Distribusi Kolmogorov-Smirnov diketahui dari uji Kolmogorov-Smirnov . Namun, itu juga distribusi supremum jembatan Brown.
Karena ini jauh dari jelas (bagi saya), saya ingin meminta Anda penjelasan intuitif tentang kebetulan ini. Referensi juga diterima.
Jawaban:
di manaZi(x)=1Xi≤x−E[1Xi≤x]
oleh CLT Anda memilikiGn=1n√∑ni=1Zi(x)→N(0,F(x)(1−F(x)))
ini adalah intuisi ...
brownian bridge memiliki varian t ( 1 - t ) http://en.wikipedia.org/wiki/Brownian_bridge ganti t dengan F ( x ) . Ini untuk satu x ...B(t) t(1−t) t F(x) x
The difficult part is to show that the distribution of the suppremum of the limit is the supremum of the distribution of the limit... Understanding why this happens requires some empirical process theory, reading books such as van der Waart and Welner (not easy). The name of the Theorem is Donsker Theorem http://en.wikipedia.org/wiki/Donsker%27s_theorem ...
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For Kolmogorov-Smirnov, consider the null hypothesis. It says that a sample is drawn from a particular distribution. So if you construct the empirical distribution function forn samples f(x)=1n∑iχ(−∞,Xi](x) , in the limit of infinite data, it will converge to the underlying distribution.
For finite information, it will be off. If one of the measurements isq , then at x=q the empirical distribution function takes a step up. We can look at it as a random walk which is constrained to begin and end on the true distribution function. Once you know that, you go ransack the literature for the huge amount of information known about random walks to find out what the largest expected deviation of such a walk is.
You can do the same trick with anyp -norm of the difference between the empirical and underlying distribution functions. For p=2 , it's called the Cramer-von Mises test. I don't know the set of all such tests for arbitrary real, positive p form a complete class of any kind, but it might be an interesting thing to look at.
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