2. (15 points) This preliminary exercise prepares you to perform and visualize confidence set
estimation using MLE. For concreteness we focus on the case d=2. We are interested in
visualizing (when d=2 ) the ellipsoidal set NC(c,):={y=(y1,,yd)Rd:(yc)C(yc)2} where C=(cij) be
a symmetric (strictly) positive definite dd matrix, cRd and >0. The boundary NC(c,) of NC(c,) is
NC(c,)={y=(y1,,yd)Rd:(yc)C(yc)=2} (a) Suppose C=[1002] is a diagonal matrix, where 1,2>0.
Find a,b>0 such that NC(c,)={y=(y1,y2)R2:a2(y1c1)2+b2(y2c2)2=1}, which is an ellipse in
"standard orientation". In R, draw NC(c,) where c=(1,2),C=[2005] and =0.5. (It is ok to plot only
the boundary which is an ellipse.) Hint: Start with the trajectory of x=acost,y=bsint,0t2.
b) Consider now the general case where C is strictly positive definite and admits the
diagonalization C=SS where S is an orthonormal matrix. Write in R a function add_boundary (c,
C, epsilon) which adds to an existing plot the boundary of NC(c,). Use this function to draw the
boundary of NC(c,) where c=(1,2) and C=[2115]. This function will be used in the next problem.
Hint: yCy=ySSy=zz where z=Sy. In R the eigenvalue decomposition is implemented by the
function eigen(). You can also check your plot against the output of WolframAlpha: Try e.g.,
https://www. wolframalpha.com/input?i= plot
+5x%5E2+%2B+2xy+%2B+10y%5E2+%3C%3D+1.
2. (15 points) This preliminary exercise prepares you to perform and .pdf
1. 2. (15 points) This preliminary exercise prepares you to perform and visualize confidence set
estimation using MLE. For concreteness we focus on the case d=2. We are interested in
visualizing (when d=2 ) the ellipsoidal set NC(c,):={y=(y1,,yd)Rd:(yc)C(yc)2} where C=(cij) be
a symmetric (strictly) positive definite dd matrix, cRd and >0. The boundary NC(c,) of NC(c,) is
NC(c,)={y=(y1,,yd)Rd:(yc)C(yc)=2} (a) Suppose C=[1002] is a diagonal matrix, where 1,2>0.
Find a,b>0 such that NC(c,)={y=(y1,y2)R2:a2(y1c1)2+b2(y2c2)2=1}, which is an ellipse in
"standard orientation". In R, draw NC(c,) where c=(1,2),C=[2005] and =0.5. (It is ok to plot only
the boundary which is an ellipse.) Hint: Start with the trajectory of x=acost,y=bsint,0t2.
b) Consider now the general case where C is strictly positive definite and admits the
diagonalization C=SS where S is an orthonormal matrix. Write in R a function add_boundary (c,
C, epsilon) which adds to an existing plot the boundary of NC(c,). Use this function to draw the
boundary of NC(c,) where c=(1,2) and C=[2115]. This function will be used in the next problem.
Hint: yCy=ySSy=zz where z=Sy. In R the eigenvalue decomposition is implemented by the
function eigen(). You can also check your plot against the output of WolframAlpha: Try e.g.,
https://www. wolframalpha.com/input?i= plot
+5x%5E2+%2B+2xy+%2B+10y%5E2+%3C%3D+1
![2. (15 points) This preliminary exercise prepares you to perform and visualize confidence set
estimation using MLE. For concreteness we focus on the case d=2. We are interested in
visualizing (when d=2 ) the ellipsoidal set NC(c,):={y=(y1,,yd)Rd:(yc)C(yc)2} where C=(cij) be
a symmetric (strictly) positive definite dd matrix, cRd and >0. The boundary NC(c,) of NC(c,) is
NC(c,)={y=(y1,,yd)Rd:(yc)C(yc)=2} (a) Suppose C=[1002] is a diagonal matrix, where 1,2>0.
Find a,b>0 such that NC(c,)={y=(y1,y2)R2:a2(y1c1)2+b2(y2c2)2=1}, which is an ellipse in
"standard orientation". In R, draw NC(c,) where c=(1,2),C=[2005] and =0.5. (It is ok to plot only
the boundary which is an ellipse.) Hint: Start with the trajectory of x=acost,y=bsint,0t2.
b) Consider now the general case where C is strictly positive definite and admits the
diagonalization C=SS where S is an orthonormal matrix. Write in R a function add_boundary (c,
C, epsilon) which adds to an existing plot the boundary of NC(c,). Use this function to draw the
boundary of NC(c,) where c=(1,2) and C=[2115]. This function will be used in the next problem.
Hint: yCy=ySSy=zz where z=Sy. In R the eigenvalue decomposition is implemented by the
function eigen(). You can also check your plot against the output of WolframAlpha: Try e.g.,
https://www. wolframalpha.com/input?i= plot
+5x%5E2+%2B+2xy+%2B+10y%5E2+%3C%3D+1](https://image.slidesharecdn.com/2-230330051803-b25f77b7/85/2-15-points-This-preliminary-exercise-prepares-you-to-perform-and-pdf-1-320.jpg)
