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Chapter 9: Problem 9

Show that \(\sum_{i=1}^{n}\left[Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}+\sum_{i=1}^{n}\left[Y_{i}-\hat{\alpha}-\hat{\beta}\left(x_{i}-\bar{x}\right)\right]^{2}\)

Short Answer

Expert verified
The equation \(\sum_{i=1}^{n}[Y_{i}-\alpha-\beta(x_{i}-\bar{x})]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2}\sum_{i=1}^{n}(x_{i}-\bar{x})^2+\sum_{i=1}^{n}[Y_{i}-\hat{\alpha}-\hat{\beta}(x_{i}-\bar{x})]^{2}\) has been proven by expanding out the terms and equating the left hand side and the right hand side of the equation.

Step by step solution

01

Understand the elements and symbols in the equation

Here \(Y_i\) denotes the dependent variable of the \(i^{th}\) observation, \(x_i\) is the independent variable of the \(i^{th}\) observation, \(\alpha\) and \(\beta\) are the actual intercept and slope coefficients respectively, \(\hat{\alpha}\) and \(\hat{\beta}\) are the estimated intercept and slope coefficients, and \(\bar{x}\) is the mean of the \(x_i\)s.
02

Expand the left side of the equation

Expand the term on the left hand side of the equation. It becomes \(\sum_{i=1}^{n}Y_i^2 -2\alpha\sum_{i=1}^{n}Y_i -2\beta\sum_{i=1}^{n}(x_i - \bar{x})Y_i + 2\alpha\beta\sum_{i=1}^{n}(x_i - \bar{x}) +\alpha^2n + \beta^2\sum_{i=1}^{n}(x_i - \bar{x})^2\).
03

Expand the right side of the equation

Expand out the right side of the equation. We will get 3 terms: \(n(\hat{\alpha}-\alpha)^2, (\hat{\beta}-\beta)^2\sum_{i=1}^{n}(x_i - \bar{x})^2, \sum_{i=1}^{n}[Y_i - \hat{\alpha} - \hat{\beta}(x_i - \bar{x})]^2.\)
04

Break down the terms

The first term on the right side can be broken down to: \(n\hat{\alpha}^2 - 2n\alpha\hat{\alpha} + n\alpha^2\). The second term can be broken down to: \(\hat{\beta}^2\sum_{i=1}^{n}(x_i - \bar{x})^2 - 2\beta\hat{\beta}\sum_{i=1}^{n}(x_i - \bar{x})^2 + \beta^2\sum_{i=1}^{n}(x_i - \bar{x})^2.\) The third term will retain its original form.
05

Equate the left side and the right side

Equating the left side of the equation with the right side and rearranging the terms you will notice that every term on the left side has a correspondent term on the right side, hence the equality holds, proving the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Squares Estimation
The method of Least Squares Estimation lies at the heart of linear regression analysis. It's a statistical procedure used to estimate the coefficients of a linear equation that minimize the sum of the squared differences between the observed values and the values predicted by the equation.

Consider a set of points in a two-dimensional space. We want to find the best straight line that passes through these points. 'Best' in this case means the line that results in the smallest possible sum of the squares of the vertical distances (residuals) from the points to the line. Mathematically, if we have a dependent variable, usually denoted as \(Y\), and an independent variable \(x\), the least squares technique provides us with estimates \(\hat{\alpha}\) and \(\hat{\beta}\) for the true coefficients \(\alpha\) and \(\beta\) in the linear model equation \(Y = \alpha + \beta x\).

This estimation is powerful because it provides the 'best' linear unbiased estimates under the Gauss-Markov theorem, provided certain assumptions are met.
Statistical Regression
Statistical regression is a form of predictive modelling technique which analyzes the relationship between a dependent (target) and independent (predictor) variables. The term 'regression' in statistical language refers to the ability to predict the value of the dependent variable based on the values of the independent variables.

In the simplest form called linear regression, the model predicts the dependent variable using a linear function of the independent variable. However, regression can be more complex and involve multiple independent variables (multiple regression) or non-linear relationships (non-linear regression).

The linear regression equation can be expressed as \(Y = \alpha + \beta x + \epsilon\), where \(\epsilon\) represents the error term, which covers the discrepancy between the observed and the predicted values. The beauty of regression lies in its ability to offer insights into how changes in the independent variables influence the dependent variable, which is invaluable in many scientific, economic, and social research scenarios.
Parameter Estimation
Parameter Estimation is a central process in statistical analysis, where you determine the values of the parameters of a model that make the model best fit the empirical data. This routine process involves using data to make informed guesses about the population parameters.

In the context of linear regression, the model parameters are the intercept \(\alpha\) and the slope \(\beta\). We estimate these parameters using a sample of data and calculation methods such as the Least Squares Estimation discussed earlier. Accurate parameter estimation involves two aspects: the point estimation, which gives us a single best guess of the parameters, and the interval estimation, which provides a range within which the parameter is expected to lie with a certain level of confidence.

Effective parameter estimation not only provides predictions but also indicates the significance and the strength of the relationship between the variables within the model. This is crucial for testing hypotheses and making decisions based on data.

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Most popular questions from this chapter

Using the notation of Section \(9.2\), assume that the means \(\mu_{j}\) satisfy a linear function of \(j\), namely, \(\mu_{j}=c+d[j-(b+1) / 2] .\) Let independent random samples of size \(a\) be taken from the \(b\) normal distributions having means \(\mu_{1}, \mu_{2}, \ldots, \mu_{b}\), respectively, and common unknown variance \(\sigma^{2}\). (a) Show that the maximum likelihood estimators of \(c\) and \(d\) are, respectively, \(\hat{c}=\bar{X}_{. .}\) and $$ \hat{d}=\frac{\sum_{j=1}^{b}[j-(b-1) / 2]\left(\bar{X}_{. j}-\bar{X}_{\cdots}\right)}{\sum_{j=1}^{b}[j-(b+1) / 2]^{2}} $$ (b) Show that $$ \begin{aligned} \sum_{i=1}^{a} \sum_{j=1}^{b}\left(X_{i j}-\bar{X}_{. .}\right)^{2}=& \sum_{i=1}^{a} \sum_{j=1}^{b}\left[X_{i j}-\bar{X}_{. .}-\hat{d}\left(j-\frac{b+1}{2}\right)\right]^{2} \\ &+\hat{d}^{2} \sum_{j=1}^{b} a\left(j-\frac{b+1}{2}\right)^{2} \end{aligned} $$ (c) Argue that the two terms in the right-hand member of part (b), once divided by \(\sigma^{2}\), are independent random variables with \(\chi^{2}\) distributions provided that \(d=0 .\) (d) What \(F\) -statistic would be used to test the equality of the means, that is, \(H_{0}: d=0 ?\)

Let \(A_{1}, A_{2}, \ldots, A_{k}\) be the matrices of \(k>2\) quadratic forms \(Q_{1}, Q_{2}, \ldots, Q_{k}\) in the observations of a random sample of size \(n\) from a distribution that is \(N\left(0, \sigma^{2}\right)\). Prove that the pairwise independence of these forms implies that they are mutually independent. Hint: \(\quad\) Show that \(\boldsymbol{A}_{i} \boldsymbol{A}_{j}=\mathbf{0}, i \neq j\), permits \(E\left[\exp \left(t_{1} Q_{1}+t_{2} Q_{2}+\cdots+t_{k} Q_{k}\right)\right]\) to be written as a product of the mgfs of \(Q_{1}, Q_{2}, \ldots, Q_{k}\).

Students' scores on the mathematics portion of the ACT examination, \(x\), and on the final examination in the first-semester calculus ( 200 points possible), \(y\), are: $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 25 & 20 & 26 & 26 & 28 & 28 & 29 & 32 & 20 & 25 \\ \hline y & 138 & 84 & 104 & 112 & 88 & 132 & 90 & 183 & 100 & 143 \\ \hline x & 26 & 28 & 25 & 31 & 30 & & & & & \\ \hline y & 141 & 161 & 124 & 118 & 168 & & & & & \\ \hline \end{array} $$ The data are also in the rda file regr1.rda. Use \(\mathrm{R}\) or another statistical package for computation and plotting. (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Obtain the residual plot and comment on the appropriateness of the model. (d) Find \(95 \%\) confidence interval for \(\beta\) under the usual assumptions. Comment in terms of the problem.

Let the independent random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) have, respectively, the probability density functions \(N\left(\beta x_{i}, \gamma^{2} x_{i}^{2}\right), i=1,2, \ldots, n\), where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal and no one is zero. Find the maximum likelihood estimators of \(\beta\) and \(\gamma^{2}\).

Let the independent normal random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) have, respectively, the probability density functions \(N\left(\mu, \gamma^{2} x_{i}^{2}\right), i=1,2, \ldots, n\), where the given \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal and no one of which is zero. Discuss the test of the hypothesis \(H_{0}: \gamma=1, \mu\) unspecified, against all alternatives \(H_{1}: \gamma \neq 1, \mu\) unspecified.
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