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Exercise 1 Suppose a new case (Yo,xo) is added to the data (Y,X). Show that the new error sum of squares will increase...
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be...
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be a limit point of X, and let yo e Y be a limit point of Y. Let f : X+Y be a function such that f(xo) = yo, and such that f is differentiable at Xo. Suppose that g:Y + R is a function which is differentiable at yo. Then the function gof:X + R is differentiable at xo, and .. (gºf)'(xo) = g'(yo)...
1. Problem 1. Given a data set (X, Y), use the least squares techniques to find the best ftting c...
1. Problem 1. Given a data set (X, Y), use the least squares techniques to find the best ftting curve y-/() within the exponential family v ab) where a,b ER. The data set is given by, where, χ-io, 0.2, 0.4, 0.6, 08, 09, 1, 12, 14, 1.6] y 2, 2.5, 3.1, 3.9, 4.8, 5.4, 6, 7.5, 9.3, 11.6] In particular: a) Going from the original data (x,Y) set to the transformed data set (X, Z) with z(Y)-In (Y), verify that...
(a) Suppose you are given the following (x, y) data pairs. x 1 2 6 y...
(a) Suppose you are given the following (x, y) data pairs. x 1 2 6 y 4 3 7 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (b) Now suppose you are given these (x, y) data pairs. x 4 3 7 y 1 2 6 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (c) In the data for parts...
(a) Suppose you are given the following (x, y) data pairs. x 1 2 5 y...
(a) Suppose you are given the following (x, y) data pairs. x 1 2 5 y 2 1 8 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (b) Now suppose you are given these (x, y) data pairs. x 2 1 8 y 1 2 5 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (c) In the data for parts (a) and (b), did...
Consider the following: (a) Suppose you are given the following x, y data pairs. x 2...
Consider the following: (a) Suppose you are given the following x, y data pairs. x 2 1 6 y 6 7 8 Find the least-squares equation for these data. (Use 3 decimal places.) = + x (b) Now suppose you are given these x, y data pairs. x 6 7 8 y 2 1 6 Find the least-squares equation for these data. (Use 3 decimal places.) = + x (c) In the data for parts (a) and (b), did we simply exchange...
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed...
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
(a) Suppose you are given the following (x, y) data pairs
(a) Suppose you are given the following (x, y) data pairs x136y217Find the least-squares equation for these data (rounded to three digits after the decimal) (b) Now suppose you are given these (x, y) data pairs. x217y136Find the least-squares equation for these data (rounded to three digits after the decimal). (c) In the data for parts (a) and (b), did we simply exchange the x and y values of each data pair? Yes No (d) Solve your answer from part (a) for x (rounded to...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose...
2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose however that we knew, for a fact, that ßı = 0. (a) What does the linear regression model look like, algebraically, if ßı = 0? (b) What does the linear regression model look like, graphically, if ßı = 0? (c) If Bi=0 the least squares "sum of squares" function becomes S(R2) = Gyi - B2x;)?. Using the data, x 1 2 3 4 5...
Question text Suppose that you have a five-point sample data set; the observations of (x, y)...
Question text Suppose that you have a five-point sample data set; the observations of (x, y) are given by (8, 3), (10, 3), (6, 2), (2, 0), and (2, 1). Fit a simple linear regression model to this data by first computing the least squares estimate of the slope parameter. Which of the following is the most accurate? Select one: a. 0.3438 b. 0.4728 d. 0.6712
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be a limit point of X, and let yo e Y be a limit point of Y. Let f : X+Y be a function such that f(xo) = yo, and such that f is differentiable at Xo. Suppose that g:Y + R is a function which is differentiable at yo. Then the function gof:X + R is differentiable at xo, and .. (gºf)'(xo) = g'(yo)...
1. Problem 1. Given a data set (X, Y), use the least squares techniques to find the best ftting curve y-/() within the exponential family v ab) where a,b ER. The data set is given by, where, χ-io, 0.2, 0.4, 0.6, 08, 09, 1, 12, 14, 1.6] y 2, 2.5, 3.1, 3.9, 4.8, 5.4, 6, 7.5, 9.3, 11.6] In particular: a) Going from the original data (x,Y) set to the transformed data set (X, Z) with z(Y)-In (Y), verify that...
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose however that we knew, for a fact, that ßı = 0. (a) What does the linear regression model look like, algebraically, if ßı = 0? (b) What does the linear regression model look like, graphically, if ßı = 0? (c) If Bi=0 the least squares "sum of squares" function becomes S(R2) = Gyi - B2x;)?. Using the data, x 1 2 3 4 5...
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