By John C. Hull
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Extra resources for Options, Futures, and other Derivatives(ISBN 97801350009949)
For illustrative purposes, suppose that for each subject, two measures of reduced stress are recorded in columns 1 and 3. Then two of the subjects belong to group 1; on the ﬁrst measure of reduced stress their scores are 7 and 8, and on the second their scores are 56 and 78. What is needed is a simple method for sorting the observations by group membership and storing the data in an R or S-PLUS variable having list mode so that functions written for this book can be applied. The function selby is supplied for accomplishing this goal.
That is, with probability , the observed value is x. The resulting distribution is Fx, = (1 − )F + x. 6) It might help to notice the similarity between Fx, and the contaminated, or mixed, normal described in Chapter 1. In the present situation, F is any distribution, including normal distributions as a special case. Also notice the similarity with the inﬂuence curve in Chapter 1. Here, interest is in how the value x affects the value of some functional when x occurs with probability . For example, if F has mean µ, then Fx, has mean (1 − )µ + x, and the difference between the mean of Fx, and the mean of F is (x − µ).
Many studies have investigated the power of Student’s t test by examining power as a function of = µ1 − µ2 , σ where σ = σ1 = σ2 is the assumed common variance. What these studies failed to take into account is that small shifts away from normality, toward a heavytailed distribution, lowers , and this can mask power problems associated with Student’s t test. The important point is that for a given difference between the means, δ = µ1 − µ2 , modern methods can have substantially more power. More recently it has been illustrated that standard conﬁdence intervals for the difference between means can be unsatisfactory and that the F test has undesirable power properties.