Problem 14 - An Introduction to Mathematical Statistics and Its Applications

Question

In some regression situations, there are a priori reasons for assuming that the \(x y\)-relationship being approximated passes through the origin. If so, the equation to be fit to the \(\left(x_{i}, y_{i}\right)\) 's has the form \(y=b x\). Use the least squares criterion to show that the "best" slope in that case is given by $$ b=\frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}} $$

Step-by-Step Solution

Answer

The slope of the best fit line through the origin, as per least squares criterion, is given by \(b = \frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}}\).

Step 1: Understand the Least Squares Criteria

The principle of least squares suggests that the sum of the squares of the residuals (the differences between the observed and predicted values) should be minimum. This is the criterion that is used to fit the best line in regression analysis. When the line passes through the origin, it simplifies the model.

Step 2: Understanding the Model Equation

Given the model equation \(y = bx\), where \(b\) is the slope or coefficient. Our task is to find an expression for \(b\) using the least squares criterion.

Step 3: Applying the Least Squares Criterion

Applying the least squares criterion, we minimize the sum of squares of the residuals, which is represented mathematically as \( \min \sum_{i=1}^{n}(y_{i} - bx_{i})^{2} \). To find the minimum, we take the derivative of the sum of squared residuals function with respect to \(b\), set it to zero, and solve for \(b\). This gives us \(\frac{d}{db} \sum_{i=1}^{n}(y_{i} - bx_{i})^{2} = 0 \). Simplifying this derivative, we find that \(b = \frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}}\).

Step 4: Confirming the Result

This last step serves to confirm that the slope of the best fit line through the origin as determined by the least squares criterion is indeed given by \(b = \frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}}\).

Key Concepts

Least Squares Method

In regression analysis, the Least Squares Method is a fundamental approach used to determine the best fit line through a set of data points. The key idea is to minimize the sum of the squares of the residuals. These residuals are essentially the differences between the observed data points and the values predicted by the model.
This approach ensures that the overall error between the data and the model is as small as possible. When the line must pass through the origin, as in the specific exercise given, the equation simplifies to a linear model of the form \( y = bx \). Here, the objective is finding the best possible slope \( b \) by minimizing the total error squared.

• The function to minimize: \( \sum_{i=1}^{n}(y_{i} - bx_{i})^{2} \)
• Take the derivative with respect to \( b \) to find the optimal slope

By applying calculus, the derivative is set to zero to find the value of \( b \) that results in the lowest sum of squares of the residuals, thus providing the most accurate possible fit.

Slope Calculation

The calculation of the slope \( b \) is a crucial step in forming our regression model. Given the formula \( y = bx \), \( b \) represents the slope or the rate of change in \( y \) for a unit change in \( x \). Unlike other scenarios where an intercept is present, here the slope is calculated directly assuming the line passes through the origin.

• The slope equation derives from the set criterion: \( b = \frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}} \)
• This formula arises by addressing the least squares condition, which accounts for both \( x \) and \( y \) values.

The numerator, \( \sum_{i=1}^{n} x_{i} y_{i} \), is the sum of the products of the data pairs \( x \) and \( y \), reflecting how each point contributes to the line in relation to their product. Meanwhile, the denominator, \( \sum_{i=1}^{n} x_{i}^{2} \), assesses the variance or dispersion of \( x \) values. Essentially, this part helps to normalize the slope considering just the variability of \( x \).

Mathematical Modeling

Mathematical modeling in the context of regression serves as a way to create an equation that accurately represents the observed data. The prime objective of a model like \( y = bx \) is to predict or analyze the relationship between variables.
Mathematical models simplify complex real-world situations into a manageable form using mathematical expressions. When assuming the line passes through the origin, the derivative equations involved further streamline the model.

• The model is defined and constrained by conditions like having no intercept.
• It uses real data to estimate a mathematical relationship between variables.

These models are instrumental because they provide predictive insights and allow for the forecasting of future observations. By crafting such a model based on regression analysis, particularly using the least squares criterion, one attains a precise depiction of trends within the data. These insights can then be utilized in fields like economics, engineering, and the natural sciences for solving real-world problems.