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Information on various topics in nonparametric statistical theory, including cubic splines with knots, natural cubic splines, finding the function that minimizes the penalized sum of squares, fixed design nonparametric regression models, and the local polynomial kernel estimator. It also covers the bias of the estimator and the concept of max-stable distribution functions.
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Thursday 7 June 2007 1.30 to 3.
Attempt THREE questions. There are FOUR questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Let n > 3 and a 6 x 1 < x 2 <... < xn 6 b. What is meant by a cubic spline with knots at x 1 ,... , xn? What is meant by a natural cubic spline?
Write S 2 [a, b] for the class of all real-valued functions on [a, b] having two continuous derivatives, and let g = (g 1 ,... , gn)T^. Find the function ˜g ∈ S 2 [a, b] that minimises
R(˜g′′) =
∫ (^) b a ˜g
′′(x) (^2) dx subject to ˜g(xi) = gi for i = 1,... , n.
[You may assume that there is a unique natural cubic spline g with knots at x 1 ,... , xn such that g(xi) = gi for i = 1,... , n.]
Consider the fixed design nonparametric regression model
Yi = g(xi) + i, i = 1,... , n,
where 1 ,... , n are independent with mean 0 and variance σ^2. Define the penalised sum of squares Sλ(˜g), for λ ∈ (0, ∞), and show that there is a unique minimiser ˆgλ of Sλ(˜g) over ˜g ∈ S 2 [a, b].
[You may assume that if g is a natural cubic spline with knots at x 1 ,... , xn, then there is a symmetric matrix ∫ K such that zT^ Kz > 0 for all z ∈ Rn^ and such that b a g
′′(x) (^2) dx = gT (^) Kg, where g = (g 1 ,... , gn)T (^) and gi = g(xi) for i = 1,... , n.]
Nonparametric statistical theory
3 What does it mean to say that a non-degenerate distribution function is max-stable? Explain without proof the relevance of this concept for determining possible limiting distributions of appropriately normalised sample maxima of independent and identically distributed random variables. Give expressions for the three extreme value types, and state a necessary and sufficient condition (in terms of these types) for a distribution function to be max-stable.
Let (Xn) be a sequence of independent and identically distributed random variables with distribution function F and let X(n) = max 16 i 6 n Xi. Let τ ∈ [0, ∞) and let (un) be a real-valued sequence. Prove that P(X(n) 6 un) → e−τ^ as n → ∞ if and only if n{ 1 − F (un)} → τ as n → ∞.
Let Sn =
∑n i=1 1 {Xi>un}^ and suppose that^ n{^1 −^ F^ (un)} →^ τ^ ∈^ [0,^ ∞). Prove that for each k ∈ { 0 , 1 , 2 ,.. .},
P(Sn 6 k) → e−τ
∑k
s=
τ s s!
as n → ∞.
[The standard result about convergence of binomial distributions to a Poisson distribution may be used without proof.]
Now suppose that there exist constants an > 0, bn and a non-degenerate distribu- tion function G such that for all x ∈ R,
( (^) X(n) − bn
an
6 x
→ G(x)
as n → ∞. Let Wn denote the second largest of X 1 ,... , Xn. By considering the events {Wn 6 un} and {Sn 6 1 }, show that whenever G(x) > 0,
( (^) Wn − bn an
6 x
→ G(x){ 1 − log G(x)}
as n → ∞.
4 Write an essay on the theory of canonical kernels and optimal kernel choice in kernel density estimation.
[You may find the following formula helpful:
AM ISE( fˆh) =
nh
h^4 μ^22 (K)R(f ′′).
Nonparametric statistical theory
