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40 Chapter 1 Introduction
Problems
(e) The sample value of the rank statistic P is 10310, and the asymptotic distribution
under the iid hypothesis (with n 200)isN 9950, 2.239 × 10 5 . Thus
|P − µP |/σP 0.76, corresponding to a computed p-value of 0.447. On the basis of
the value of P there is therefore not sufficient evidence to reject the iid hypothesis at
level .05.
(f) The minimum-AICC Yule–Walker autoregressive model for the data is of
order seven, supporting the evidence provided by the sample ACF and Ljung–Box
tests against the iid hypothesis.
Thus, although not all of the tests detect significant deviation from iid behavior,
the sample autocorrelation, the Ljung–Box statistic, and the fitted autoregression provide
strong evidence against it, causing us to reject it (correctly) in this example.
The general strategy in applying the tests described in this section is to check
them all and to proceed with caution if any of them suggests a serious deviation
from the iid hypothesis. (Remember that as you increase the number of tests, the
probability that at least one rejects the null hypothesis when it is true increases. You
should therefore not necessarily reject the null hypothesis on the basis of one test
result only.)
1.1. Let X and Y be two random variables with E(Y) µ and EY 2 < ∞.
a. Show that the constant c that minimizes E(Y − c) 2 is c µ.
b. Deduce that the random variable f(X)that minimizes E (Y − f(X)) 2 |X is
f(X) E[Y |X].
c. Deduce that the random variable f(X)that minimizes E(Y − f(X)) 2 is also
f(X) E[Y |X].
1.2. (Generalization of Problem 1.1.) Suppose that X1,X2,...is a sequence of random
variables with E(X2 t )