PROBLEM 3 3. Considerthefollowingmodelfor{(xi,yi)}: Р Yi = Bo+Bjoj (xi) + ɛi for i j=1 = 1, n for some basis functions 1(x), ..., Op(x) where the parameters Bo, B₁, · Р Bo+Bjoj (xi) ≥ 0 for i j=1 = 1, ... n " To estimate ẞo, ẞ1, ..., ßp, we used constrained least squares: Minimize subject to the constraints n Р Σ Yi βο - Σ β;φ; (α;) Σβ,Φ,(;) i=1 Bo+Bjoj (xi) ≥ 0 for i = j=1 = 1, 2 ... n Bp are such that The parameter estimates can be computed using an interior point algorithm whereby we define (Bo(r), · · ·‚ßp(r)) to minimize - Bo - Р 2 n Р ½Σ ( − − £¾(x)* −Ë (+Ź 8,6,(z) j=1 ―r In (xi) i=1 j=1 for r > 0 and let r↓0. (Note that we are implicitly assuming that the unconstrained LS estimates of the parameters violate the constraints.) (a) Derive an IRLS algorithm for computing (Bo(r),···, ßp(r)) for r > 0. 3 (b) The file problem3-data.txt on Quercus contains 1000 observations (xi, yi) where 0 ≤ X1,X1000 ≤ 1. Take p 10 and define 1(x), …, 10(x) to be B-spline functions; in R, = these can be defined as follows: > library(splines) > xx <- bs(x, df=10) The object xx will be matrix with 10 columns with the number of rows equal to the length of x. Compute the constrained LS estimates for these data. (The unconstrained LS estimates violate some of the non-negativity constraints.) 4

Solved: PROBLEM 3 3. Considerthefollowingmodelfor{(xi,yi)}: Р Yi = Bo+Bjoj (x

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problem 3 3 considerthefollowingmodelfor xi yi r yi bo bjoj xi i for i

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PROBLEM 3 3. Considerthefollowingmodelfor{(xi,yi)}: Р Yi = Bo+Bjoj (xi) + ɛi for i j=1 = 1, n for some basis functions 1(x), ..., Op(x) where the parameters Bo, B₁, · Р Bo+Bjoj (xi) ≥ 0 for i j=1 = 1, ... n " To estimate ẞo, ẞ1, ..., ßp, we used constrained least squares: Minimize subject to the constraints n Р Σ Yi β