Arnab Chakraborty's classical nonparametric statistics notes

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Last updated on Fri May 21 11:52:16 IST 2010.

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Asymptotic distribution of KS statistics

Computing the exact distribution of the KS statistic is prohibitively difficult for large n. In this case we use the asymptotic distributions of D, D₊ and D_-.

Theorem Suppose we have computed D based on iid sample of size n. Then

P(sqrt(n) D ≤ x) → H(x),

where

H(x) = 1-2 ∑ _j>0 (-1)^j-1 exp(-2 j² x²) if x > 0

0 else

Proof: Not to be done in this course.

H(x) is indeed a valid continuous distribution function. It appears quite difficult to prove that it is indeed so. The simplest proof that I know of is by Feller. It is a few page long (however, it actually proves something stronger.) H(x) is an example of what mathematicians call a theta function.

Theorem Let D₊ and D_- be computed based on iid sample of ize n. Then

P(sqrt(n) D₊) →	1-exp(-2x²)	if x ≥ 0
	0	else

P(sqrt(n) D_- ≤ x) has the same limit, as well.

Proof: Not to be done in this course.

Exercise 6.4: Show that nD₊² is asymptotically distributed as Expo(2).

Exercise 6.5: The file data.txt contains iid data from some unknown continuous distribution, F. We want to test
H₀: F = N(0,1) Vs. H₁: F N(0,1).
Perform KS test using the asymptotic distribution. Report the P-value.

Confidence intervals using KS procedure

Suppose that X₁,...,X_n are iid with unknown continuous distribution G. We want to get a &alpha-level CI for G. For this consider the random variable

D = sup_x |F_n(x)-G(x)|

Exercise 6.6: Can you compute D if G is not known? Does the distribution of D depend on G?

Find a constant c such that

	P(D≤c)	= 1-&alpha
i.e.,	P(\|F_n(x)-G(x)\| ≤ c for all x)	= 1- &alpha
i.e.,	P(F_n(x)-c ≤ G(x) ≤ F_n(x)+c for all x)	= 1- &alpha

This provides a CI for G.

Exercise 6.7: The file data.txt contains iid data from some unknown continuous distribution, F. Obtain a 95% CI for F using 2-sided KS statistic. Plot the CI.
[Hint: You need to compute cut-off point c based on the infinite series given earlier. Use Matlab.]

More on KS procedure

Asymptotic distribution of KS statistics

Confidence intervals using KS procedure