10.14 One way to show that the ordinary least squares estimate is the best linearunbiased estimator (BLUE) is to verify that the condition of Zyskind (1967)applies. The Zyskind condition states that for a model y X3 + ε withE(e) = 0 and cov(e) = V, the ordinary least squares estimate (X'X)¯¹X'yis BLUE if and only if there exists a matrix Q such that VX = = XQ.

The given information is about the Zyskind condition, which is a criterion to determine whether the…

Answered: One way to show that the ordinary least…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 30E

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Suppose the x-coordinates of the data (x1, V1), .., (x,, Vn) are in mean deviation form, so that E x; = 0. Show that if X is the design matrix for the least-squares line in this case, then XTX is a diagonal matrix.
5. Using the least squares approximation, the best fit an exponential curve y = ea(x-b) to the given data: (2, 7), (3, 17), (5, 49), (6, 71), defined by the form a) Y = Cx² + Bx + A b) Y = Bx + A c) Y = Cx² + A d) None of the above a O b C O d
5. Using the least squares approximation, the best fit an exponential curve y = e(x-b) to the given data: (2, 7), (3, 17), (5, 49), (6, 71), defined by the form a) Y = Cx² + Bx + A b) Y = Bx + A c) Y = Cx² + A d) None of the above
Q.2: The population of the germs grown in a specific lab environment has been measured to be 600, 700, and 1000 at times t 0, 1, and 2 hours. Find the curve which gives the best least squares fit to the given population points. Find the coefficients C and D of the best curve. Also proceed to find the error vector that would demonstrate the measurement error / tolerance in population sizem.
y y 90 54 50 53 80 91 35 41 60 48 35 61 60 71 40 56 60 71 55 68 40 47 65 36 55 53 35 11 50 68 60 70 65 57 90 79 50 79 35 59 A data set consist of dependent variable (y) and independent variable (x) as shown above. It is claims that the relationship between the x and y can be modelled through a regression model as follows: ŷ = a+bx where a and b are the estimated values for a and ß (refer to Appendix). (i) Determine the equation of the regression line to predict the y value from the x value. (ii) If the x value is 75, what is the value of y? (iii) Test the hypothesis that a=10 against the alternative a <10. Use a 0.05 level of significance. (iv) Construct a 95% prediction interval for the y with x=35.
(5.4) (10) Use the least-squares method to find the orthogonal projection of b = [−2 2 1]T onto the column space of the matrix A. A = 0 1 1 1 1 2 projs b = ??
6. Consider the regression model Yi = βXi + ui, where ui and Xi satisfy the least squares assumptions in Key Concept 4.3. Let β ̄ denote an estimator of β that is constructed as β ̄ = Y ̄/X ̄, where Y ̄ and X ̄ are the sample means of Yi and Xi, respectively. (a) Show that β ̄ is a linear function of Y1 , ..., Yn . (b) Show that β ̄ is conditionally unbiased.
Show that the first least squares assumption, E(ui | Xi) = 0, implies thatE(Yi | Xi) = β0 + β1Xi.
2. Suppose you are to fit a linear model which is forced to have an intercept equal to 5, of the form given by: Y₁ = 5 + B₁X₁ + &; for i = 1,2,.., n where & has mean 0 and variance o² for any value of X. a) Derive the least squares estimator, ₁ of the slope ₁. b) Find E(₁) and o² (B₁). c) Derive the maximum likelihood estimator, ₁ of the slope ₁. Is it the same as the least squares estimator? d) What is the estimator of the variance of the random error term? e) Prove whether or not it is still true that Σ₁ e₁ = 0.
A certain experiment produces the data (1,7.9), (2,5.4), and (3, –.9) · Describe the model that produces a least-squares fit of these points by a function of the form y = A cos x + B sin x
(10] 16. Find the least squares solution of As-b where A=01 [11] 119
14.3 (2.14) Consider the two points p₁ = (1,0), P2 = (2, 1) in R². (a) Let m € R. Find a point p3 € R² so that the least squares approximation to {P₁, P2, P3} has slope m. (b) Let a ≥ 3 € R. Find points p3, P4 € R² so that the degree 2 least squares approxi- mation to {P₁, P2, P3, P4} has its vertex on the line x = a.

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