Goodness-of-fit tests for the functional linear model based on randomly projected empirical processes

1.4 MB·

by  Juan A. Cuesta-Albertos

Goodness-of-fit tests for the functional linear model based on
randomly projected empirical processes
Juan A. Cuesta-Albertos1 Eduardo García-Portugués2,4 Manuel Febrero-Bande3
Wenceslao González-Manteiga3
Abstract
7 We consider marked empirical processes, indexed by a randomly projected functional co-
1 variate, to construct goodness-of-fit tests for the functional linear model with scalar response.
0
The test statistics are built from continuous functionals over the projected process, resulting
2
in computationally efficient tests that exhibit root-n convergence rate and circumvent the curse

r
of dimensionality. The weak convergence of the process is obtained conditionally on a random
a
M direction, whilst it is proved the almost surely equivalence between the testing for significance
expressedontheoriginalandontheprojectedfunctionalcovariate. Thecomputationofthetest

4 in practise involves the calibration by wild bootstrap resampling and the combination of several
2 p-values arising from different projections by means of the false discovery rate method. The

finite sample properties of the test are illustrated in a simulation study for a variety of linear
]
E models, underlying processes and alternatives. The software provided implements the tests and
M allows the replication of simulations and data applications.
.
t Keywords: Empirical process; Functional data; Functional linear model; Functional principal components;
a
t Goodness-of-fit; Random projections.
s
[

2 1 Introduction
v
3
Since Karl Pearson introduced the term “goodness-of-fit” at the beginning of the twentieth century
6
3 there has been an enormous amount of papers on this topic. First concentrated in fitting a model
8
for one distribution function, and later, especially after the papers of Bickel and Rosenblatt (1973)
0
and Durbin (1973), in more general models related with the regression function. The literature is
.
1
vast, and we refer to González-Manteiga and Crujeiras (2013) for an updated review of the topic.
0
7
1 Theideasofgoodness-of-fitfordensityanddistributionhavebeennaturallyextendedinthenineties
:
v ofthelastcenturytoregressionmodels. Considering, asareference, aregressionmodelwithrandom
i
X design Y = m(X)+ε the goal is to test
r
a H : m ∈ M = {m : θ ∈ Θ ⊂ Rq} vs. H : m ∈/ M
0 Θ θ 1 Θ
inanomnibuswayfromasample{(X ,Y )}n of(X,Y). Herem(x) = E[Y|X = x]istheregression
i i i=1
function of Y over X and ε is a random error centred and such that E[ε|X] = 0.
Following the ideas on smoothing for testing about the density function (Bickel and Rosenblatt,
1973), the usual pilot estimator for m was a nonparametric one, for example, the Nadaraya-
Watson estimator (Nadaraya (1964), Watson (1964)): mˆ (x) = (cid:80)n W (x)Y , with W (x) =
h i=1 ni i ni
1Department of Mathematics, Statistics and Computer Science, University of Cantabria (Spain).
2Department of Statistics, Carlos III University of Madrid (Spain).
3DepartmentofStatistics,MathematicalAnalysisandOperationsResearch,UniversityofSantiagodeCompostela
(Spain).
4Corresponding author. e-mail: [email protected].
1
K(cid:0)(x − X )/h(cid:1)(cid:14)(cid:80)n K(cid:0)(x − X )/h(cid:1), where K is a kernel function and h is a bandwidth pa-
i j=1 j
rameter. Other possible weights such as the ones from local linear estimation, k-nearest neigh-
bours or splines were also used in different studies. Using these kind of pilot estimators, statis-
tical tests were given by T = d(cid:0)mˆ,m (cid:1), being d some functional distance and θˆ an estimator
√ n θˆ
of θ such that n(θˆ − θ) = OP(1) under H0. In an alternative way, following the paper by
Durbin (1973) for testing about the distribution, the pilot estimator in the regression case was
given by I (x) = n−1(cid:80)n Y 1 , the empirical estimation of the integrated regression func-
tion I(x) =n E(cid:2)Y1 (cid:3)i=.1Hiär{dXlei≤axn}d Mammen (1993) using mˆ and Stute (1997) using I are
{X≤x} h n
key references for these two approximations in the literature, which were only the beginning of more
thantwohundredpaperspublishedinthelasttwodecades(González-ManteigaandCrujeiras,2013).
More recently, it has been of interest testing about a possible structure in a regression setting where
we have functional covariates:
Y = m(X)+ε, (1)
being now X a random element in a functional space, for example in the Hilbert space H = L2[0,1],
and Y a scalar response. This is the context of “Functional Data Analysis”, which has received
an increasing attention in the last decade (see for example Ramsay and Silverman (2005), Ferraty
and Vieu (2006) and Horváth and Kokoszka (2012)), specially motivated by the practical needs of
analysing data generated from high-resolution measuring devices.
A very simple null hypothesis H considered in the literature for the model (1) is H : m(X) = c,
0 0
wherec ∈ Risafixedconstant: thetestingofsignificanceofthecovariateXoverY. Followingsome
of the ideas from Ferraty and Vieu (2006) on considering pseudometrics for performing smoothing
with functional data, the test by Härdle and Mammen (1993) was adapted by Delsol et al. (2011a):
(cid:90)
T = (cid:0)mˆ (x)−Y¯(cid:1)2ω(x)dP (x),
n h X
n (cid:20) (cid:18) (cid:19) (cid:30) n (cid:18) (cid:19)(cid:21)
(cid:88) d(x,Xi) (cid:88) d(Xi,Xj)
mˆ (x) = K Y K ,
h i
h h
i=1 j=1
being d a functional pseudometric, K a kernel function adapted to this situation, h a bandwidth
parameter, w a weight function and P the probability measure induced by X over the functional
X
space of the covariate. The testing about H has been also considered by Cardot et al. (2003) or,
0
more recently, Hilgert et al. (2013), not in an omnibus way but inside a Functional Linear Model
(FLM):m(X) = (cid:104)X,ρ(cid:105), where(cid:104)·,·(cid:105)representstheinnerproductinH andρ ∈ H istheFLMparam-
eter. For both approximations, omnibus or not, there has been also some recent papers considering
the case of functional response; see for example, Cardot et al. (2007), Chiou and Müller (2007),
Kokoszka et al. (2008) and Bücher et al. (2011).
The generalization of the previous functional hypothesis to the general case
H : m ∈ M = {m : ρ ∈ P} vs. H : m ∈/ M , (2)
0 P ρ 1 P
where P can be either of finite or infinite dimension, was the focus of very few papers, specially in
the context of omnibus goodness-of-fit tests. In Delsol et al. (2011b) a discussion is given, without
theoretical results, for the extension of the checking of a more complex null hypothesis such as a
FLM.OnlyonepaperisknownforuswheretheFLMhypothesisisanalysedwiththeoreticalresults.
In Patilea et al. (2012), motivated by the smoothing test statistic considered by Zheng (1996) for
finite dimensional covariates, one test based on
(cid:18) (cid:19)
1 (cid:88) 1 Fn,h((cid:104)Xi,h(cid:105))−Fn,h((cid:104)Xj,h(cid:105))
T = (Y −mˆ (X ))(Y −mˆ (X )) K ,
n,h n(n−1) i H0 i j H0 j h h
1≤i(cid:54)=j≤n
2
is employed for checking the null hypothesis of linearity with mˆ (X) = (cid:104)X,ρˆ(cid:105), with ρˆ a suitable
H0
estimator of ρ and F the empirical distribution function of {(cid:104)X ,h(cid:105)}n . In the same spirit,
n,h i i=1
Lavergne and Patilea (2008) gave a test for the finite dimensional context and Patilea et al. (2016)
forfunctionalresponse. Fromadifferentperspective,andmotivatedbythetestofEscanciano(2006)
for finite dimensional predictors, in García-Portugués et al. (2014) a test was constructed from the
marked empirical process I (x) = 1 (cid:80)n Y 1 , with x ∈ R and h ∈ H. This approach
n,h n i=1 i {(cid:104)Xi,h(cid:105)≤x}
circumvents the technical difficulties that a marked empirical process indexed by x ∈ H, a possible
functional extension of the process in Stute (1997), would represent.
Inthispaperweconsidermarkedempiricalprocessesindexedbyrandomprojectionsofthefunctional
covariate. Themotivationstemsfromthealmostsurely(a.s.) characterizationofthenullhypothesis
(2)viaaprojected hypothesis thatarisesfromtheconditionalexpectationontheprojectedfunctional
covariate. This allows, conditionally on a randomly chosen h, the study of the weak convergence of
the process I (x) for hypothesis testing of infinite dimension. As a by-product, we obtain root-n
n,h
goodness-of-fit tests that evade the curse of dimensionality and, contrary to smoothing-based tests,
do not rely on a tuning parameter. Particularly, we focus on the testing of the aforementioned
hypothesis of functional linearity where, contrary to the finite dimensional situation, the functional
estimator has a non-trivial effect on the limiting process and requires a careful regularization. The
test statistics are built by a continuous functional (Kolmogorov-Smirnov or Cramér-von Mises) over
the empirical process and are effectively calibrated by a wild bootstrap on the residuals. To account
for a higher power and less influence from h, we consider a number K (not to confuse with a kernel
function) of different random projections and merge the resulting p-values into a final p-value by
means of the False Discovery Rate (FDR) of Benjamini and Yekutieli (2001). The empirical anal-
ysis reports a competitive performance of the test in practice, with a low impact of the choice of
K above a certain bound, and an expedient computational complexity of O(n) that yields notable
speed improvements over García-Portugués et al. (2014).
Therestofthepaperisorganizedasfollows. Thecharacterizationofthenullhypothesisthroughthe
projected predictor is addressed in Section 2, together with an application for the testing of the null
hypothesis H : m = m (Subsection 2.1). Section 3 is devoted to testing the composite hypothesis
0 0
H : m ∈ {(cid:104)·,ρ(cid:105) : ρ ∈ H}. To that aim, the regularized estimator for ρ of Cardot et al. (2007),
0
ρˆ, is reviewed in Subsection 3.1. The pointwise asymptotic distribution of the projected process is
studied in Subsection 3.2, whereas Subsection 3.3 gives its weak convergence. Section 4 describes
the implementation of the test and other practicalities. Section 5 illustrates the finite sample prop-
erties of the test by a simulation study and with some real data applications. Some final comments
and possible extensions are given in Section 6. Appendix A presents the main proofs, whereas the
supplementary material contains the auxiliary lemmas and further results for the simulation study.
Some general setting and notation are introduced now. The random variable (r.v.) X belongs to
a separable Hilbert space H endowed with the inner product (cid:104)·,·(cid:105) and associated norm (cid:107)·(cid:107). The
space H is a general real Hilbert space but for simplicity can be regarded as H = L2[0,1]. Y and
X are assumed to be centred r.v.’s, and ε is a centred, independent from X, r.v. with variance σ2.
ε
The independence between ε and X is a technical assumption required for proving Lemmas A.4 and
A.5, while for the rest of the paper it suffices with E[ε|X] = 0. Given the H-valued r.v. X and
h ∈ H, we denote by Xh = (cid:104)X,h(cid:105) to the projected X in the direction h. Bold letters are used for
vectors either in H (mainly) or in Rp, and its kind is clearly determined by the context. Capital
letters represent r.v.’s defined on the same probability space (Ω,σ,ν). Weak convergence is denoted
by (cid:32)L and D(R) represents the Skorohod’s space of càdlàg functions defined on R. Finally, we shall
implicitly assume that the null hypotheses stated hold a.s.
3
2 Hypothesis projection
The pillar of the goodness-of-fit tests we present is the a.s. characterization of the null hypothesis
(2), re-expressed as H : E[Y −m (X)|X] = 0 for some ρ ∈ P, by means of the associated projected
0 ρ
hypothesis on h ∈ H, defined as Hh : E(cid:2)Y −m (X)|Xh(cid:3) = 0. We identify Y −m (X) by Y for the
0 ρ ρ
sake of simplicity in notation. We give in this section two necessary and sufficient conditions based
on the projections of X to that E[Y|X] = 0 holds a.s.
The first condition only requires the integrability of Y, but the condition needs to be satisfied for
every direction h.
Proposition 2.1. Assume that E[|Y|] < ∞. Then,
E[Y|X] = 0 a.s. ⇐⇒ E(cid:2)Y|Xh(cid:3) = 0 a.s. for every h ∈ H.
The second condition, more adequate for application, somehow generalizes Proposition 2.1, as it
only needs to be satisfied for a randomly chosen h. In exchange, it holds only under some additional
conditions on the moments of X and Y. Before stating it we need some preliminary results, being
the first one included here for the sake of completeness.
Lemma 2.2 (Theorem 4.1 in Cuesta-Albertos et al. (2007)). Let µ be a non-degenerate Gaussian
measure on H, let X ,X be two H-valued r.v.’s and denote by X ∼ X if they are identically
1 2 1 2
distributed. Assume that:
(a) m := (cid:82) (cid:107)X (cid:107)kdν < ∞, for all k ≥ 1, and (cid:80)∞ m−1/k = ∞.
k 1 k=1 k
(b) The set {h ∈ H : Xh ∼ Xh} is of positive µ-measure.
1 2
Then X ∼ X .
1 2
Remark 2.2.1. It is not strictly needed that µ be a Gaussian distribution in Lemma 2.2 and this
can be replaced by assuming a certain smoothness condition on µ (see, for instance, Theorem 2.5
and Example 2.6 in Cuesta-Albertos et al. (2007)).
Remark 2.2.2. Assumption (a) in Lemma 2.2 is not of technical nature. According to Theorem
3.6 in Cuesta-Albertos et al. (2007), it becomes apparent that a similar condition is required. This
assumption is satisfied if the tails of P are light enough or if X has a finite moment generating
X1 1
function in a neighbourhood of zero.
Lemma 2.3. If E(cid:2)Y2(cid:3) < ∞ and X satisfies (a) in Lemma 2.2, then l := E(cid:2)(cid:107)X(cid:107)k|Y|(cid:3) < ∞, for
k
all k ≥ 1, and (cid:80)∞ l−1/k = ∞.
k=1 k
The second condition and most important result in this section is given as follows.
Theorem 2.4. Let µ be a non-degenerate Gaussian measure on H. Assume that X satisfies (a) in
Lemma 2.2 and that E(cid:2)Y2(cid:3) < ∞. Then,
E[Y|X] = 0 a.s. ⇐⇒ H := (cid:8)h ∈ H : E(cid:2)Y|Xh(cid:3) = 0 a.s.(cid:9) has positive µ-measure.
0
Corollary 2.5. Under the assumptions of the previous theorem,
E[Y|X] = 0 a.s. ⇐⇒ µ(cid:0)H (cid:1) = 1.
0
According to this corollary, it happens that if we are interested in testing the simple null hypothesis
H : E[Y|X] = 0 we can do it as follows: i) select at random with µ a direction h ∈ H; ii)
0
conditionally on h, test the projected null hypothesis Hh : E(cid:2)Y|Xh(cid:3) = 0. The rationale is simple
0
yet powerful: if H holds, then Hh also holds; if H fails, then Hh also fails µ-a.s. In this case, with
0 0 0 0
probability one we have chosen a direction h for which Hh fails. Of course, the main advantage
0
to test Hh over testing H directly is that in Hh the conditioning r.v. is real, which simplifies the
0 0 0
problem substantially.
4
2.1 Testing a simple null hypothesis
An immediate application of Corollary 2.5 is the testing of the simple null hypothesis H : m = m
0 0
via the empirical process of Stute (1997). Recall that other testing alternatives can be considered
on the projected covariate due to the µ-a.s. characterization. We refer to González-Manteiga and
Crujeiras (2013) for a review on alternatives.
For a random sample {(X ,Y )}n from (X,Y), we can consider the empirical process of the regres-
i i i=1
sion conditioned on the direction h,
n
(cid:88)
R (x) := n1/2I (x) = n−1/2 1 Y , x ∈ R
n,h n,h {Xh≤x} i
i
i=1
and then the following result is trivially satisfied from Theorem 1.1 in Stute (1997).
Corollary 2.6. Under Hh and E(cid:2)Y2(cid:3) < ∞, R (cid:32)L G in D(R), being G a Gaussian process
0 n,h 1 1
with zero mean and covariance function K (s,t) := (cid:82)s∧tVar(cid:2)Y|Xh = u(cid:3) dF (u), where F is the
1 −∞ h h
distribution function of Xh.
Different statistics for the testing of Hh can be built from continuous functionals on R (x). We
0 n,h
shall cover this with more detail in Section 3.
Example 2.7. Consider the FLM Y = (cid:104)X,ρ(cid:105) + ε in H = L2[0,1], with X a Gaussian process
with associated Karhunen-Loéve expansion (4) and ε independent from X. Then Xh and Xρ are
centred Gaussians with variances σ2 and σ2, respectively, and Cov[Xh,Xρ] = (cid:80)∞ h ρ λ , with
h ρ j=1 j j j
h = (cid:104)h,e (cid:105), ρ = (cid:104)ρ,e (cid:105). Hence,
j j j j
(cid:90) s∧t(cid:32)σ2σ2 −(cid:0)(cid:80)∞ h ρ λ (cid:1)2 (cid:33)
K1(s,t) = ρ h j=1 j j j +σ2 φ(u/σ )/σ du
σ2 ε h h
−∞ h
(cid:32)σ2σ2 −(cid:0)(cid:80)∞ h ρ λ (cid:1)2 (cid:33)
= ρ h j=1 j j j +σ2 Φ((s∧t)/σ ),
σ2 ε h
h
where φ and Φ are the density and distribution functions of a N(0,1), respectively.
3 Testing the functional linear model
We focus now in testing the composite null hypothesis, expressed as
H : m(X) = (cid:104)X,ρ(cid:105) = Xρ, for some ρ ∈ H.
0
According to Corollary 2.5, it happens that testing H is µ-a.s. equivalent to test
0
Hh : E(cid:104)(Y −Xρ)(cid:12)(cid:12)Xh(cid:105) = 0, for some ρ ∈ H,
0
where h is sampled from a non-degenerate Gaussian law µ. Again, we construct the associated
empirical regression process indexed by the projected covariate following Stute (1997). Therefore,
given an estimate ρˆ of ρ under H , we have
0
n
T (x) := a (cid:88)1 (cid:16)Y −Xρˆ(cid:17) = a (cid:0)T1 (x)+T2 (x)+T3 (x)(cid:1), (3)
n,h n {Xh≤x} i i n n,h n,h n,h
i
i=1
5
where a → 0 is a normalizing positive sequence to be determined later and
n
n
T1 (x) := (cid:88)1 (cid:0)Y −Xρ(cid:1),
n,h {Xh≤x} i i
i
i=1
n
(cid:88)(cid:68) (cid:104) (cid:105) (cid:69)
T2 (x) := 1 X −E 1 X ,ρ−ρˆ ,
n,h {Xh≤x} i {Xh≤x}
i
i=1
(cid:68) (cid:104) (cid:105) (cid:69)
T3 (x) :=n E 1 X ,ρ−ρˆ .
n,h {Xh≤x}
The selection of the right estimator ρˆ has a crucial role in the weak convergence of T3 , which
n,h
poses a substantially complexer proof than for the simple hypothesis. We consider the regularized
estimate proposed in Sections 2 and 3 of Cardot et al. (2007) (denoted by CMS in the sequel), whose
construction is sketched here for the sake of exposition of our results.
3.1 Construction of the estimator of ρ
Consider the so-called Karhunen-Loéve expansion of X:
∞
(cid:88) 1/2
X = λ ξ e , (4)
j j j
j=1
where {e }∞ is a sequence of orthonormal eigenfunctions associated to the covariance operator
j j=1
of X, Γz := E[(X⊗X)(z)] = E[(cid:104)z,X(cid:105)X], z ∈ H and the ξ ’s are centred real r.v.’s (because X
j
is centred) such that E(cid:2)ξ ξ (cid:3) = δ , where δ is the Kronecker’s delta. We assume that the
j j(cid:48) j,j(cid:48) j,j(cid:48)
multiplicity of each eigenvalue is one, so λ > λ > ... > 0.
1 2
The functional coefficient ρ is determined by the equation ∆ = Γρ, with ∆ the cross-covariance
operator of X and Y, ∆z := E[(X⊗Y)(z)] = E[(cid:104)z,X(cid:105)Y], z ∈ H. To ensure the existence and
uniqueness of a solution to ∆ = Γρ we require the next basic assumptions:
A1. X and Y satisfy (cid:80)∞ 1 (cid:104)E[XY],e (cid:105)2 < ∞.
j=1 λ2 j
j
A2. Ker(Γ) = {0}.
The estimation of ρ requires the inversion of Γ := 1 (cid:80)n X ⊗X , but since Γ is a.s. a finite rank
n n i=1 i i n
operator, its inverse does not exist. CMS proposed a regularization yielding a family of continuous
estimators for Γ−1. We employ the one from Example 1 in CMS, which provides an empirical
finite rank inverse of Γ denoted Γ† (we use Γ† for the population version). Consider a sequence of
n
thresholdsc ∈ (0,λ ),n ∈ N,withc → 0. Then: i)computetheFunctionalPrincipalComponents
n 1 n
(FPC) of Γ , i.e., calculate its eigenvalues {λˆ } and eigenfunctions {eˆ }; ii) define the sequences
n j j
{δ }, with δ := λ −λ and δ := min(λ −λ ,λ −λ ) for j > 1, and set
j 1 1 2 j j j+1 j−1 j
k := sup{j ∈ N : λ +δ /2 ≥ c };
n j j n
iii) compute Γ† (respectively Γ†) as the finite rank operator with the same eigenfunctions as Γ
n n
(resp. Γ) and associated eigenvalues equal to λˆ−1 (resp. λ−1) if j ≤ k and 0 otherwise. The
j j n
regularized estimator of ρ is
ρˆ := Γ†∆ = 1 (cid:88)kn (cid:88)n (cid:104)Xi⊗Yi,eˆj(cid:105)eˆ. (5)
n n n λˆ j
j=1 i=1 j
Note that (5) is not readily computable in practise, since {λ } is usually unknown (and hence k ).
j n
As in CMS, we consider the (random) finite rank
d := sup{j ∈ N : λˆ ≥ c }
n j n
6
as a replacement in practise for the deterministic k . As seen in Lemma A.2, ν[k = d ] → 1, hence
n n n
the estimator (5) has the same asymptotic behaviour with either k or d . Therefore, we consider
n n
k in (5) due to the convenient probabilistic tractability.
n
The following assumptions allow to obtain meaningful asymptotic convergences:
A3. E(cid:2)(cid:107)X(cid:107)2(cid:3) < ∞.
A4. (cid:80)∞ |(cid:104)ρ,e (cid:105)| < ∞.
l=1 l
A5. For j large, λ = λ(j) with λ(·) a convex positive function.
j
A6. λnn4 = O(1).
logn
(cid:110) (cid:111)
A7. inf |(cid:104)ρ,e (cid:105)|, √ λkn = O(n−1/2).
kn knlogkn
(cid:110) (cid:111)
A8. sup max(cid:0)E(cid:2)ξ4(cid:3),E(cid:2)|ξ |5(cid:3)(cid:1) ≤ M < ∞, for M ≥ 1.
j j j
A9. c = O(cid:0)n−1/2(cid:1).
n
A brief summary of these assumptions is given as follows. A3 is standard to obtain asymptotic
distributions, allows decomposition (4) and implies E(cid:2)Y2(cid:3) < ∞, required in Theorem 1.1 of Stute
(1997). A4 and A5 are A.1 and A.2 in CMS. A6 is very similar to one assumption in the second
part of Theorem 2 in CMS. A7 is the minimum requirement to control (cid:104)X,L (cid:105) when Lemma 7 in
n
CMS is used to prove Lemma A.7. A8 is a reinforcement of A.3 in CMS, where only fourth order
moments are used. The reason is because we handle inner products of ρˆ times a non independent
r.v., while in CMS the r.v. is not used to estimate ρ. A9 is useful, mainly (but also see the final
part of Lemma A.6) to control the behaviour of k . We show this fact in Proposition A.1, with a
n
conclusion very close to the assumption (8) in CMS and coinciding with one of the conditions of
their Theorem 3 if lim t < ∞ (the term t is defined in (6)). Finally, we point out that
n n,E n,E
x,h x,h
in CMS the assumptions aim to control the behaviour of k while here we have targeted to control
n
the threshold c , as this can be modified by the statistician.
n
3.2 Pointwise asymptotic distribution of T
n,h
Corollary 2.6 gives the weak convergence of n−1/2T1 . We analyse now the pointwise behaviour of
n,h
T2 (x) and T3 (x) for a fixed x ∈ R. We will show that T2 (x) = oP(n1/2) and that the rate of
n,h n,h n,h
T3 (x) depends on the key normalizing sequence {t }, where
n,h n,Ex,h
(cid:118)
tn,x := (cid:117)(cid:117)(cid:116)(cid:88)kn (cid:104)x,λej(cid:105)2 and Ex,h := E(cid:104)1{Xh≤x}X(cid:105). (6)
j
j=1
Theorem 3.1. Under Hh and A1–A9, for a fixed x ∈ R, it happens that:
0
(a) n−1/2t−1 T3 (x) (cid:32)L N(0,σ2).
n,E n,h ε
x,h
(b) If lim t = ∞, then with a = n−1/2t−1 in (3) the asymptotic distribution of T (x)
n n,Ex,h n n,Ex,h n,h
is the one of n−1/2t−1 T3 (x).
n,E n,h
x,h
(c) If lim t < ∞, then with a = n−1/2 in (3) the asymptotic distribution of T (x) is the
n n,Ex,h n n,h
one of n−1/2(cid:0)T1 (x)+T3 (x)(cid:1).
n,h n,h
7
The behaviour of the sequence {t }, indexed by n ∈ N and with arbitrary h ∈ H and x ∈ R, is
n,E
x,h
crucial for the convergence of T . Since {t } is non-decreasing, it has always a limit (finite or
n,h n,Ex,h
infinite). Its asymptotic behaviour is described next.
(cid:0) 1/2(cid:1)
Proposition 3.2. The sequence {t } has asymptotic orders between O(1) and O k . In
n,E n
x,h
addition, if X is Gaussian and satisfies A3, then σ2 := Var(cid:2)Xh(cid:3) < ∞ and lim t = φ(x/σ ).
h n n,Ex,h h
3.3 Weak convergence of T and of the test statistics
n,h
The result given in Theorem 3.1 holds for every x ∈ R. For the case (c) of Theorem 3.1 (where the
estimation of ρ is not dominant) and under an additional assumption, the result can be generalized
to functional weak convergence.
Theorem 3.3. Under Hh, A1–A9 and (c) in Theorem 3.1, it happens that:
0
(a) The finite dimensional distributions of T converge to a multivariate Gaussian with covari-
n,h
ance function K (s,t) := K (s,t)+C(s,t)+C(t,s)+V(s,t), where
2 1
(cid:90)
C(s,t) := Var[Y|X = u](cid:10)E ,Γ−1u(cid:11) dP (u),
t,h X
{uh≤s}
(cid:90)
V(s,t) := Var[Y|X = u](cid:10)E ,Γ−1u(cid:11)(cid:10)E ,Γ−1u(cid:11) dP (u).
s,h t,h X
(b) If E(cid:2)||ρˆ −ρ||4(cid:3) = O(n−2), then T (cid:32)L G in D(R), being G a Gaussian process with zero
n,h 2 2
mean and covariance function K .
2
Remark 3.3.1. According to Theorem 1 in CMS, it is impossible for ρˆ −ρ to converge to a non-
degenerate random element in the topology of H. In order to circumvent this issue, we make the
assumption E(cid:2)||ρˆ −ρ||4(cid:3) = O(n−2) which implies ||ρˆ −ρ|| = OP(n−1/2), thus a finite-dimensional
parametric convergence rate for ρˆ. In practice this means that ρ lives in a finite dimensional subspace
of H.
The next result gives the convergence of the Kolmogorov-Smirnov (KS) and Cramér-von Mises
(CvM) statistics for testing the FLM.
Corollary 3.4. Under the assumptions in Theorem 3.3 and E(cid:2)||ρˆ −ρ||4(cid:3) = O(n−2), if ||T || :=
n,h KS
sup |T (x)| and ||T || := (cid:82) T (x)2dF (x), then
x∈R n,h n,h CvM R n,h n,h
(cid:90)
||T || (cid:32)L ||G || and ||T || (cid:32)L G (x)2dF (x).
n,h KS 2 KS n,h CvM 2 h
R
Remark 3.4.1. An alternative to (b) and Corollary 3.4 is to consider a deterministic discretization
of the statistics, for which the convergence in law is trivial from (a). For example, if ||T || :=
n,h K(cid:102)S
max |T (x )| for a grid {x ,...,x }, then ||T || (cid:32)L ||Z || , where Z ∼ N (0,Σ),
k=1,...,G n,h k 1 G n,h K(cid:102)S 2 K(cid:102)S 2 G
Σ = K (x ,x ).
ij 2 i j
4 Testing in practise
The major advantage to test Hh over H is that in Hh the conditioning r.v. is real. The potential
0 0 0
drawbacksofthisuniversalmethodareapossiblelossofpowerandthattheoutcomeofthetestmay
vary for different projections. Both inconveniences can be alleviated by sampling several directions
h ,...,h , testing the projected hypotheses Hh1,...,HhK and selecting an appropriate way to
1 K 0 0
mix the resulting p-values. For example, by the FDR method proposed in Benjamini and Yekutieli
(2001) (see Section 2.2.2 of Cuesta-Albertos and Febrero-Bande (2010)) it is possible to control the
final rejection rate to be at most α under H . The procedure is described in the following generic
0
algorithm.
8
Algorithm 4.1 (Testing procedure for H ). Let T denote a test for checking Hh with h chosen by
0 n 0
a non-degenerate Gaussian measure µ on H.
i) For i = 1,...,K, set by p the p-value of Hhi obtained with the test T .
i 0 n
ii) Set the final p-value of H as min Kp , where p ≤ ... ≤ p .
0 i=1,...,K i (i) (1) (K)
The calibration of the test statistic for Hh is done by a wild bootstrap resampling. The next
0
algorithm states the steps for testing the FLM. The particular case of the simple null hypothesis
corresponds to ρ = 0, so its calibration corresponds to setting ρˆ = ρˆ∗ = 0 in the algorithm.
Algorithm 4.2 (Bootstrap calibration in FLM testing). Let {(X ,Y )}n be a random sample from
i i i=1
(1). To test H : m ∈ {(cid:104)·,ρ(cid:105) : ρ ∈ H} proceed as follows:
0
i) Estimate ρ by FPC for a given d and obtain εˆ = Y −(cid:104)X ,ρˆ(cid:105).
n i i i
(cid:12)(cid:12) (cid:12)(cid:12)
ii) Compute ||T || = (cid:12)(cid:12)n−1/2(cid:80)n 1 εˆ(cid:12)(cid:12) with N either KS or CvM.
n,h N (cid:12)(cid:12) i=1 {Xh≤x} i(cid:12)(cid:12)
i N
iii) Bootstrap resampling. For b = 1,...,B, do:
√ √
a) Draw binary i.i.d. r.v.’s V∗,...,V∗ such that P(cid:8)V∗ = (1− 5)/2(cid:9) = (5+ 5)/10 and
√ 1 √ n
P(cid:8)V∗ = (1+ 5)/2(cid:9) = (5− 5)/10.
b) Set Y∗ := (cid:104)X ,ρˆ(cid:105)+ε∗ from the bootstrap residuals ε∗ := V∗εˆ.
i i i i i i
c) Estimate ρ∗ from {(X ,Y∗)}n by FPC using the same d of i).
i i i=1 n
d) Obtain the estimated bootstrap residuals εˆ∗ := Y∗−(cid:104)X ,ρˆ∗(cid:105).
i i i
(cid:12)(cid:12) (cid:12)(cid:12)
e) Compute ||T∗b || := (cid:12)(cid:12)n−1/2(cid:80)n 1 εˆ∗(cid:12)(cid:12) .
n,h N (cid:12)(cid:12) i=1 {Xh≤x} i(cid:12)(cid:12)
i N
iv) Approximate the p-value by 1 (cid:80)B 1 .
B b=1 {||Tn∗,bh||N≤||Tn,h||N}
The choice of an adequate d for the estimation of ρ can be done in a data-driven way, for exam-
n
ple by the corrected Schwartz Information Criterion (McQuarrie, 1999), denoted by SICc. Besides,
Steps c) and d) can be easily computed using the properties of the linear model, see Section 3.3 of
García-Portugués et al. (2014).
The bootstrap process we are considering is given by (we consider a = n−1/2):
n
n n n
T∗ (x) := n−1/2(cid:88)1 εˆ∗ = n−1/2(cid:88)1 εˆV∗+n−1/2(cid:88)1 Xρˆ−ρˆ∗
n,h {Xh≤x} i {Xh≤x} i i {Xh≤x} i
i i i
i=1 i=1 i=1
which is estimating the distribution of
n n
T (x) = n−1/2(cid:88)1 εˆ +n−1/2(cid:88)1 Xρ−ρˆ.
n,h {Xh≤x} i {Xh≤x} i
i i
i=1 i=1
The bootstrap consistency could be obtained as an adaptation of: Lemma A.1 of Stute et al. (1998)
forthefirsttermofT∗ ; LemmaA.2ofthesamepaperforthesecondterm,usingthedecomposition
n,h
of ρˆ −ρ given in (11) of CMS.
The drawing of the random directions is clearly influential in the power of the test. For example,
in the extreme case where the projections were orthogonal to the data, that is Xh = 0, then
T (x) = (n−1/2(cid:80)n εˆ)1 and ||T || = ||T∗b || = 0 under H . Therefore, Algorithm 4.2
n,h i=1 i {0≤x} n,h N n,h N 0
would fail to calibrate the level of the test and potentially yield spurious results due to numerical
inaccuracies in ||T∗b || ≤ ||T || . A data-driven compromise to avoid drawing projections in
n,h N n,h N
9
subspaces almost orthogonal to the data is the following: i) compute the FPC of X ,...,X , i.e.,
1 n
the eigenpairs {(λˆ ,eˆ )}; ii) choose j := min(cid:8)k = 1,...,n−1 : ((cid:80)k λˆ2)/((cid:80)n−1λˆ2) ≥ r(cid:9) for a
j j n j=1 j j=1 j
variancethresholdr, e.g. r = 0.95; iii)generatethedata-drivenGaussianprocessh := (cid:80)jn η eˆ ,
jn j=1 j j
with η ∼ N(0,s2) and s2 the sample variance of the scores in the j-th FPC. Formally, the Gaussian
j j j
measure µ associated to h does not respect the assumptions in Theorem 2.4, since it is degenerate
jn
(but recall that µ does not have to be independent from X). A non-degenerate Gaussian process
can be obtained as h + G, with G a Gaussian process tightly concentrated around zero, albeit
jn
employing h or h +G has negligible effects in practise.
jn jn
0
1 10 3
2
8 2 8 2
5
6 6 1
1
4 4
0 0 0 0
2
2 1
−
0 −1
0 2 2
− −2 −5 −
2 2
− − 3
4 −
−
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
6 4 6 4 4.5 2
4
4 4
2 2 3.5
0
2 2 3
0 0 5
2.
0 0
−2 −2 52 −2
2 2 1.
− −
−4 −4 1 4
4 4 −
− −
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
4
4 5
6 1
5
1. 2
1 1 2 4
0.5 0.5 10
0 2
0 0
5 0
0.
1.5−1− −2 1−0.5 −2 −20 5
− −
4
− −4 −4 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Figure 1: From left to right and up to down, functional coefficients ρ (black, right scale) and underlying
processes (grey, left scale) for the nine different scenarios, labelled S1 to S9. Each graph contains a sample
of 100 realizations of the functional covariate X and ρˆ (red) with d selected by SICc.
n
10