Posted: November 5th, 2015

Problem solving

Problem solving

Matrix X should be easy to get if you just write the given equations in matrix notation. So, again, FIRST write the system of the three given equations and then rewrite it using matrix-vector notation. It is always in this order, not in the reverse order. If B is the vector of parameters, and Y is the vector with the data, then the coefficient matrix of B should be X.

Regarding the second question, for two estimable linear combinations, once you get the best linear unbiased estimates there are two r vectors (r1 and r2). The covariance (covered in IE 516) of ?1Tß(hat) and ?2Tß(hat) is equal to E[(?1Tß(hat) – ?1Tß) (?2Tß(hat) – ?2Tß)] = E[(r1TXTY – ?1Tß) (r2TXTY – ?2Tß)] = … =E[(r1TXTe)(eT Xr2)]. You can simplify this by noting that E[eeT] = s2. The variance of the estimate of a linear combination is the covariance with itself. I hope this helps.

PROBLEM 1
Find the rank of the matrices (show all your work) given below:

PROBLEM 2
Consider the model Y = Xß+e, which is explicitly written as
y1= t1 + t2 + t3 + e1
y2= t1         + t3 + e2
y3 =        t2    + e3
(a)    Find X.
(b)    Find XTX.
(c)    Find the rank of X.
(d)    Write the normal equations (no solution required).
(e)    Show that t1 + t2 – 2t3 is estimable.
(f)    From the normal equations find the best linear unbiased estimate of t1 + t2 – 2t3.
(g)    Find a full-rank reparametrization of the model.
(h)    Find the variance of the estimates of t1 – 2 t2 + t3 and t1 + 2 t2 + t3.
(i)    Find the covariance of the estimates of t1 – 2 t2 + t3 and t1 + 2 t2 + t3.
PROBLEM 3
An experiment was run on three ovens, each at four temperatures, to ascertain the strength of the final product.  Suppose that three runs were made in each oven at each temperature.   Assume the data shown in the following table (rows correspond to temperatures and columns to ovens).
1    2    3
1    3
3
2    4
3
2    3
4
6
2    6
3
4    6
2
7    8
5
9
3    3
4
4    3
6
7    5
6
9
4    4
5
4    3
7
9    7
8
10

Test the significance of the effects of ovens, temperature and their interaction using a 5% level of significance.   Show all your work.

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