Stat 5132 (Geyer) Old First Midterm Solutions

Stat 5132 (Geyer) Old First Midterm Solutions - Up: Stat...

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Up: Stat 5132 Midterm 1 Problem 1 This density is symmetric about because [p. 94 in Lindgren]. The center of symmetry is the population median (and also the population mean, though that is irrelevant in this problem). From the discussion at the top of p. 218 in Lindgren or Theorem A.1 in the handouts, is asymptotically normal with mean (the population median) and variance Alternative ways to write this are and Problem 2 From Theorem 11 on p. 213 in Lindgren Since is equivalent to the probability to be calculated is where is a random variable.
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In the case n = 10, the degrees of freedom is n - 1 = 9 and the probability to be calculated is ,which Table Va says is 0.035. Problem 3 The joint p. d. f. is Hence the log likelihood is To simplify notation, define so the log likelihood becomes If we prefer, we can drop the last term, which does not contain the parameter, obtaining The score function is Solving the ``likelihood equation'' gives At first sight this looks like an impossible solution, since must be positive. On a second look, however, we
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Stat 5132 (Geyer) Old First Midterm Solutions - Up: Stat...

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