS involves showing that the RSS is distributed as $\sigma^2 \chi^2_{n-p-1}$. This can be done by using the properties of the residuals and the normal distribution.

A.2 Derivation of the Distribution of $\hat{\sigma}^2$

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Frequently Asked Questions

In this article, we will address some of the most common questions related to deriving the distribution of $\hat{\sigma}^2$. Whether you are a student, researcher, or practitioner, this Q&A section will provide you with a deeper understanding of the topic.

Q: What is the significance of the distribution of $\hat{\sigma}^2$?

A: The distribution of $\hat{\sigma}^2$ is crucial in linear regression as it provides a measure of the variability of the error term. Understanding its distribution is essential for making inferences about the population parameters and for conducting hypothesis tests.

Q: How is the distribution of $\hat{\sigma}^2$ related to the chi-squared distribution?

A: The distribution of $\hat{\sigma}^2$ is related to the chi-squared distribution through the residual sum of squares (RSS). Specifically, the RSS is distributed as $\sigma^2 \chi^2_{n-p-1}$, where $\chi^2_{n-p-1}$ denotes the chi-squared distribution with $n-p-1$ degrees of freedom.

Q: What is the formula for the distribution of $\hat{\sigma}^2$?

A: The formula for the distribution of $\hat{\sigma}^2$ is:

$$\hat{\sigma}^2 = \frac{\sigma^2 \chi^2_{n-p-1}}{n - p - 1}$$

Q: What are the assumptions required for the derivation of the distribution of $\hat{\sigma}^2$?

A: The derivation of the distribution of $\hat{\sigma}^2$ requires the following assumptions:

• The residuals are normally distributed with mean 0 and variance $\sigma^2$.
• The design matrix $\mathbf{X}$ has full column rank.

Q: How can I apply the distribution of $\hat{\sigma}^2$ in practice?

A: The distribution of $\hat{\sigma}^2$ can be applied in practice by using it to make inferences about the population parameters. For example, you can use the distribution of $\hat{\sigma}^2$ to:

• Conduct hypothesis tests about the population variance.
• Construct confidence intervals for the population variance.
• Make predictions about the future values of the response variable.

Q: What are some common mistakes to avoid when deriving the distribution of $\hat{\sigma}^2$?

A: Some common mistakes to avoid when deriving the distribution of $\hat{\sigma}^2$ include:

• Failing to check the assumptions required for the derivation.
• Using an incorrect formula for the distribution of $\hat{\sigma}^2$.
• Ignoring the degrees of freedom when applying the distribution of $\hat{\sigma}^2$.

Q: Where can I find more information about deriving the distribution of $\hat{\sigma}^2$?

A: You can find more information about deriving the distribution of $\hat{\sigma}^2$ in the following resources:

• Hastie, T., Tibshirani, R., & Friedman, J. (2009). Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer.
• Seber, G. A. F. (1977). Linear Regression Analysis. Wiley.
• Online tutorials and courses on linear regression and statistical inference.

Conclusion

In this Q&A article, we have addressed some of the most common questions related to deriving the distribution of $\hat{\sigma}^2$. We hope that this article has provided you with a deeper understanding of the topic and has helped you to apply the distribution of $\hat{\sigma}^2$ in practice.