Jackknife Method for Simple Linear Regression: A Step-by-Step Mathematical Derivation

The Jackknife method is a powerful resampling technique used to estimate the bias and variance of an estimator, thereby improving the accuracy and reliability of parameter estimates in statistical models. In the context of simple linear regression, the Jackknife method can be effectively employed to estimate the intercept, slope, and error variance. This article provides a detailed, step-by-step derivation of the Jackknife method for simple linear regression, complete with mathematical formulas and explanations.

Introduction to the Simple Linear Regression Model

A simple linear regression model is a statistical model used to analyze the relationship between a dependent variable (response) and an independent variable (predictor). The model can be represented as:

Y β0 β1X ε

Where:

Y: The dependent variable or response variable.

X: The independent variable or predictor variable.

β0: The intercept, an unknown parameter to be estimated.

β1: The slope, another unknown parameter to be estimated.

ε: The error term, assumed to be normally distributed with mean 0 and variance σ2.

Step-by-Step Derivation of Jackknife Estimators

Step 1: Set up the Simple Linear Regression Model

The first step involves formulating the simple linear regression model:

Y β0 β1X ε

Step 2: Obtain the Original Parameter Estimates

Using the least squares method, the original parameter estimates can be obtained as follows:

β0 mean(Y) - β1 mean(X)

β1 Σ(Xi - mean(X))(Yi - mean(Y)) / Σ(Xi - mean(X))2

Where:

β0: Estimated intercept.

β1: Estimated slope.

Xi: The ith observation of the independent variable X.

Yi: The ith observation of the dependent variable Y.

mean(X): The mean of the independent variable X.

mean(Y): The mean of the dependent variable Y.

Step 3: Compute the Leave-One-Out Residuals

In the Jackknife method, leave-one-out resampling is performed. For each data point, one data point is removed from the dataset, and the regression coefficients are recomputed using the remaining data points. The residuals are then calculated as the difference between the observed values and the predicted values based on the reduced model.

Let n be the total number of data points. For each i from 1 to n:

Remove the ith data point Xi, Yi from the dataset.

Compute the reduced parameter estimates β-i and β-i using the remaining n-1 data points.

Calculate the leave-one-out residual ε-i for the ith observation:

ε-i  Yi - β-i - β-i Xi

Step 4: Calculate the Jackknife Estimators

The Jackknife estimators for the intercept, slope, and error variance are given by:

Jackknife intercept estimator: β0 (n-1) * mean(β-i) - β0

Jackknife slope estimator: β1 (n-1) * mean(β-i) - β1

Jackknife error variance estimator: σ2 (n-1) * mean(ε-i2) - σ2

Where:

β-i and β-i: Reduced parameter estimates.

ε-i: Leave-one-out residual.

mean(β-i): Mean of the reduced parameter estimates.

mean(ε-i2): Mean of the squared leave-one-out residuals.

σ2: Estimated error variance.

Step 5: Compute the Bias and Variance of the Parameter Estimates

The bias and variance of the parameter estimates can be calculated as follows:

Bias β0 mean(β-i) - β0

Bias β1 mean(β-i) - β1

Variance β0 (n-1) * mean((β-i - Jackknife intercept estimator)2)

Variance β1 (n-1) * mean((β-i - Jackknife slope estimator)2)

Variance σ2 (n-1) * mean((ε-i - Jackknife error variance estimator)2)

By following these steps and formulas, you can estimate the intercept, slope, and error variance of a simple linear regression model using the Jackknife method.