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i. Based on features in this plot, would you choose a cubic smoothing spline approach to estimate of thefunctionf(x) over the intervalx∈[0,1]?Justify your answer, using cubic smoothing spline assump-tions.This is the figure from a simulation assigned in one of your homeworks - Chapter 14 Q6.No - the data seem to indicate a point of discontinuity in functionf(or very rapid change at ascale smaller than the smallest interval betweenxi’s). Cubic smoothing spline assumptions includecontinuity of the functionf(x) (and other assumptions - see definition of functionAin slides. Basedon the figure, it does not seem reasonable to assume thatf(x) is continuous.ii. It is possible to fit a Gaussian linear regression model that would capture the features in the data above.Write down the form of an “X-matrix”with full column rankthat you could create that would be ableto capture the features of the figure above if you fitted the modelY=Xβ+with∼Nn(0, σ2In).Hint: You may assume that the “jump” occurs atx= 0.5in your answer.Note: there are multiple ways of creating such an “X” matrix. I only ask you to give one of these.7
iii. Suppose you use ordinary least squares (i.e., minimizing the sum of squared residuals) to obtain estimatorˆβof the parameter vectorβin part (5(a)ii). Write down the expected value and the variance ofˆβ.You do not need to show any working for this part if you know the answers - you can immediately writethem down.You can answer this and the remainder of the question even if you could not finish part(5(a)ii), by simply writing your answers using generic matrix “X” notation.The question does not ask for the formula in blue, but it is included for completeness.