Introduction Testing Equality of Mean Functions Testing Equality of Covariance Operators Bibliography Two Sample Problem for Functional Data Radek Hendrych Stochastic Modelling in Economics and Finance 1 November 25, 2013 Radek Hendrych Two Sample Problem for Functional Data
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IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Two Sample Problem for Functional Data
Radek Hendrych
Stochastic Modelling in Economics and Finance 1
November 25, 2013
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Outline
1 Introduction
2 Testing Equality of Mean Functions
3 Testing Equality of Covariance Operators
4 Bibliography
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Two Sample Problem for Functional Data
testing the equality of the means in two independent samples
testing the equality of the covariance operators in two independentsamples
Asymptotic procedures will be introduced.
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Outline
1 Introduction
2 Testing Equality of Mean Functions
3 Testing Equality of Covariance Operators
4 Bibliography
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Testing Equality of Mean Functions
Model
Consider two samples X1, . . . ,XN and X ∗1 , . . . ,X∗M satisfying the model
Xi (t) = µ(t) + εi (t), i = 1 . . . ,N, (1)
X ∗j (t) = µ∗(t) + ε∗j (t), j = 1 . . . ,M. (2)
Assumptions
(A1) the two samples are independent
(A2) ε1, . . . , εN are i.i.d. with Eε1(t) = 0 and E||ε1||4 <∞(A3) ε∗1 , . . . , ε
∗M are i.i.d. with Eε∗1(t) = 0 and E||ε∗1 ||4 <∞
The ε1 and ε∗1 do not have to follow the same distribution.
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Main Goal
Testing Hypothesis
H0 : µ = µ∗ in L2 against H1 : ¬H0
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Method I
XN(t) =1
N
N∑i=1
Xi (t) and X ∗M(t) =1
M
M∑j=1
X ∗j (t)
are unbiased estimators for µ(t) and µ∗(t), respectively.
It is natural to reject the null hypothesis if
UN,M =NM
N + M
∫ 1
0
(XN(t)− X ∗M(t)
)2dt (3)
is large.
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Convergence of UN,M under H0
Theorem
If H0 and the assumptions (A1), (A2) and (A3) hold, and
N
N + M→ θ, for some θ ∈ [0, 1], as N →∞,
then
UN,Md−→∫ 1
0
Γ2(t)dt, N,M →∞, (4)
where {Γ(t), t ∈ [0, 1]} is a Gaussian process satisfying EΓ(t) = 0 and
E [Γ(t)Γ(s)] = (1− θ)c(t, s) + θc∗(t, s),
with c(t, s) = cov(X1(t),X1(s)) and c∗(t, s) = cov(X ∗1 (t),X ∗1 (s)).
Proof. See [1].
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
The limit distribution of UN,M in (4) depends on the unknown covariancefunctions c and c∗.
According to the Karhunen-Loeve expansion, one can suppose that
Γ(t) =∞∑k=1
τ1/2k Nkϕk(t),
where Nk , k ∈ N, are independent N(0, 1) random variables,τ1 ≥ τ2 ≥ . . . and ϕ1, ϕ2, . . . are the eigenvalues and eigenfunctions ofthe operator determined by (1− θ)c + θc∗.
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Since ∫ 1
0
Γ2(t)dt =∞∑k=1
τkN2k ,
to provide a reasonable approximation for∫ 1
0Γ2(t)dt, one only need to
estimate τk .
This can be done using τk , the eigenvalues of the empirical covariancefunction
zN,M(t, s) =M
N + M
1
N
N∑i=1
(Xi (t)− XN(t)
) (Xi (s)− XN(s)
)+
N
N + M
1
M
M∑j=1
(X ∗j (t)− X ∗M(t)
) (X ∗j (s)− X ∗M(s)
).
Thus, the sum∑d
k=1 τkN2k offers an approximation to the limit
distribution in (4) if d is large enough.
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Asymptotic Consistency of Method I
Theorem
If the assumptions (A1), (A2) and (A3) hold,
N
N + M→ θ, for some θ ∈ [0, 1], as N →∞,
and ∫ 1
0
(µ(t)− µ∗(t))2 dt > 0,
then UN,MP−→∞, as N,M →∞.
Proof. See [1].
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
Method II
The method is a projection version of the first procedure based on UN,M .
It does not require the numerical evaluation of the integral in thedefinition of UN,M (⇒ an easier implementation).
Consider projections onto the space determined by the leadingeigenfunctions of the operator Z = (1− θ)C + θC∗.
In particular, assume that the eigenvalues of Z satisfy
τ1 > τ2 > · · · > τd > τd+1. (5)
One wants to project observations onto the space spanned by ϕ1, . . . , ϕd ,i.e. the corresponding eigenfunctions.
Radek Hendrych Two Sample Problem for Functional Data
IntroductionTesting Equality of Mean Functions
Testing Equality of Covariance OperatorsBibliography
In fact, the functions ϕ1, . . . , ϕd are unknown.
⇓
The corresponding eigenfunctions of ZN,M , denoted by ϕi , are used.
This delivers a projection of XN − X ∗M into the linear space spanned byϕ1, . . . , ϕd . Let
a = (a1, . . . , ad)> , where ai = 〈XN − X ∗M , ϕi 〉.
Under the conditions of the first theorem, it can be shown that√NM/(N + M)a has approximately d-variate normal distribution (up to
some random signs) with the asymptotic variance Q = {Q(i , j)}di,j=1,