Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Exam of Statistical Science which includes Fixed Point Theorem, Erlang Fixed Point, Loss Network, Fixed Routing, Carefully, Relating, Approximation, Mathematical, Wardrop Equilibrium etc. Key important points are: Distribution Function, Independent, Empirical Distribution Function, Variables, Brownian Bridge, Central Limit Theorem, Continuous Mapping, Theorem, Convergence, Confidence Band
Typology: Exams
1 / 3
This page cannot be seen from the preview
Don't miss anything!
Monday, 7 June, 2010 9:00 am to 11:00 am
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
2
1
Let X 1 ,... , Xn be independent and identically distributed random variables with distribution function F : R → [0, 1]. Define the empirical distribution function Fn : R → [0, 1]. Define the (standard) Brownian bridge G and the F -Brownian bridge process GF. Carefully state Donsker’s central limit theorem for Fn − F.
Assuming that F is continuous, use Donsker’s central limit theorem to prove that √ n sup t ∈ R
|Fn(t) − F (t)| →d^ max t ∈ [0,1]
|G(t)|
n → ∞. [You may use the continuous mapping theorem for convergence in distribution in metric spaces.]
Explain how this result can be used to construct an asymptotic confidence band for F centered at Fn.
Let X 1 ,... , Xn be independent and identically distributed random variables with probability density function f : R → [0, ∞). Define the kernel density estimator f (^) nK (x, h) of f with bandwidth h.
What is a kernel of order l? Let {φm}m ∈ N be the orthonormal basis of Legendre polynomials defined by
φ 0 (x) := 2 −^1 /^2 , φm(x) =
2 m + 1 2
2 m^ m!
dm dxm^
[(x^2 − 1)m]
for x ∈ [− 1 , 1] and m ∈ N. Using these polynomials (or otherwise), construct a kernel of order l. [You may use standard properties of spaces of polynomials in your answer.]
Suppose f is three times differentiable and that f and D^3 f are bounded functions. Devise a kernel K and a bandwidth hn depending on n such that for every x ∈ R
E
f (^) nK (x, h) − f (x)
6 Cn−^
3 7
for some constant C independent of n. Give an example of a probability density function f : R → [0, ∞) for which E
∣f (^) nK (x, h) − f (x)
∣ (^6) Cn−^12
for every x ∈ R.
Nonparametric Statistical Theory
