Let yi denote the observed data point and ?! denote the predicted value. Suppose we are...
Let yi denote the observed data point and ?! denote the predicted value. Suppose we are going to model-predicted value using an equation of a straight line.
That means yi = ? + bxi
Using “Least-square Criteria”, derive formulas for coefficients ? and ? showing all the steps in the derivation.
Homework Answers
and
b = 
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Let yi denote the observed data point and ?! denote the
predicted value. Suppose we are...
(e) Let Wi denote the random response for the ith observation and denote the associated random error. Suppose the c...
(e) Let Wi denote the random response for the ith observation and denote the associated random error. Suppose the constant variance assumption has been violated such that Var(e) is proportional to (y) = H . That is, Var(€;) = Var(yi) = Chw where c is some constant. Consider a transformed response variable, g(x). We can write the first-order Taylor expansion of g() at Hy, as 9(y) (y) + (y) (yi - My), (1) where g'( ) is the first derivative...
Consider simple linear regression with n pairs of numbers xi, yi . Let β0 + gx...
Consider simple linear regression with n pairs of numbers xi, yi . Let β0 + gx be the least squares line where Ao--gx and β,-rosy/s . În terms of the summary statistics, derive a simple expression for the residual standard deviation [Σ- n-2) 12 where For a question like this one that involves a derivation, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances,...
Suppose we have collected a random sample from our population, denoted by (xi , yi), i...
Suppose we have collected a random sample from our population, denoted by (xi , yi), i = 1, . . . , n. We now fit a least-squares line: yˆi = βˆ 0 + βˆ 1xi (i = 1, . . . , n). What additional assumption do we need in order to carry out statistical inference on our least square estimators βˆ 0 and βˆ 1? c. Using the results we’ve derived in class, prove that the sum of...
1. Given data on (yi, xi) for i = 1, , n, consider the following least...
1. Given data on (yi, xi) for i = 1, , n, consider the following least square problem for a imple linear regression bo,b We assume the four linear regression model assumptions dicussed in class hold (i) Compute the partial derivatives of the objective function. (ii) Put the derived partial derivatives in (i) equal to zeros. Explain why the resulting equa tions are called normal equation'. (Hin wo n-dimesional vectors (viand (wi)- are normal-orthogonal ) if Σ-1 ui wi-0. )...
Problem 1. Suppose we have a finite population of bivariate data, so the population consists of...
Problem 1. Suppose we have a finite population of bivariate data, so the population consists of N pairs (You can think of these as pairs of measurements on a particular object-height and age of an individual, for instance.) Let μΖ and μy be the population means of the z-and y-measurements, respectively, and σ and σ, denote the population variances of the r-and y-measurements. So, for instance, and the corresponding formulas hold for Finally, let and σ ー1 denote the population...
R STUDIO Create a simulated bivariate data set consisting of n 100 (xi, yi) pairs: Generate...
R STUDIO
Create a simulated bivariate data set consisting of n 100 (xi, yi) pairs: Generate n random a-coordinates c from N(0, 1) Generate n random errors, e, from N(0, o), using o 4. Set yiBoB1x; + , Where Bo = 2, B1 = 3, and eN(0, 4). (That is, y is a linear function of , plus some random noise.) (Now we have simulated data. We'll pretend that we don't know the true y-intercept Bo 2, the true slope...
2 2. Suppose we are given data on n observations (i, Y),, and we have a...
2
2. Suppose we are given data on n observations (i, Y),, and we have a linear model, so that E(X)-A, + ßiri-Let呙-SXY /SXX and β') = F-β,2 be the least-square estimates given in lecture (a) Show that E(SXY)-ASXX and E (T)-A] + β,7. (b) Use (a) to show that E(角)-βι and E(A) = 3). In other words, these are unbiased estimators. (c) The fitted values Yt = Atari are used as estimates of E(A), and the residuals e.-Yi for...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
Suppose we are given data on n observations (zi, Y4), i-1, . . . , n,...
Suppose we are given data on n observations (zi, Y4), i-1, . . . , n, and we have a linear model so that E(X)-β0+B1zi. Let A-SXY/SXX and A-Y-Aī be the least-square estimates given in lecture. (a) Show that E(Sxy)-ASxx and E(Y)-A +AF (b) Use (a) to show that E(BB and E(B)In other words, these are unbiased estimators (c) The fitted values Yi-A+Azi are used as estimates of %), and the residuals e,-x-Y; are used as surrogates for the unobservable...
Question 1 Suppose we wanted to predict the selling price of a house using its size...
Question 1 Suppose we wanted to predict the selling price of a house using its size in a certain area of a city. A random sample of six houses were selected from the area. The data is presented in the following table with size given in hundreds of square feet, and sale price in thousands of dollars. Size (Xi) 12 15 18 21 24 27 Price (Yi) 60 85 75 105 120 110 a) Find the least squares estimate for the...
(e) Let Wi denote the random response for the ith observation and denote the associated random error. Suppose the constant variance assumption has been violated such that Var(e) is proportional to (y) = H . That is, Var(€;) = Var(yi) = Chw where c is some constant. Consider a transformed response variable, g(x). We can write the first-order Taylor expansion of g() at Hy, as 9(y) (y) + (y) (yi - My), (1) where g'( ) is the first derivative...
Consider simple linear regression with n pairs of numbers xi, yi . Let β0 + gx be the least squares line where Ao--gx and β,-rosy/s . În terms of the summary statistics, derive a simple expression for the residual standard deviation [Σ- n-2) 12 where For a question like this one that involves a derivation, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances,...
1. Given data on (yi, xi) for i = 1, , n, consider the following least square problem for a imple linear regression bo,b We assume the four linear regression model assumptions dicussed in class hold (i) Compute the partial derivatives of the objective function. (ii) Put the derived partial derivatives in (i) equal to zeros. Explain why the resulting equa tions are called normal equation'. (Hin wo n-dimesional vectors (viand (wi)- are normal-orthogonal ) if Σ-1 ui wi-0. )...
Problem 1. Suppose we have a finite population of bivariate data, so the population consists of N pairs (You can think of these as pairs of measurements on a particular object-height and age of an individual, for instance.) Let μΖ and μy be the population means of the z-and y-measurements, respectively, and σ and σ, denote the population variances of the r-and y-measurements. So, for instance, and the corresponding formulas hold for Finally, let and σ ー1 denote the population...
R STUDIO
Create a simulated bivariate data set consisting of n 100 (xi, yi) pairs: Generate n random a-coordinates c from N(0, 1) Generate n random errors, e, from N(0, o), using o 4. Set yiBoB1x; + , Where Bo = 2, B1 = 3, and eN(0, 4). (That is, y is a linear function of , plus some random noise.) (Now we have simulated data. We'll pretend that we don't know the true y-intercept Bo 2, the true slope...
2
2. Suppose we are given data on n observations (i, Y),, and we have a linear model, so that E(X)-A, + ßiri-Let呙-SXY /SXX and β') = F-β,2 be the least-square estimates given in lecture (a) Show that E(SXY)-ASXX and E (T)-A] + β,7. (b) Use (a) to show that E(角)-βι and E(A) = 3). In other words, these are unbiased estimators. (c) The fitted values Yt = Atari are used as estimates of E(A), and the residuals e.-Yi for...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
Suppose we are given data on n observations (zi, Y4), i-1, . . . , n, and we have a linear model so that E(X)-β0+B1zi. Let A-SXY/SXX and A-Y-Aī be the least-square estimates given in lecture. (a) Show that E(Sxy)-ASxx and E(Y)-A +AF (b) Use (a) to show that E(BB and E(B)In other words, these are unbiased estimators (c) The fitted values Yi-A+Azi are used as estimates of %), and the residuals e,-x-Y; are used as surrogates for the unobservable...
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