/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 The angles of the triangle \(\ma... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 3

The angles of the triangle \(\mathrm{ABC}\) are measured with \(\mathrm{A}\) and \(\mathrm{B}\) each measured twice and \(\mathrm{C}\) three times. All the measurements are independent and unbiased with common variance \(\sigma^{2}\). Find the least squares estimates of the angles \(\mathrm{A}\) and \(\mathrm{B}\) based on the seven measurements and calculate the variance of these estimates.

Short Answer

Expert verified
The estimates are \(\hat{A} = \frac{1}{2}(Y_1 + Y_2)\), \(\hat{B} = \frac{1}{2}(Y_3 + Y_4)\), with variances \(\frac{\sigma^2}{2}\).

Step by step solution

01

State the problem mathematically

We are given angles \(A, B,\) and \(C\) of a triangle where the sum of all angles is \(180^\circ\). Angles \(A\) and \(B\) are measured twice, and angle \(C\) is measured three times. The problem is to find the least squares estimates for \(A\) and \(B\) based on these seven measurements, knowing they have a common variance \(\sigma^2\).
02

Define the measurements

Let each independent measurement of \(A\), \(B\), and \(C\) be represented as follows:\[ Y_1 = A + \epsilon_1, \quad Y_2 = A + \epsilon_2 \]\[ Y_3 = B + \epsilon_3, \quad Y_4 = B + \epsilon_4 \]\[ Y_5 = C + \epsilon_5, \quad Y_6 = C + \epsilon_6, \quad Y_7 = C + \epsilon_7 \]where \(\epsilon_i\) are independent with \(E[\epsilon_i] = 0\) and \(Var(\epsilon_i) = \sigma^2\).
03

Use Sum of Angles Property

Since \(A + B + C = 180^\circ\), we use this relationship to inform our least squares estimation approach.
04

Formulate the Least Squares Estimation

The least squares estimation problem can be written in matrix form as:\[Y = X\beta + \epsilon\]where:- \(Y = \begin{bmatrix} Y_1 & Y_2 & Y_3 & Y_4 & Y_5 & Y_6 & Y_7 \end{bmatrix}^T\)- \(X = \begin{bmatrix} 1 & 0 & 0 \ 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ 0 & 0 & 1 \ 0 & 0 & 1 \end{bmatrix}\)- \(\beta = \begin{bmatrix} A \ B \ C \end{bmatrix}\)- \(\epsilon = \begin{bmatrix} \epsilon_1 & \epsilon_2 & \epsilon_3 & \epsilon_4 & \epsilon_5 & \epsilon_6 & \epsilon_7 \end{bmatrix}^T\)
05

Calculate Least Squares Estimates

The least squares estimator \(\hat\beta\) is given by:\[\hat\beta = (X^TX)^{-1}X^TY\]Calculating:\[X^TX = \begin{bmatrix} 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix}\]\[(X^TX)^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & 0 \ 0 & \frac{1}{2} & 0 \ 0 & 0 & \frac{1}{3} \end{bmatrix}\]Hence:\[\hat\beta = \begin{bmatrix} \frac{1}{2}(Y_1 + Y_2) \ \frac{1}{2}(Y_3 + Y_4) \ \frac{1}{3}(Y_5 + Y_6 + Y_7) \end{bmatrix}\]
06

Compute Variances of Estimates

The variances of these estimates can be obtained by:\[Var(\hat\beta_i) = \sigma^2 [(X^TX)^{-1}]_{ii}\]This gives:- \(Var(\hat A) = \frac{\sigma^2}{2}\)- \(Var(\hat B) = \frac{\sigma^2}{2}\)
07

Conclusion

The least squares estimates for angles \(A\) and \(B\) are:\[\hat{A} = \frac{1}{2}(Y_1 + Y_2), \quad \hat{B} = \frac{1}{2}(Y_3 + Y_4)\]The variances of these estimates are \(\frac{\sigma^2}{2}\).

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Key Concepts

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

Variance Calculation
Understanding variance is crucial when working with data, particularly when evaluating estimates like the angles in a triangle problem. Variance measures how much the values of a dataset differ from the mean of that dataset. In the context of least squares estimation, variance helps us determine the precision of our estimates.
By knowing the variance, we can assess how much the estimates \(\hat{A}\) and \(\hat{B}\) might fluctuate due to variability in the measurements.
  • The variance of each measurement is given as \(\sigma^2\).
  • The variance for the estimates \(\hat{A}\) and \(\hat{B}\) is reduced due to averaging: \(Var(\hat{A}) = \frac{\sigma^2}{2}\).
  • Variance for \(\hat{B}\) follows the same formula: \(Var(\hat{B}) = \frac{\sigma^2}{2}\).
Notice that reducing variance in \(\hat{A}\) and \(\hat{B}\) means more reliable estimates, showing less deviation from their true values. Integrating variance into least squares estimation provides a better grasp of data reliability.
Independent Measurements
In statistical analyses, independent measurements are key for ensuring accurate results. When measurements are independent, they do not influence each other, which is essential for reducing biases and errors.
This property ensures that the estimated values for the triangle's angles are unbiased and accurate.
  • Each measurement of angles \(A, B,\) and \(C\) was taken independently, contributing individually to the estimate.
  • Measurement independence is denoted by separate error terms \(\epsilon_i\) with zero mean and variance \(\sigma^2\).
  • Independence simplifies variance calculations for estimates like \(\hat{A}\) and \(\hat{B}\), ensuring unbiased error distribution.
Understanding independent measurements is crucial because they uphold the integrity of the estimation process, which is vital when calculating least squares estimates.
Sum of Angles in a Triangle
The sum of angles in a triangle is always \(180^\circ\). This geometric principle is not just a fact but a useful tool when calculating least squares estimates as seen in the exercise.
By relying on this property, we can utilize all measurements efficiently and build logical constraints into calculations.
  • This rule helps tie the estimates of three independent angles, forming a constraint: \(A + B + C = 180^{\circ}\).
  • In the context of least squares, using this property ensures the estimates respect the geometry of a triangle.
  • Along with independent measurements, this principle complements our approach to find least squares estimates for \(A\) and \(B\).
Acknowledging the sum of angles constraint reinforces the validity and correctness of estimation methods, leading to more precise and reliable outcomes in geometrical problems.

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Most popular questions from this chapter

In the normal straight-line regression model it is thought that a power transformation of the covariate may be needed, that is, the model $$ y=\beta_{0}+\beta_{1} x^{(\lambda)}+\varepsilon $$ may be suitable, where \(x^{(\lambda)}\) is the power transformation $$ x^{(\lambda)}= \begin{cases}\frac{x^{\lambda}-1}{\lambda}, & \lambda \neq 0 \\\ \log x, & \lambda=0\end{cases} $$ (a) Show by Taylor series expansion of \(x^{(\lambda)}\) at \(\lambda=1\) that a test for power transformation can be based on the reduction in sum of squares when the constructed variable \(x \log x\) is added to the model with linear predictor \(\beta_{0}+\beta_{1} x\). (b) Show that the profile log likelihood for \(\lambda\) is equivalent to \(\ell_{\mathrm{p}}(\lambda) \equiv-\frac{n}{2} \log \operatorname{SS}\left(\widehat{\beta}_{\lambda}\right)\), where \(S S\left(\widehat{\beta}_{\lambda}\right)\) is the residual sum of squares for regression of \(y\) on the \(n \times 2\) design matrix with a column of ones and the column consisting of the \(x_{j}^{(\lambda)}\). Why is a Jacobian for the transformation not needed in this case, unlike in Example \(8.23 ?\) (Box and Tidwell, 1962 )

Data \(\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)\) satisfy the straight-line regression model (5.3). In a calibration problem the value \(y_{+}\)of a new response independent of the existing data has been observed, and inference is required for the unknown corresponding value \(x_{+}\)of \(x\). (a) Let \(s_{x}^{2}=\sum\left(x_{j}-\bar{x}\right)^{2}\) and let \(S^{2}\) be the unbiased estimator of the error variance \(\sigma^{2}\). Show that $$ T\left(x_{+}\right)=\frac{Y_{+}-\widehat{\gamma}_{0}-\widehat{\gamma}_{1}\left(x_{+}-\bar{x}\right)}{\left[S^{2}\left\\{1+n^{-1}+\left(x_{+}-\bar{x}\right)^{2} / s_{x}^{2}\right\\}\right]^{1 / 2}} $$ is a pivot, and explain why the set $$ \mathcal{X}_{1-2 \alpha}=\left\\{x_{+}: t_{n-2}(\alpha) \leq T\left(x_{+}\right) \leq t_{n-2}(1-\alpha)\right\\} $$ contains \(x_{+}\)with probability \(1-2 \alpha\). (b) Show that the function \(g(u)=(a+b u) /\left(c+u^{2}\right)^{1 / 2}, c>0, a, b \neq 0\), has exactly one stationary point, at \(\tilde{u}=-b c / a\), that sign \(g(\tilde{u})=\operatorname{sign} a\), that \(g(\tilde{u})\) is a local maximum if \(a>0\) and a local minimum if \(a<0\), and that \(\lim _{u \rightarrow \pm \infty} g(u)=\mp b .\) Hence sketch \(g(u)\) in the four possible cases \(a, b<0, a, b>0, a<0

Suppose that random variables \(Y_{g j}, j=1, \ldots, n_{g}, g=1, \ldots, G\), are independent and that they satisfy the normal linear model \(Y_{g j}=x_{g}^{\mathrm{T}} \beta+\varepsilon_{g j}\). Write down the covariate matrix for this model, and show that the least squares estimates can be written as \(\left(X_{1}^{\mathrm{T}} W X_{1}\right)^{-1} X_{1}^{\mathrm{T}} W Z\), where \(W=\operatorname{diag}\left\\{n_{1}, \ldots, n_{G}\right\\}\), and the \(g\) th element of \(Z\) is \(n_{g}^{-1} \sum_{j} Y_{g j} .\) Hence show that weighted least squares based on \(Z\) and unweighted least squares based on \(Y\) give the same parameter estimates and confidence intervals, when \(\sigma^{2}\) is known. Why do they differ if \(\sigma^{2}\) is unknown, unless \(n_{g} \equiv 1 ?\) Discuss how the residuals for the two setups differ, and say which is preferable for modelchecking.

Suppose that the straight-line regression model \(y=\beta_{0}+\beta_{1} x+\varepsilon\) is fitted to data in which \(x_{1}=\cdots=x_{n-1}=-a\) and \(x_{n}=(n-1) a\), for some positive \(a .\) Show that although \(y_{n}\) completely determines the estimate of \(\beta_{1}, C_{n}=0 .\) Is Cook's distance an effective measure of influence in this situation?

Consider a normal linear regression \(y=\beta_{0}+\beta_{1} x+\varepsilon\) in which the parameter of interest is \(\psi=\beta_{0} / \beta_{1}\), to be estimated by \(\widehat{\psi}=\widehat{\beta}_{0} / \widehat{\beta}_{1} ;\) let \(\operatorname{var}\left(\widehat{\beta}_{0}\right)=\sigma^{2} v_{00}, \operatorname{cov}\left(\widehat{\beta}_{0}, \widehat{\beta}_{1}\right)=\sigma^{2} v_{01}\) and \(\operatorname{var}\left(\widehat{\beta}_{1}\right)=\sigma^{2} v_{11}\) (a) Show that $$ \frac{\widehat{\beta}_{0}-\psi \widehat{\beta}_{1}}{\left\\{s^{2}\left(v_{00}-2 \psi v_{01}+\psi^{2} v_{11}\right)\right\\}^{1 / 2}} \sim t_{n-p} $$ and hence deduce that a \((1-2 \alpha)\) confidence interval for \(\psi\) is the set of values of \(\psi\) satisfying the inequality $$ \widehat{\beta}_{0}^{2}-s^{2} t_{n-p}^{2}(\alpha) v_{00}+2 \psi\left\\{s^{2} t_{n-p}^{2}(\alpha) v_{01}-\beta_{0} \beta_{1}\right\\}+\psi^{2}\left\\{\widehat{\beta}_{1}^{2}-s^{2} t_{n-p}^{2}(\alpha) v_{11}\right\\} \leq 0 $$ How would this change if the value of \(\sigma\) was known? (b) By considering the coefficients on the left-hand-side of the inequality in (a), show that the confidence set can be empty, a finite interval, semi- infinite intervals stretching to \(\pm \infty\), the entire real line, two disjoint semi-infinite intervals - six possibilities in all. In each case illustrate how the set could arise by sketching a set of data that might have given rise to it. (c) A government Department of Fisheries needed to estimate how many of a certain species of fish there were in the sea, in order to know whether to continue to license commercial fishing. Each year an extensive sampling exercise was based on the numbers of fish caught, and this resulted in three numbers, \(y, x\), and a standard deviation for \(y, \sigma\). A simple model of fish population dynamics suggested that \(y=\beta_{0}+\beta_{1} x+\varepsilon\), where the errors \(\varepsilon\) are independent, and the original population size was \(\psi=\beta_{0} / \beta_{1}\). To simplify the calculations, suppose that in each year \(\sigma\) equalled 25 . If the values of \(y\) and \(x\) had been \(\begin{array}{cccccc}y: & 160 & 150 & 100 & 80 & 100 \\ x: & 140 & 170 & 200 & 230 & 260\end{array}\) after five years, give a \(95 \%\) confidence interval for \(\psi\). Do you find it plausible that \(\sigma=25\) ? If not, give an appropriate interval for \(\psi\).
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