| Quantile function |
| Written by theoretic |
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The generalized inverse of the distribution function F, i.e., i?<_(t) = inf{x G R : F(x) >t}, 0 < t < 1, is called the quantile function of the distribution function F. The quantity xt = F^(t) defines the t-quantile of F. If F is monotone increasing (such as the distribution function Ф of the standard normal distribution), we see that F^~ = F^1 on the image of F, i.e., the ordinary inverse of F. An illustration of the quantile function is given in Figure 3.2.1. Notice that intervals where F is constant turn into jumps of F^, and jumps of F turn into intervals of constancy for F^. In this way we can define the generalized inverse of the empirical distribution function Fn of a sample Xi,. .., Xn, i.e., Fn(x) = i=i I (-oo,x] (X i ) X G (3.2.8) It is easy to verify that Fn has all properties of a distribution function: • limj^-oo Fn(x) = 0 and lim^^oo Fn(x) = 1. • Fn is non-decreasing: Fn(x) < Fn(y) for x < y. • Fn is right-continuous: limyix Fn(y) = Fn(x) for every x G R. Let X(i) < < -X"(n) be the ordered sample of Xi,..., Xn. In what follows, we assume that the sample does not have ties, i.e., X(i) < < X(n) a.s. For example, if the Xj’s are iid with a density the sample does not have ties; see the proof of Lemma 2.1.9 for an argument. |