69. Exercise 63 from Chapter 7 introduced regression through the origin to relate a dependent variable y to an independent variable x. The assumption there was that for any xed x value, the dependent variable is a random variable Y with mean value bx and variance s2 (so that Y has mean value zero when x  0). The data consists of n independent (xi, Yi)

pairs, where each Yi is normally distributed with mean bxi and variance s2. The likelihood is then a product of normal pdf s with different mean values but the same variance.

a. Show that the mle of bis .

b. Verify that the mle of

(a) is unbiased.

c. Obtain an expression for and then for .

d. For purposes of obtaining a precise estimate of b, is it better to have the xi s all close to 0 (the origin)

or spread out quite far above 0? Explain your reasoning.

e. The natural prediction of Yi is . Let which is analogous to our earlier sample variance for a univariate sample X1, . . . , Xn

(in which case is a natural prediction for each Xi). Then it can be shown that T  1b ˆ  b2/

X 1n  12 S2  1Xi  X22/

1Yi  b ˆ xi 22/ 1n  12 S2  b ˆ xi sb V1b ˆ 2 b ˆ

 xiYi /x2i and use this to derive a 100(1  a)% CI for u.

b. Verify that P(a1/n Y/u 1)  1 

a, and derive a 100(1  a)% CI for u based on this probability statement.

c. Which of the two intervals derived previously is shorter? If my waiting time for a morning bus is uniformly distributed and observed waiting times are x1  4.2, x2  3.5, x3  1.7, x4  1.2, and x5  2.4, derive a 95% CI for u by using the shorter of the two intervals.