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

Introduction

In linear regression, the estimated variance, denoted as $\hat{\sigma}^2$, is a crucial component in assessing the goodness of fit of the model and making predictions. The formula for $\hat{\sigma}^2$ 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.

Understanding the Formula

To derive the distribution of $\hat{\sigma}^2$, we need to understand the underlying assumptions of linear regression. The formula for $\hat{\sigma}^2$ is based on the normality assumption, which states that the residuals, $\mathbf{Y} - \mathbf{X} \hat{\beta}$, follow a normal distribution with mean 0 and variance $\sigma^2$. The formula can be rewritten as:

$$\hat{\sigma}^2 = \frac{1}{n - p - 1} \mathbf{e}^\top \mathbf{e}$$

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

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

To derive the distribution of $\hat{\sigma}^2$, we can use the following steps:

1. Show that $\hat{\sigma}^2$ is a quadratic form: We can rewrite the formula for $\hat{\sigma}^2$ as a quadratic form:

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

where $\mathbf{y}$ is the vector of response variables.

2. Show that $\hat{\sigma}^2$ is a function of a chi-squared random variable: We can show that $\hat{\sigma}^2$ is a function of a chi-squared random variable with $n - p - 1$ degrees of freedom.

3. Derive the distribution of $\hat{\sigma}^2$: Using the properties of the chi-squared distribution, we can derive the distribution of $\hat{\sigma}^2$.

Derivation

Let $\mathbf{e} = \mathbf{y} - \mathbf{X} \hat{\beta}$ be the vector of residuals. Then, we can show that:

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

Using the properties of the matrix $\mathbf{X}$, we can show that:

$$\mathbf{X}^\top \mathbf{X} = \sum_{i=1}^n \mathbf{x}_i \mathbf{x}_i^\top$$

where $\mathbf{x}_i$ is the $i$th row of the design matrix $\mathbf{X}$.

Using the properties of the chi-squared distribution, we can show that:

$$\frac{1}{n - p - 1} \mathbf{e}^\top \mathbf{e} \sim \frac{1}{n - p - 1} \chi^2_{n - p - 1}$$

where $\chi^2_{n - p - 1}$ is a chi-squared random variable with $n - p - 1$ degrees of freedom.

Conclusion

In this article, we have derived the distribution of the estimated variance in linear regression, denoted as $\hat{\sigma}^2$. We have shown that $\hat{\sigma}^2$ is a function of a chi-squared random variable with $n - p - 1$ degrees of freedom. This result is useful in assessing the goodness of fit of the model and making predictions.

References

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

Further Reading

• Linear Regression Analysis by G. A. F. Seber
• The Elements of Statistical Learning by T. Hastie, R. Tibshirani, and J. Friedman
• Elements of Statistical Learning by R. Tibshirani, E. Gelman, and J. Friedman
  Frequently Asked Questions (FAQs) about Deriving the Distribution of the Estimated Variance in Linear Regression ==============================================================================================

Q: What is the estimated variance in linear regression?

A: The estimated variance in linear regression, denoted as $\hat{\sigma}^2$, is a measure of the spread of the residuals from the fitted model. It is an important component in assessing the goodness of fit of the model and making predictions.

Q: What is the formula for the estimated variance in linear regression?

A: The formula for the estimated variance in linear regression 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.

Q: What is the distribution of the estimated variance in linear regression?

A: The distribution of the estimated variance in linear regression is a chi-squared distribution with $n - p - 1$ degrees of freedom.

Q: Why is the distribution of the estimated variance in linear regression important?

A: The distribution of the estimated variance in linear regression is important because it allows us to make inferences about the population variance, $\sigma^2$. It also provides a way to assess the goodness of fit of the model and make predictions.

Q: How can I use the distribution of the estimated variance in linear regression to make predictions?

A: To make predictions using the distribution of the estimated variance in linear regression, you can use the following steps:

1. Estimate the model: Estimate the linear regression model using the given data.
2. Calculate the estimated variance: Calculate the estimated variance, $\hat{\sigma}^2$, using the formula above.
3. Use the distribution of the estimated variance: Use the distribution of the estimated variance to make predictions about the population variance, $\sigma^2$.

Q: What are some common applications of the distribution of the estimated variance in linear regression?

A: Some common applications of the distribution of the estimated variance in linear regression include:

1. Hypothesis testing: The distribution of the estimated variance is used to test hypotheses about the population variance, $\sigma^2$.
2. Confidence intervals: The distribution of the estimated variance is used to construct confidence intervals for the population variance, $\sigma^2$.
3. Prediction: The distribution of the estimated variance is used to make predictions about the population variance, $\sigma^2$.

Q: What are some common mistakes to avoid when working with the distribution of the estimated variance in linear regression?

A: Some common mistakes to avoid when working with the distribution of the estimated variance in linear regression include:

1. Ignoring the degrees of freedom: Failing to account for the degrees of freedom when using the distribution of the estimated variance can lead to incorrect results.
2. Using the wrong distribution: Using the wrong distribution, such as a normal distribution, can lead to incorrect results.
3. Not accounting for non-normality: Failing to account for non-normality in the residuals can lead to incorrect results.

Q: What are some common tools and software used to work with the distribution of the estimated variance in linear regression?

A: Some common tools and software used to work with the distribution of the estimated variance in linear regression include:

1. R: R is a popular programming language and software environment for statistical computing and graphics.
2. Python: Python is a popular programming language and software environment for statistical computing and graphics.
3. SAS: SAS is a popular software environment for statistical analysis and data management.

Q: What are some common resources for learning more about the distribution of the estimated variance in linear regression?

A: Some common resources for learning more about the distribution of the estimated variance in linear regression include:

1. Textbooks: Textbooks such as "Elements of Statistical Learning" by Hastie, Tibshirani, and Friedman provide a comprehensive introduction to the distribution of the estimated variance in linear regression.
2. Online courses: Online courses such as "Linear Regression" on Coursera provide a comprehensive introduction to the distribution of the estimated variance in linear regression.
3. Research papers: Research papers such as "The Distribution of the Estimated Variance in Linear Regression" by Seber provide a comprehensive introduction to the distribution of the estimated variance in linear regression.